What are the different subgroups of $A_4 \times \mathbb Z_2$? Is there a place where I can find the subgroups of $A_4 \times \mathbb Z_2$ or is there a way I can list them completely? In particular, given the set of elements in terms of the generators, i.e.,
$\langle a,x \mid a^3 = x^2 = (ax)^3 = e \rangle$, that generates $A_4$
and $\langle b \mid b^2 = e \rangle $ that generates $\mathbb Z_2$, I would like to find all the possible ways to get the different subgroups.
For example, subgroup $\mathbb Z_2 = \{e,x\}, \{e,b \}, \{e, a x a^2 \}, \ldots$ .
I am a physicist so I apologize if I have phrased this incorrectly. 
 A: I'll describe a classification of subgroups of $G\times \Bbb Z_2$ for any finite group $G$. This leaves you the task of understanding why the classification is true and then implementing it for $G=A_4$. But I think the more general version is more illuminating and will lead to better understanding down the road.
So let $H$ be a subgroup of $G\times \Bbb Z_2$. Define $G_0 = \{(g,0)\colon g\in G\}$, which is isomorphic to $G$ itself. Since the intersection of two subgroups is again a subgroup, we know that $H_0 = H\cap G_0$ is a subgroup of $G_0$, hence is isomorphic to a subgroup of $G$ (and we'll abuse notation and call $H_0$ a subgroup of $G$). Moreover, since $[G\times \Bbb Z_2:G_0] = 2$, we have $[H:H_0] \mid 2$ and so $[H:H_0]$ equals either $1$ or $2$.
The case $[H:H_0]=1$ (equivalently, $H\subseteq G_0$) is easy: there's one such example for each subgroup of $G$.
The case $[H:H_0]=2$ has two flavors. Let $H_1 = H \setminus H_0$, and let $\bar H_j$ ($i=0,1$) be the projection of $H_j$ onto $G$ (that is, forget the second coordinates). Then choose any $x\in \bar H_1$; then $(x,1)^{-1}H_1 = H_0$, and so $H = H_0 \times \langle(x,1)\rangle$. (This includes the case $x=e$, when $H = \bar H_0 \times \langle(e,1)\rangle = \bar H_0 \times \Bbb Z_2$.)
