Let $H\leq_cG=SL_2(Z_p)$ and $K_n$ be the kernel of the epi $\pi_n:G\rightarrow SL_2(Z/p^nZ)$ defined by reducing the entries modulo $p^n$. Show that: $|K_1/K_n|=p^3$ and $K_1$ is pro-$p$.
If $p\geq 5$ then $\pi_1(H)=\pi_1(G)$ implies $H=G$, thus $K_1\subseteq \Phi(G)$, where $\Phi(G)$ is the Frattini subgroup of $G$.