For the following group how to prove no two of them are isomorphic?

Question

The groups are $A_3 \times \mathbb Z_2$, $A_4 \times \mathbb Z_2$, $\mathbb Z_8$, $S_3 \times \mathbb Z_4$. I know that if the order of the groups are not same it is not isomorphic. And the groups with huge order is not isomorphic to small groups. But I dont know how to determine whether the above groups are isomorphic to each other.

Answer 1

Hint: $S_3\times \mathbb{Z}_4$ has an element of order 12.

Answer 2

Only two of the groups you seem to have written down (hard to understand) have the same order: $\;A_4\times\Bbb Z_2\;,\;\;S_3\times\Bbb Z_4\;$. Yet the first group has no element of order $\;4\;$ , whereas the second group has.

The other two groups have order $\;6\;,\;\;8\;$ .