This question is related to this math.se question.
Consider the dihedral group $D_n = \langle r,s \rangle.$
Two subgroups $G, H \leq D_n$ are said to be ''isomorphic'' if there is an $f \in \rm{Aut}(D_n)$ such that $f(G) = H.$
I would like to describe all non-isomorphic subgroups of $D_n.$ I can describe all subgroups of $D_n$ up to conjugation but don't see how to do the final step of checking how these list collapses with respect to outer automorphisms of $D_n.$
What I do know
- Every automorphism $f$ of $D_n$ sends $f(r) = r^k$ and $f(s) = sr^l$ for $1 \le l,k < n$ such that $\gcd(k,l) = 1.$
The inner automorphisms of $D_n$ are of the form $f(r) = r^{\pm 1}, f(s) = sr^{2k}$
For $n$ odd the subgroups of $D_n$ are (up to conjugacy): $$\{ \langle r^m, s \rangle \quad m | 2n \quad \mbox{ and } m \equiv 1 \pmod{2} \} \cup \{ \langle r^{m/2} \rangle | m |2n \mbox{ and } m \equiv 0 \pmod{2}\}.$$
- For $n$ even the subgroups of $D_n$ up to conjugation are $$ \{ \langle r^m, s \rangle \mid m |2n \mbox{ and } m \equiv 1 \pmod{2}\} \cup \{ \langle r^{m/2} \rangle \mid m \not | n \mbox{ and } m \equiv 0 \pmod{2} \}$$ and $$\{ \langle r^{m/2} \rangle, \langle r^m,s \rangle , \langle r^m,rs\rangle \mid m | n \mbox{ and } m \equiv 0 \pmod{2}\}.$$
My question now is, how to take into account the isomorphisms of $\rm{Out}(D_n)$ and the listed classes of subgroups?