Past exam of the mathematics course of the University of Cambridge / 2004 / ia / Paper 3 / 5D / Solution

Use $D_{12}$ for the dihedral group of order twelve and write each element uniquely as $g^j$ or $g^{j}h$, $0\le j<6$. The classification of subgroups of a dihedral group organizes the complete list: rotational subgroups are cyclic, and a subgroup containing a reflection is generated by its rotational part and one reflection. The individual lists and quotients follow below.