Example of a space group which does not contain the point group as a subgroup

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