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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.

Since $|D_5|=10$, Lagrange's theorem permits proper subgroup orders only $1,2,5$. The complete list isTo show completeness, an order-two subgroup must be generated by its unique nonidentity element, necessarily one of the five reflections. An order-five subgroup is cyclic, since its order is prime; its generator cannot be a reflection, and every nonidentity rotation generates the same rotation subgroup. There is only one order-one subgroup.

Thus there are seven proper subgroups when the trivial subgroup is included, or six nontrivial proper subgroups. These are the prime-sided case of the classification of subgroups of a dihedral group.

All six reflections square to the identity, and the rotation $R^3$ is the half-turn. These exhaust the nonidentity involutions: $R^k$ has square one only when $2k\equiv 0(\mathrm{mod}6)$, giving $k=3$ apart from the identity. Thus the order-two subgroups areA reflection cannot have order three. The rotations of order three are $R^2$ and $R^4$, and they generate the same subgroup:All these subgroups are cyclic, because a group of prime order is cyclic; the displayed nonidentity elements supply their generators. This gives seven subgroups of order two and one of order three, consistently with involutions in an even dihedral group and the classification of subgroups of a dihedral group.

Let $n=p\ge 3$ be prime. Any subgroup contained in $⟨R⟩\cong C_p$ is either $\{I\}$ or the whole rotation subgroup. Both are normal: the trivial subgroup always is, and the rotation subgroup has index two, or equivalently is preserved by $mR{m}^{-1}=R^{-1}$.

If a normal subgroup contains any reflection, it contains its entire conjugacy class. Part (b) shows that this is every reflection for odd $p$. Products of reflections include a generator of the rotations, since $(Rm)m=R$. Such a normal subgroup must therefore be the whole dihedral group.

Thus the complete list of proper normal subgroups, including the trivial subgroup, isThis also agrees with classification of subgroups of a dihedral group: the reflection subgroups of order two are proper but are not normal when $p$ is odd.