Symmetry group

Up to conjugacy the discrete point groups in two-dimensional space are the following classes:

Within each of these symmetry types, there are two degrees of freedom for the center of rotation, and in the case of the dihedral groups, one more for the positions of the mirrors.

The remaining isometry groups in two dimensions with a fixed point are:

Non-bounded figures may have isometry groups including translations; these are:

For spherical symmetry, there is no such distinction: any patterned object has planes of reflection symmetry.