Let $G={\rm PSL}(2,8)$. So $|G|=2^3 \times 3^2 \times 7$. Let $ P \in {\rm Syl}_2(G)$, $Q \in {\rm Syl}_3(G)$ and $R \in {\rm Syl}_7(G)$. I know $P \cong C_2 \times C_2 \times C_2$ and $Q,R$ are cyclic of order $9,7$, respectively. Now i want to find the structure of $N_G(H)$, where $ H \cong C_2 \times C_2$.
${\bf My Try}$: Since $P$ is abelian, $P \le C_G(H) \le N_G(H)<G$. Thus $8 \mid |N_G(H)| \mid 504$.