I need to list all the subgroups of the symmetry group of the cube that are isomorphic to either $\mathbb{Z}_n$ or $\mathbb{D}_n$ (the dihedral group) for some $n \in \mathbb{N}$.
I know that the actual group is isomorphic to $S_4 \times \mathbb{Z}_2$. Hence, all the subgroups of $S_4$ are subgroups of the symmetry group of the cube, and I should check whether they are isomorphic or not to the two groups of interest.
Some questions: Is that it? And I can see here a thorough classification of all such subgroups. Is there any other way of reducing the amount of possibilities (given that we are interested in only certain kind of subgroups) than using Lagrange's theorem? Can Sylow be of further help in reducing this range of options? (So far in my course we haven't covered his theorem).
Thanks in advance!
