Set H=<(12)(34),(123)>. Then H=A4: 2 and 3 both divide the order of H but since A4 has no subgroup of order 6 (if it did a Sylow 3 subgroup of A4 would be normal in A4), H=A4.
Here's a Cayley diagram for A4 and the given generators. The products of cycles are formed "left to right" and the elements are post multiplied by the generators. If you prefer products "right to left", just think of the elements being pre multiplied by the generators.