Let V be a set of given matrices. Prove that V is a group under matrix multiplication, and find all subgroups of V. Let $V=\left\lbrace A_1=\begin{bmatrix}1&0\\0&1\end{bmatrix},
A_2=\begin{bmatrix}-1&0\\0&1\end{bmatrix},
A_3=\begin{bmatrix}1&0\\0&-1\end{bmatrix},
A_4=\begin{bmatrix}-1&0\\0&-1\end{bmatrix}
\right\rbrace $. Prove that $V$ is a group under matrix multiplication, and find all subgroups of $V$.
I tried and got proof that $V$ is a group under matrix multiplication . So, let's skip to the my second question.
Let $M\subseteq V$ and $M=\lbrace A_2,A_3,A_4 \rbrace$, since $e_V = I = A_1$ is trivial subgroup of V. We'll show that $M$ is a subgroup of $V$. Consider the multiplication of every two matrices in $M$. Well, since $A_2A_3^{-1}, A_2A_4^{-1}, \text{ and } A_3A_4^{-1} \in M$,   $M$ is a non-trivial subgroup of $V$.
But, how do I find the other non-trivial subgroups of $V$? Thanks for your help and attention.

Non-trivial proper subgroups of $V$ are $\{A_1, A_2\}$, $\{A_1,A_3\}$, and $\{A_1,A_4\}$.
Any subgroup must contain the identity element $A_1$.
