Point Groups in Three Dimensions - Maximal Symmetries

Maximal Symmetries

There are two discrete point groups with the property that no discrete point group has it as proper subgroup: O_h and I_h. Their largest common subgroup is T_h. The two groups are obtained from it by changing 2-fold rotational symmetry to 4-fold, and adding 5-fold symmetry, respectively. Alternatively the two groups are generated by adding for each a reflection plane to T_h.

There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup: O_h and D_6h. Their maximal common subgroups, depending on orientation, are D_3d and D_2h.