Hint: for a permutation of objects then has order precisely . Prove this, then apply it to your questions.

To find the subgroups isomorphic to note that , the cyclic group of 2 elements. So, how many elements in this group have order 2?...

For , your groups are just the permutations of 3 elements, . This is because is merely the group of all possible permutations of 3 objects, by it's "rawest" definition. If you take all possible permutations of 3 objects you will get a group isomorphic to .

I suspect someone here can provide you with a hint towards a much neater proof than I have. Sorry. However, I would advise you to try this for smaller values of to see how it works. Note that is generated by a transposition and an -cycle for all