The general formula for the action of an element of a space group is
: y = M.x + D
where M is its matrix, D is its vector, and where the element transforms point x into point y. In general, D = D (lattice) + , where  is a unique function of M that is zero for M being the identity. The matrices M form a point group that is a basis of the space group; the lattice must be symmetric under that point group, but the crystal structure itself may not be symmetric under that point group as applied to any particular point (that is, without a translation). For example, the diamond cubic structure does not have any point where the cubic point group applies.
The lattice dimension can be less than the overall dimension, resulting in a "subperiodic" space group. For (overall dimension, lattice dimension):
* (1,1): One-dimensional line groups
* (2,1): Two-dimensional line groups: frieze groups
* (2,2): Wallpaper groups
* (3,1): Three-dimensional line groups; with the 3D crystallographic point groups, the rod groups
* (3,2): Layer groups
* (3,3): The space groups discussed in this article