Prove that a a normal subgroup $\displaystyle H $ of the group $\displaystyle G $ is maximal if and only if the Quotient Group $\displaystyle \frac{G}{H} $ is simple.
Remember the correspondence theorem for groups: every subgroup of the quotient group $\displaystyle G/H$ is of the form $\displaystyle K/H$ , where $\displaystyle K$ is a subgroup of $\displaystyle G$ , with the following characteristics:
1) $\displaystyle H < K$
2) $\displaystyle [G/H:K/H]=[G:K]$
3)$\displaystyle K/H$ is normal in $\displaystyle G/H$ iff $\displaystyle K$ is normal in $\displaystyle G$
Now solve your problem.
Tonio