How many of the 230 crystallographic groups are realized in nature? All of them or only a subset? 
This is a famous and fundamental result in solid state physics. 
 A: 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.
A: 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.
A: 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.
A: 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.
A: Probably all of them and more besides.  The 230 does not include quasicrystals, while nature does.
