I understand that the automorphism group of an $n$ cycle graph is the dihedral group $D_n$ of order $2 n$. From the comment of @Christian, I also understand that $S_n$ is the automorphism group of the complete graph $K_n$ which has $D_n$.
Other than these two obvious cases, I would like to know what are the other known automorphism groups of graphs which have $D_n$ as a subgroup. Of course, the easiest are the graphs which have multiple $n$ cycle graphs as subgraphs or multiple $K_n$ as subgraphs. But, are there other graphs?