Problem re: Sylow's Theorems

Hi there,

I had this question on an exam today and I'm not sure how to do it.

"Let G = $\displaystyle \mathbb{Z}_3$ x $\displaystyle A_4$. List all the Sylow 2-subgroups of G, and all the Sylow 3-subgroups of G, giving reasons that your lists are complete"

I found that G has order 36, and so I determined that $\displaystyle n_2$ = 1, 3 or 9 and that $\displaystyle n_3$ = 1 or 4 (where $\displaystyle n_i$ denotes the number of Sylow i-subgroups) but from there I couldn't do much. By brute force I managed to find one subgroup of order 4 and one of order 9, so in the end I just put that $\displaystyle n_2$ = 1 and $\displaystyle n_3$= 1.

Can anyone verify, or correct this?

Many thanks