Example of a space group which does not contain the point group as a subgroup In my current understanding, a space group $G_\infty$ is a combination of a crystallographic point group $G_0$ and translations in the following way: $(R_1,\vec{t_1})(R_2,\vec{t_2}):=(R_1R_2,\vec{t_1}+R_1\vec{t_2})\in G_\infty$. Now I have heard that there are space groups, which do not contain $G_0$ as a subgroup i.e. $\exists R\in G_0:(R,0)\not\in G_\infty$. I would love to see an example. So can anyone make (or provide a link to) an example of such a space group?
Edit: I assume that I misunderstood the person telling me the statement above. Probably he wanted to tell me about the existence of cases like the one illustrated below. So the question properly rephrased should sound like:
Can anyone provide an example of a space group which has an element g, for which the element $gt^-1\not\in G_\infty$ where t is the "translational contribution" to g.
 A: I'm not sure this is an answer to your question, but per your comment I will leave it anyway. Consider the Kagome lattice, shown below [ref]. One can see in the image that the Kagome lattice has the same translational symmetries of a triangular lattice: each of the red, blue, and green sublattices form a triangular lattice, connected by the Bravais lattice vectors $\mathbf{b}_1$, $\mathbf{b}_2$, $\mathbf{b}_3$. We also have point group symmetries, which are equivalent to those of the hexagonal honeycomb lattice, which the Kagome is the bond-dual lattice of (place a vertex and the center of each triangle and you will see they form a honeycomb lattice with the vertices of the Kagome lying on the bonds of the honeycomb).
Notice that the translations $\mathbf{a}_1$, $\mathbf{a}_2$, $\mathbf{a}_3$ are not symmetries of the lattice by themselves, but that we can construct new symmetry operations by combining a translation by $\mathbf{a}_i$ followed by a reflection across the line which that vector lies along. Take the two green arrows on the triangle labeled (1) and translate them by $\mathbf{a}_1$, then reflect them across the horizontal line forming the base of the triangle (1) and you will get the dashed green arrows. This is a glide symmetry, and it cannot be written as a combination of one of the Bravais translations and a point group operation. Neither the translation by $\mathbf{a}_i$ nor the reflection perpendicular to $\mathbf{a}_i$ are elements of either the translation or point symmetry groups of the lattice, but together they are an element of the space symmetry group.
Edit:
Some more info on space groups is available on the Wikipedia page here. Notice that it lists "Symmetry fixing a point" (the point group), then Translations, then Glide and Screw symmetries, which are called non-symmorphic symmetries. They are present in the space group but they are not a simple composition of a translation and point symmetry operation.
For more info on space symmetry groups one may consult the books:
"Group Theory: Application to the Physics of Condensed Matter" by Dresselhaus & Dresselhaus
The mathematical theory of symmetry in solids. Representation theory for point groups and space groups by C. J. Bradley and A. P. Cracknell

