Yeah, that's the idea. You can use the fact that if

and

then

where

and

. Essentially, that every subgroup of

is a direct product of subgroups from these respective groups.

Why is this true in this case? This is definitively not so in other cases, for example: in the Klein group , the sbgp. is not the direct sum of sbgps. of Tonio
However, you should probably look at your proof of

not having a subgroup of order 6 as

, and 6 divides 12 (of course, I may be getting the wrong end of your stick here)...(if I was proving it, I'd look at what the elements of

look like).

Why is your thread called "normal subgroups"?