How many of the 230 crystallographic groups are realized in nature?

Question

All of them or only a subset?

This is a famous and fundamental result in solid state physics.

Answer 1

According to Fizicheskaya Entsiklopediya (Physical Encyclopedia, in Russian, http://www.femto.com.ua/articles/part_2/3634.html ), no real crystals had been found for 4 space groups (Pcc2 and three others) as of the encyclopedia's publication in 1988-1999.

Answer 2

If there were only a few that had not been seen, I suspect that Ashcroft and Mermin would have gleefully pointed that out. However, that is not proof.

On the other hand, I see no way to prove that a given space group definitively could not exist in nature, given the essentially infinite number of molecules that could form crystals. We still spend lots of synchrotron beam time determining crystal structures of proteins - who is to say the next one would not have the magic unseen space group symmetry?

At best, one might say that, given the number of elements identified so far is less than 230, the STP equilibrium elemental crystals do not cover all of the space groups.

Answer 3

The first idea would be to do it ourselves but this is not so easy. Sure enough, one may come to the idea of exploring the Crystallographic Open Database (COD) as they provide a free access to their SQL database (or alternatively querying the Cambridge Structure Database (CSD) for those who have paid a subscription). So choose your preferred MySQL client and connect to host www.crystallography.net with user cod_reader, and issue the following SQL query:

select sg, count(*) from data group by sg

They do not provide the space group number unfortunately, but a bit of scripting (see the listing at the end of this answer) with the Computational Crystallographic ToolBox (cctbx) easily remedies that, and at the time of writing the answer, the only missing spacegroup is number 93. However the frequencies show that some are much rarer than others, as shown in the list at the end of this answer. One may seriously doubt whether the spacegroups appearing less than 0.5% of the time are actually mistakes or not. Here lies the problem. It is therefore necessary to painstakingly comb through the data and assess whether we have a genuine rare group or a mistake. No need to say that I don't quite have the time on my hands to do that!

But others have done precisely that. Urusov and Nadezhina [UN09} have analysed 100,444 mineral and inorganic structures from the CSD and they found 72 empty spacegroups. For organic compounds, the classic study is that of Mighell and Himes [MH83], which is pretty old by now but I haven't searched for a more up-to-date study. They studied about 30,000 organic structures in the NBS Crystal Data Identification File and they concluded that 35 spacegroups had no entry and 29 had only one entry.

[UN09] V. S. Urusov and T. N. Nadezhina. Frequency distribution and selection of space groups in inorganic crystal chemistry. Journal of Structural Chemistry, 50(1):22–37, Dec 2009.

[MH83] A. D. Mighell, V. L. Himes, and J. R. Rodgers. Space-group frequencies for organic compounds. Acta Crystallographica Section A, 39(5):737–740, Sep 1983.

Frequencies of spacegroups in the COD

• 14: 114611 (P 1 21/c 1)
• 2: 85079 (P -1)
• 15: 30821 (C 1 2/c 1)
• 19: 19734 (P 21 21 21)
• 4: 14607 (P 1 21 1)
• 61: 10839 (P b c a)
• 62: 7879 (P n m a)
• 33: 4886 (P n a 21)
• 12: 4522 (C 1 2/m 1)
• 225: 4398 (F m -3 m)
• 227: 3830 (F d -3 m :2)
• 9: 3660 (C 1 c 1)
• 148: 3132 (R -3 :H)
• 60: 3046 (P b c n)
• 1: 2892 (P 1)
• 5: 2815 (C 1 2 1)
• 13: 2627 (P 1 2/c 1)
• 29: 2574 (P c a 21)
• 11: 2362 (P 1 21/m 1)
• 194: 2212 (P 63/m m c)
• 139: 1912 (I 4/m m m)
• 221: 1773 (P m -3 m)
• 166: 1754 (R -3 m :H)
• 88: 1625 (I 41/a :2)
• 167: 1578 (R -3 c :H)
• 63: 1572 (C m c m)
• 7: 1543 (P 1 c 1)
• 18: 1285 (P 21 21 2)
• 43: 1272 (F d d 2)
• 176: 1246 (P 63/m)
• 56: 1176 (P c c n)
• 191: 1033 (P 6/m m m)
• 123: 901 (P 4/m m m)
• 36: 795 (C m c 21)
• 229: 760 (I m -3 m)
• 129: 742 (P 4/n m m :2)
• 64: 738 (C m c a)
• 164: 732 (P -3 m 1)
• 205: 687 (P a -3)
• 92: 680 (P 41 21 2)
• 216: 671 (F -4 3 m)
• 82: 650 (I -4)
• 58: 638 (P n n m)
• 160: 637 (R 3 m :H)
• 96: 616 (P 43 21 2)
• 141: 613 (I 41/a m d :2)
• 87: 608 (I 4/m)
• 70: 604 (F d d d :2)
• 20: 604 (C 2 2 21)
• 186: 595 (P 63 m c)
• 147: 573 (P -3)
• 146: 566 (R 3 :H)
• 230: 563 (I a -3 d)
• 161: 563 (R 3 c :H)
• 136: 555 (P 42/m n m)
• 140: 538 (I 4/m c m)
• 173: 536 (P 63)
• 55: 532 (P b a m)
• 86: 515 (P 42/n :2)
• 198: 494 (P 21 3)
• 57: 486 (P b c m)
• 52: 473 (P n n a)
• 220: 462 (I -4 3 d)
• 74: 432 (I m m a)
• 71: 428 (I m m m)
• 152: 424 (P 31 2 1)
• 223: 422 (P m -3 n)
• 122: 422 (I -4 2 d)
• 114: 410 (P -4 21 c)
• 85: 380 (P 4/n :2)
• 142: 369 (I 41/a c d :2)
• 31: 368 (P m n 21)
• 41: 366 (A b a 2)
• 47: 362 (P m m m)
• 154: 349 (P 32 2 1)
• 59: 348 (P m m n :2)
• 189: 335 (P -6 2 m)
• 130: 331 (P 4/n c c :2)
• 165: 314 (P -3 c 1)
• 127: 314 (P 4/m b m)
• 204: 305 (I m -3)
• 113: 301 (P -4 21 m)
• 217: 295 (I -4 3 m)
• 76: 294 (P 41)
• 163: 285 (P -3 1 c)
• 8: 283 (C 1 m 1)
• 193: 281 (P 63/m c m)
• 78: 273 (P 43)
• 65: 266 (C m m m)
• 155: 259 (R 3 2 :H)
• 72: 256 (I b a m)
• 144: 246 (P 31)
• 206: 240 (I a -3)
• 145: 230 (P 32)
• 159: 224 (P 3 1 c)
• 187: 220 (P -6 m 2)
• 169: 213 (P 61)
• 192: 210 (P 6/m c c)
• 54: 210 (P c c a)
• 170: 207 (P 65)
• 45: 202 (I b a 2)
• 218: 201 (P -4 3 n)
• 150: 196 (P 3 2 1)
• 121: 175 (I -4 2 m)
• 68: 158 (C c c a :2)
• 34: 156 (P n n 2)
• 197: 149 (I 2 3)
• 182: 149 (P 63 2 2)
• 143: 146 (P 3)
• 137: 144 (P 42/n m c :2)
• 110: 143 (I 41 c d)
• 69: 143 (F m m m)
• 51: 134 (P m m a)
• 128: 133 (P 4/m n c)
• 66: 130 (C c c m)
• 26: 130 (P m c 21)
• 203: 126 (F d -3 :2)
• 46: 126 (I m a 2)
• 190: 123 (P -6 2 c)
• 178: 119 (P 61 2 2)
• 53: 119 (P m n a)
• 174: 116 (P -6)
• 99: 116 (P 4 m m)
• 40: 116 (A m a 2)
• 199: 114 (I 21 3)
• 126: 114 (P 4/n n c :2)
• 10: 114 (P 1 2/m 1)
• 215: 113 (P -4 3 m)
• 179: 111 (P 65 2 2)
• 79: 111 (I 4)
• 80: 107 (I 41)
• 73: 107 (I b c a)
• 44: 107 (I m m 2)
• 226: 103 (F m -3 c)
• 201: 95 (P n -3 :2)
• 185: 94 (P 63 c m)
• 138: 94 (P 42/n c m :2)
• 81: 93 (P -4)
• 38: 93 (A m m 2)
• 180: 91 (P 62 2 2)
• 131: 86 (P 42/m m c)
• 23: 85 (I 2 2 2)
• 118: 83 (P -4 n 2)
• 135: 82 (P 42/m b c)
• 162: 78 (P -3 1 m)
• 107: 78 (I 4 m m)
• 213: 75 (P 41 3 2)
• 212: 70 (P 43 3 2)
• 32: 70 (P b a 2)
• 156: 68 (P 3 m 1)
• 224: 67 (P n -3 m :2)
• 100: 66 (P 4 b m)
• 219: 64 (F -4 3 c)
• 202: 64 (F m -3)
• 67: 63 (C m m a)
• 222: 60 (P n -3 n :2)
• 157: 60 (P 3 1 m)
• 120: 60 (I -4 c 2)
• 98: 59 (I 41 2 2)
• 50: 59 (P b a n :2)
• 95: 57 (P 43 2 2)
• 119: 56 (I -4 m 2)
• 94: 56 (P 42 21 2)
• 228: 54 (F d -3 c :2)
• 124: 54 (P 4/m c c)
• 84: 54 (P 42/m)
• 42: 54 (F m m 2)
• 200: 52 (P m -3)
• 125: 52 (P 4/n b m :2)
• 151: 51 (P 31 1 2)
• 6: 50 (P 1 m 1)
• 175: 47 (P 6/m)
• 3: 47 (P 1 2 1)
• 158: 46 (P 3 c 1)
• 196: 45 (F 2 3)
• 104: 44 (P 4 n c)
• 91: 44 (P 41 2 2)
• 37: 43 (C c c 2)
• 211: 42 (I 4 3 2)
• 90: 42 (P 4 21 2)
• 30: 42 (P n c 2)
• 181: 41 (P 64 2 2)
• 188: 40 (P -6 c 2)
• 83: 40 (P 4/m)
• 21: 40 (C 2 2 2)
• 106: 39 (P 42 b c)
• 75: 37 (P 4)
• 214: 36 (I 41 3 2)
• 17: 36 (P 2 2 21)
• 132: 34 (P 42/m c m)
• 77: 34 (P 42)
• 133: 32 (P 42/n b c :2)
• 134: 31 (P 42/n n m :2)
• 102: 31 (P 42 n m)
• 25: 31 (P m m 2)
• 210: 30 (F 41 3 2)
• 117: 30 (P -4 b 2)
• 112: 30 (P -4 2 c)
• 108: 30 (I 4 c m)
• 35: 30 (C m m 2)
• 24: 30 (I 21 21 21)
• 149: 28 (P 3 1 2)
• 97: 27 (I 4 2 2)
• 39: 27 (A b m 2)
• 116: 26 (P -4 c 2)
• 22: 25 (F 2 2 2)
• 48: 22 (P n n n :2)
• 28: 21 (P m a 2)
• 115: 20 (P -4 m 2)
• 109: 20 (I 41 m d)
• 209: 19 (F 4 3 2)
• 171: 17 (P 62)
• 195: 15 (P 2 3)
• 103: 15 (P 4 c c)
• 172: 14 (P 64)
• 111: 13 (P -4 2 m)
• 49: 13 (P c c m)
• 27: 13 (P c c 2)
• 16: 12 (P 2 2 2)
• 207: 11 (P 4 3 2)
• 183: 11 (P 6 m m)
• 153: 10 (P 32 1 2)
• 208: 9 (P 42 3 2)
• 184: 6 (P 6 c c)
• 177: 6 (P 6 2 2)
• 105: 6 (P 42 m c)
• 89: 6 (P 4 2 2)
• 168: 5 (P 6)
• 101: 4 (P 42 c m)

CCTBX script to analyse the data

# Obtained from the COD and simply formatted as a Python dictionary
d={
'''NULL''':204,
'''?''':10,
'''???''':5,
'''????''':1,
'''?P?''':5,
'''(000 ???''':1,
'''A -1''':33,
'''A 1 1 2''':7,
'''A 1 1 2/a''':5,
'''A 1 1 2/m''':4,
'''A 1 1 m''':1,
'''A 1 2 1''':21,
'''A 1 2/a 1''':139,
'''A 1 2/m 1''':45,
'''A 1 2/n 1''':18,
'''A 1 a 1''':12,
'''A 1 n 1''':3,
'''A 2 1 1''':1,
'''A 2 a a''':2,
'''A 2 m m''':6,
'''A 2/m''':2,
'''A 21 2 2''':1,
'''A 21 a m''':64,
'''A 21 m a''':1,
'''A b a 2''':325,
'''A b m 2''':24,
'''A b m a''':23,
'''A b m m''':7,
'''A c 2 a''':1,
'''A c a m''':13,
'''A c m m''':4,
'''A e 2 a''':1,
'''A e a 2''':18,
'''A e a m''':1,
'''A e m a''':2,
'''A m 2 m''':1,
'''A m a 2''':99,
'''A m a 2 (b,-a,c)''':1,
'''A m a a''':9,
'''A m a m''':20,
'''A m m 2''':69,
'''A m m 2 (b,c,a)''':1,
'''A m m a''':18,
'''A m m m''':28,
'''A2/m''':1,
'''B -1''':8,
'''B 1''':1,
'''B 1 1 2''':5,
'''B 1 1 2/b''':30,
'''B 1 1 2/m''':23,
'''B 1 1 2/n''':3,
'''B 1 1 b''':9,
'''B 1 1 m''':11,
'''B 1 21 1''':26,
'''B 1 21/d 1''':4,
'''B 1 21/m 1''':1,
'''B 2 c b''':2,
'''B 2 m b''':2,
'''B 2 m m''':2,
'''B 21/c''':1,
'''B 21/d''':1,
'''B b 21 m''':27,
'''B b a 2''':3,
'''B b c b :2''':1,
'''B b c m''':8,
'''B b e m''':2,
'''B b m 2''':3,
'''B b m m''':53,
'''B m 2 m''':2,
'''B m a b''':59,
'''B m a m''':1,
'''B m m 2''':1,
'''B m m b''':12,
'''B2/m''':1,
'''Bmmm(0\b0)s00''':1,
'''C -1''':244,
'''C -4 2 b''':3,
'''C 1''':30,
'''C 1 1 2/a''':1,
'''C 1 1 21''':4,
'''C 1 1 21/d''':3,
'''C 1 2 1''':2636,
'''C 1 2 1 (a+2*c,a,b)''':6,
'''C 1 2/c 1''':29452,
'''C 1 2/c 1 (-a,a+c,b)''':1,
'''C 1 2/c 1 (-a,c,b)''':5,
'''C 1 2/c 1 (-a+c,c,b)''':2,
'''C 1 2/c 1 (c,a+c,b)''':1,
'''C 1 2/c 1 (c,b,-a)''':3,
'''C 1 2/c 1 (c,b,-a+c)''':19,
'''C 1 2/c 1 S''':1,
'''C 1 2/m 1''':4183,
'''C 1 2/m 1 (-a,c,b)''':4,
'''C 1 2/m 1 (a,b,a+2*c)''':4,
'''C 1 2/m 1 (a+c,b,c)''':21,
'''C 1 2/m 1 (c,b,-a)''':7,
'''C 1 2/n 1''':14,
'''C 1 21 1''':1,
'''C 1 c 1''':3522,
'''C 1 m 1''':248,
'''C 2 2 2''':40,
'''C 2 2 21''':603,
'''C 2 c b''':13,
'''C 2 c m''':11,
'''C 2 e b''':1,
'''C 2 m m''':6,
'''C 2/c''':1,
'''C 2/m 1 1''':1,
'''C 2/n 1 1''':1,
'''C 4 2 21''':1,
'''C c 2 a''':3,
'''C c c 2''':40,
'''C c c 2 (b,c,a)''':1,
'''C c c a''':6,
'''C c c a :2''':139,
'''C c c a :2 (-a,c,b)''':2,
'''C c c a (C c c e)''':2,
'''C c c b :1''':7,
'''C c c b :2''':1,
'''C c c e :1''':2,
'''C c c m''':119,
'''C c c m (b,c,a)''':2,
'''C c m 21''':25,
'''C c m b''':13,
'''C c m e''':1,
'''C c m m''':35,
'''C m 2 a''':3,
'''C m 2 m''':13,
'''C m c 21''':677,
'''C m c 21 (b,c,a)''':1,
'''C m c a''':560,
'''C m c a (-a,c,b)''':1,
'''C m c a (b,c,a)''':3,
'''C m c a (c,a,b)''':1,
'''C m c e''':51,
'''C m c m''':1420,
'''C m c m (-a,c,b)''':1,
'''C m c m (b,-a,c)''':5,
'''C m c m (b,c,a)''':4,
'''C m c m (c,a,b)''':2,
'''C m c m (c,b,-a)''':2,
'''C m m 2''':21,
'''C m m 2 (2*c,a,b)''':1,
'''C m m a''':47,
'''C m m a (b,c,a)''':2,
'''C m m a (c,a,b)''':1,
'''C m m b''':1,
'''C m m m''':218,
'''C m m m (c,a,b)''':20,
'''C-1''':1,
'''C-1(\a\b\g)0''':1,
'''C1(\a\b\g)0''':1,
'''C2:b1''':1,
'''C2/c(0\b0)s0''':2,
'''C2/m(\a0\g)-1s''':2,
'''C2/m(\a0\g)00''':2,
'''C2/m(\a0\g)0s''':5,
'''C2/m(0\b0)s0''':5,
'''Cc(\a0\g)0''':1,
'''Ccmm(\a00)00s''':1,
'''Cmca(00\g)s00''':3,
'''Cmcm(00\g)000''':1,
'''Cmmm(00\g)0s0''':1,
'''Cubic''':1,
'''F''':1,
'''F -1''':4,
'''F -4 3 c''':64,
'''F -4 3 m''':671,
'''F -4 d 2''':2,
'''F 1''':3,
'''F 1 -1 1''':1,
'''F 1 1 2''':1,
'''F 1 2/d 1''':2,
'''F 1 2/m 1''':3,
'''F 1 d 1''':6,
'''F 2 2 2''':25,
'''F 2 3''':45,
'''F 2 d d''':32,
'''F 2 m m''':1,
'''F 4 3 2''':19,
'''F 4/m m m''':4,
'''F 41 3 2''':30,
'''F 41/a d c''':1,
'''F d -3''':5,
'''F d -3 :1''':5,
'''F d -3 :2''':115,
'''F d -3 c''':10,
'''F d -3 c :2''':44,
'''F d -3 m''':88,
'''F d -3 m :1''':1136,
'''F d -3 m :2''':2604,
'''F d -3 m {origin @ -3 m}''':1,
'''F d -3 m {origin @ centre (-3m)}''':1,
'''F d 2 d''':9,
'''F d 3 m''':1,
'''F d d''':2,
'''F d d 2''':1229,
'''F d d 2 S''':2,
'''F d d d''':59,
'''F d d d :1''':44,
'''F d d d :2''':501,
'''F d d d {origin @ -1 @ d d d}''':2,
'''F d d d {origin @ 2 2 2}''':1,
'''F m -3''':64,
'''F m -3 c''':103,
'''F m -3 m''':4385,
'''F m -3m''':2,
'''F m 2 m''':5,
'''F m m 2''':46,
'''F m m 2 (b,c,a)''':2,
'''F m m m''':143,
'''F_d_d_2''':1,
'''F2(\a0\g)0''':4,
'''F2/m(\a0\g)0s''':4,
'''Fd-3m:1''':1,
'''Fd3''':1,
'''Fdd2''':1,
'''Fddd(00\g)ss0''':2,
'''Fm-3m''':2,
'''Fm3m''':9,
'''Fmmm(\a00)0s0''':3,
'''I''':1,
'''I -1''':54,
'''I -4''':649,
'''I -4 2 d''':421,
'''I -4 2 d S''':1,
'''I -4 2 m''':175,
'''I -4 3 d''':462,
'''I -4 3 m''':295,
'''I -4 c 2''':60,
'''I -4 m 2''':56,
'''I -4(\a\-b0,\b\a0)00''':1,
'''I 1''':1,
'''I 1 1 2''':1,
'''I 1 1 2/a''':3,
'''I 1 1 2/b''':13,
'''I 1 1 2/m''':4,
'''I 1 1 b''':2,
'''I 1 2 1''':139,
'''I 1 2/a 1''':955,
'''I 1 2/a 1 S''':1,
'''I 1 2/c 1''':141,
'''I 1 2/m 1''':207,
'''I 1 a 1''':103,
'''I 1 c 1''':9,
'''I 1 m 1''':23,
'''I 2 1 1 2''':1,
'''I 2 2 2''':85,
'''I 2 3''':149,
'''I 2 c b''':4,
'''I 2 c m''':6,
'''I 2 m b''':23,
'''I 2 m m''':4,
'''I 2/b 1 1''':2,
'''I 2/c 1 1''':12,
'''I 2/m''':11,
'''I 21 21 21''':30,
'''I 21 3''':113,
'''I 4''':111,
'''I 4 1/a (origin at -1)''':1,
'''I 4 2 2''':27,
'''I 4 3 2''':42,
'''I 4 c m''':30,
'''I 4 m m''':78,
'''I 4/m''':608,
'''I 4/m c m''':538,
'''I 4/m c m S''':1,
'''I 4/m m m''':1910,
'''I 4/m m m (a+b,-a+b,c)''':1,
'''I 41''':107,
'''I 41 2 2''':59,
'''I 41 3 2''':36,
'''I 41 c d''':143,
'''I 41 m d''':20,
'''I 41/a''':157,
'''I 41/a :1''':79,
'''I 41/a :2''':1389,
'''I 41/a (origin @ -1 on glide pla''':1,
'''I 41/a c d''':30,
'''I 41/a c d :1''':25,
'''I 41/a c d :2''':314,
'''I 41/a m d''':31,
'''I 41/a m d :1''':175,
'''I 41/a m d :2''':407,
'''I 41/a m d 1''':1,
'''I 41/a Z1''':1,
'''I 43''':1,
'''I a -3''':240,
'''I a -3 d''':562,
'''I b a 2''':197,
'''I b a m''':247,
'''I b a m (b,c,a)''':1,
'''I b a m S''':1,
'''I b c a''':106,
'''I b m 2''':10,
'''I b m m''':17,
'''I c 2 a''':1,
'''I c a b''':1,
'''I c m a''':1,
'''I c m a S''':1,
'''I c m m''':18,
'''I m -3''':305,
'''I m -3 m''':759,
'''I m 2 a''':4,
'''I m 2 m''':10,
'''I m a 2''':82,
'''I m a 2 (b,-a,c)''':1,
'''I m a m''':14,
'''I m c b''':7,
'''I m c m''':3,
'''I m m 2''':92,
'''I m m 2 (b,c,a)''':1,
'''I m m a''':374,
'''I m m a (-a,c,b)''':2,
'''I m m a (c,a,b)''':2,
'''I m m b''':2,
'''I m m m''':428,
'''I-4''':1,
'''I-42d''':1,
'''I2(1)3''':1,
'''I2/a''':1,
'''I2/b(\a\b0)00''':1,
'''I2/m''':5,
'''I4/m m m''':1,
'''Ia3d''':1,
'''Im-3m''':1,
'''Imma(00\g)s00''':2,
'''P -1''':85046,
'''P -1 (-a+b+c,a-b+c,a+b-c)''':2,
'''P -3''':573,
'''P -3 1 c''':285,
'''P -3 1 c S''':1,
'''P -3 1 m''':78,
'''P -3 c 1''':313,
'''P -3 m 1''':732,
'''P -4''':93,
'''P -4 2 c''':30,
'''P -4 2 m''':13,
'''P -4 21 c''':410,
'''P -4 21 m''':301,
'''P -4 21/c''':1,
'''P -4 3 m''':113,
'''P -4 3 n''':201,
'''P -4 b 2''':30,
'''P -4 c 2''':26,
'''P -4 m 2''':20,
'''P -4 n 2''':83,
'''P -6''':116,
'''P -6 2 c''':123,
'''P -6 2 m''':335,
'''P -6 c 2''':40,
'''P -6 m 2''':220,
'''P 1''':2888,
'''P 1 (-a+c,-b,a+c)''':1,
'''P 1 (b+c,a+c,a+b)''':2,
'''P 1 1 2''':5,
'''P 1 1 2/a''':1,
'''P 1 1 2/b''':5,
'''P 1 1 2/m''':7,
'''P 1 1 2/n''':3,
'''P 1 1 21''':56,
'''P 1 1 21/a''':33,
'''P 1 1 21/b''':75,
'''P 1 1 21/m''':30,
'''P 1 1 21/n''':44,
'''P 1 1 a''':1,
'''P 1 1 b''':2,
'''P 1 1 m''':6,
'''P 1 2 1''':41,
'''P 1 2/a 1''':98,
'''P 1 2/c 1''':1599,
'''P 1 2/c 1 (a,2*b,c)''':1,
'''P 1 2/m 1''':106,
'''P 1 2/n 1''':917,
'''P 1 21 1''':14543,
'''P 1 21 1 (a-1/4,b,c)''':1,
'''P 1 21 1 (c,2*a+c,b)''':1,
'''P 1 21/a 1''':2929,
'''P 1 21/c 1''':65464,
'''P 1 21/c 1 (2*a+c,b,c)''':3,
'''P 1 21/c 1 (2*c,2*a+c,b)''':14,
'''P 1 21/m 1''':2330,
'''P 1 21/m 1 S''':1,
'''P 1 21/n 1''':45963,
'''P 1 a 1''':25,
'''P 1 c 1''':889,
'''P 1 m 1''':41,
'''P 1 n 1''':624,
'''P 2 1 1''':1,
'''P 2 2 2''':12,
'''P 2 2 21''':36,
'''P 2 21 21''':62,
'''P 2 3''':15,
'''P 2 a n''':5,
'''P 2 c b''':3,
'''P 2 c m''':2,
'''P 2 m m''':8,
'''P 2 n n''':6,
'''P 2/c 1 1''':1,
'''P 2/m 1 1''':1,
'''P 2/n''':1,
'''P 21''':1,
'''P 21 1 1''':5,
'''P 21 2 21''':14,
'''P 21 21 2''':1209,
'''P 21 21 21''':19726,
'''P 21 21 21 (origin shift x,y,z+1''':1,
'''P 21 3''':493,
'''P 21 a b''':13,
'''P 21 a m''':8,
'''P 21 c a''':30,
'''P 21 c n''':79,
'''P 21 m a''':16,
'''P 21 m n''':8,
'''P 21 n b''':58,
'''P 21 n m''':29,
'''P 21/b''':8,
'''P 21/b 1 1''':60,
'''P 21/c''':3,
'''P 21/c 1 1''':3,
'''P 21/m 1 1''':2,
'''P 21/N''':9,
'''P 21/n 1 1''':8,
'''P 3''':146,
'''P 3 1 2''':28,
'''P 3 1 c''':224,
'''P 3 1 m''':60,
'''P 3 2 1''':196,
'''P 3 c 1''':45,
'''P 3 m 1''':68,
'''P 31''':246,
'''P 31 1 2''':51,
'''P 31 2 1''':424,
'''P 31 2 1 S''':1,
'''P 32''':230,
'''P 32 1 2''':10,
'''P 32 2 1''':349,
'''P 32 2 1 S''':3,
'''P 3c1''':1,
'''P 4''':37,
'''P 4 2 2''':6,
'''P 4 21 2''':42,
'''P 4 3 2''':11,
'''P 4 b m''':64,
'''P 4 b m (a,b,2*c)''':2,
'''P 4 c c''':15,
'''P 4 m m''':115,
'''P 4 n c''':44,
'''P 4/m''':40,
'''P 4/m b m''':312,
'''P 4/m c c''':54,
'''P 4/m m m''':901,
'''P 4/m n c''':133,
'''P 4/mbm''':2,
'''P 4/n''':45,
'''P 4/n :1''':18,
'''P 4/n :2''':317,
'''P 4/n b m''':5,
'''P 4/n b m :1''':11,
'''P 4/n b m :2''':36,
'''P 4/n c c''':31,
'''P 4/n c c :1''':24,
'''P 4/n c c :2''':276,
'''P 4/n m m''':21,
'''P 4/n m m :1''':285,
'''P 4/n m m :2''':436,
'''P 4/n n c''':20,
'''P 4/n n c :1''':5,
'''P 4/n n c :2''':89,
'''P 4/n n c {origin @ -1 @ n n (n,''':2,
'''P 4/n n c Z1''':1,
'''P 41''':294,
'''P 41 2 2''':44,
'''P 41 21 2''':680,
'''P 41 3 2''':75,
'''P 42''':34,
'''P 42 21 2''':56,
'''P 42 3 2''':9,
'''P 42 b c''':39,
'''P 42 c m''':4,
'''P 42 m c''':6,
'''P 42 n m''':31,
'''P 42/m''':54,
'''P 42/m b c''':82,
'''P 42/m c m''':34,
'''P 42/m m c''':86,
'''P 42/m m c S''':1,
'''P 42/m n m''':554,
'''P 42/mnm''':1,
'''P 42/n''':47,
'''P 42/n :1''':28,
'''P 42/n :2''':440,
'''P 42/n b c''':6,
'''P 42/n b c :1''':1,
'''P 42/n b c :2''':25,
'''P 42/n c m''':9,
'''P 42/n c m :1''':10,
'''P 42/n c m :2''':75,
'''P 42/n m c''':13,
'''P 42/n m c :1''':44,
'''P 42/n m c :2''':85,
'''P 42/n m c S''':2,
'''P 42/n n m''':5,
'''P 42/n n m :1''':9,
'''P 42/n n m :2''':17,
'''P 43''':273,
'''P 43 2 2''':57,
'''P 43 21 2''':616,
'''P 43 3 2''':70,
'''P 4mm''':1,
'''P 6''':5,
'''P 6 2 2''':6,
'''P 6 c c''':6,
'''P 6 m m''':11,
'''P 6/m''':47,
'''P 6/m c c''':210,
'''P 6/m c c S''':1,
'''P 6/m m m''':1033,
'''P 61''':213,
'''P 61 2 2''':119,
'''P 62''':17,
'''P 62 2 2''':91,
'''P 63''':533,
'''P 63 2 2''':149,
'''P 63 c m''':94,
'''P 63 m c''':595,
'''P 63 m c S''':1,
'''P 63/m''':1242,
'''P 63/m c m''':281,
'''P 63/m m c''':2198,
'''P 64''':14,
'''P 64 2 2''':41,
'''P 65''':207,
'''P 65 2 2''':111,
'''P a -3''':686,
'''P b -3''':1,
'''P b 2 n''':1,
'''P b 21 a''':12,
'''P b 21 m''':8,
'''P b a 2''':67,
'''P b a a''':3,
'''P b a m''':521,
'''P b a n''':10,
'''P b a n :1''':4,
'''P b a n :2''':42,
'''P b c 21''':91,
'''P b c a''':10582,
'''P b c b''':4,
'''P b c m''':440,
'''P b c n''':2826,
'''P b m 2''':3,
'''P b m a''':19,
'''P b m m''':12,
'''P b m n''':19,
'''P b n 21''':55,
'''P b n a''':58,
'''P b n b''':11,
'''P b n m''':1193,
'''P b n m S''':1,
'''P b n n''':11,
'''P c 21 b''':37,
'''P c 21 n''':89,
'''P c a 2 1''':1,
'''P c a 21''':2388,
'''P c a a''':1,
'''P c a b''':256,
'''P c a m''':11,
'''P c a n''':40,
'''P c c 2''':13,
'''P c c a''':201,
'''P c c b''':1,
'''P c c m''':13,
'''P c c n''':1133,
'''P c m 21''':4,
'''P c m a''':3,
'''P c m b''':9,
'''P c m n''':80,
'''P c n 2''':1,
'''P c n b''':27,
'''P c n m''':1,
'''P c n n''':12,
'''P m -3''':52,
'''P m -3 m''':1770,
'''P m -3 n''':422,
'''P m 1 1''':3,
'''P m 2 a''':3,
'''P m 2 m''':4,
'''P m 21 b''':3,
'''P m 21 n''':14,
'''P m 3 m''':1,
'''P m a 2''':13,
'''P m a b''':5,
'''P m a m''':9,
'''P m a n''':10,
'''P m c 21''':91,
'''P m c a''':2,
'''P m c b''':7,
'''P m c m''':4,
'''P m c n''':258,
'''P m c n S1''':1,
'''P m c n S2''':1,
'''P m m 2''':19,
'''P m m a''':99,
'''P m m a (2*b,c,a)''':1,
'''P m m a (2*b+1/4,c,a-1/3)''':4,
'''P m m b''':5,
'''P m m m''':361,
'''P m m m (2*a,2*b,c)''':1,
'''P m m n''':21,
'''P m m n :1''':48,
'''P m m n :2''':228,
'''P m n 21''':291,
'''P m n 21 S''':1,
'''P m n a''':74,
'''P m n b''':75,
'''P m n b S''':3,
'''P m n m :1''':8,
'''P m n m :2''':20,
'''P m n n''':27,
'''P n -3''':4,
'''P n -3 :1''':29,
'''P n -3 :2''':62,
'''P n -3 m :1''':22,
'''P n -3 m :2''':45,
'''P n -3 n''':6,
'''P n -3 n :1''':1,
'''P n -3 n :2''':53,
'''P n 1 1''':2,
'''P n 2 n''':3,
'''P n 21 a''':119,
'''P n 21 m''':15,
'''P n a 21''':4484,
'''P n a a''':32,
'''P n a b''':28,
'''P n a m''':344,
'''P n a n''':16,
'''P n c 2''':35,
'''P n c a''':67,
'''P n c b''':1,
'''P n c b :1''':2,
'''P n c m''':14,
'''P n c n''':20,
'''P n m 21''':11,
'''P n m a''':5925,
'''P n m a (c,a-1/4,b)''':3,
'''P n m b''':1,
'''P n m m''':1,
'''P n m m :1''':1,
'''P n m m :2''':21,
'''P n m n''':25,
'''P n n 2''':147,
'''P n n a''':412,
'''P n n b''':2,
'''P n n m''':582,
'''P n n n''':6,
'''P n n n :1''':1,
'''P n n n :2''':15,
'''P_-1''':1,
'''P_1_21/c_1''':1,
'''p_21_c_a''':1,
'''p_21/a''':1,
'''P_63''':3,
'''P_63/m''':3,
'''P-1''':30,
'''P-1(\a\b\g)0''':1,
'''P-3c1''':1,
'''P-4-h(10)g(2)dcba''':1,
'''P-42_1m(a,a,0)00s(-a,a,0)000''':1,
'''P1''':1,
'''P112/n''':1,
'''P2_1/n''':1,
'''P2(1)2(1)2(1)''':2,
'''P2/m(\a1/2\g)00''':1,
'''P21 21 21''':1,
'''P21 or P21/m''':1,
'''P21.a''':1,
'''P21(\a0\g)0''':2,
'''P21/c(\a0\g)00''':1,
'''P21/c(0\b0)s0''':1,
'''P21/m(\a0\g)00''':4,
'''P21/m(\a0\g)0s''':1,
'''P212121''':5,
'''P213''':1,
'''P3c&''':1,
'''P42 mnm''':1,
'''P43-n''':1,
'''P63/m''':1,
'''P63/mmc''':14,
'''Pa3''':1,
'''Pbam''':1,
'''Pbc2''':2,
'''Pbca''':1,
'''Pca21''':1,
'''Pm-3m''':1,
'''Pm21n(\a00)000''':1,
'''Pm3m''':1,
'''Pmcn(00\g)ss0''':2,
'''Pmmm(\a,1/2,0)000(1/2,\b,0)000''':1,
'''Pmn2''':3,
'''Pna2(1)''':1,
'''Pna21''':1,
'''Pnma''':1,
'''Pnnm''':4,
'''R -3''':193,
'''R -3 :H''':2761,
'''R -3 :R''':176,
'''R -3 c''':94,
'''R -3 c :H''':1416,
'''R -3 c :R''':66,
'''R -3 c RS''':7,
'''R -3 cr {rhombohedral axes}''':1,
'''R -3 HR''':1,
'''R -3 m''':52,
'''R -3 m :H''':1530,
'''R -3 m :R''':172,
'''R -3 m HR''':1,
'''R -3:H''':1,
'''R -3:r''':1,
'''R 1 2/c 1''':1,
'''R 3''':52,
'''R 3 :H''':481,
'''R 3 :R''':32,
'''R 3 2''':24,
'''R 3 2 :H''':221,
'''R 3 2 :R''':13,
'''R 3 c''':19,
'''R 3 c :H''':527,
'''R 3 c :R''':17,
'''R 3 m''':15,
'''R 3 m :H''':580,
'''R 3 m :R''':42,
'''R 3:r''':1,
'''R 32''':1,
'''R-3(00\g)0''':13,
'''r-3ch''':2,
'''R-3m(00\g)''':1,
'''R-3m(00\g)00''':3,
'''R-3m(00\g)0s''':1,
'''R???''':1,
'''R3(00\g)t''':2,
'''R31(00\g)ts''':8,
'''tetragonal''':1,
'''unknown''':1,
'''X c''':1,
'''X-3(00\g)0''':1,
'''X-3c1(00\g)000''':1,
'''X2/m''':2,
'''X2/m(\a0\g)0s''':13,
'''X21(000)0''':1,
'''X21(1/21/2\g)0''':4,
'''X3(00\g)0''':1,
'''X4bm''':8,
'''Xmc21''':1,
'''Xmcm(0\b0)s0s''':1,
'''Xmcm(00\g)000''':3,
'''Xmmm(\a00)000''':1,
'''Xmmm(00\g)0s0''':2,
'''Xmnm(\a00)00s''':1,
}

from cctbx import sgtbx
n = {}
for hm, count in d.iteritems():
  try:
    sgi = sgtbx.space_group_info(hm)
    nb = sgi.type().number()
  except Exception, err:
    nb = 0
  n.setdefault(nb, 0)
  n[nb] += count
print "missing: {}".format(set(xrange(1,231))-set(n.keys()))
print "buggy: {}".format(n[0])
for count, nb in sorted(((count, nb) for nb, count in n.iteritems()
                         if nb != 0), 
                        reverse=True):
  sgi = sgtbx.space_group_info(number=nb)
  print "- {:3d}: {:>7} ({})".format(
    nb, count, sgi.type().universal_hermann_mauguin_symbol())

Answer 4

Actually for all 230 of them, according to this Q/A on Earth Science.SE , which refers to a nice table given at this blog post.

Answer 5

I would be surprised if there wasn't an example of each.

For instance, you can search the American Mineralogist Crystal Structure Database by space group. Click on Cell Parameters and Symmetry and select whichever you like. I'll leave it to you to go through all 230 of them.

Answer 6

Probably all of them and more besides. The 230 does not include quasicrystals, while nature does.