Let $p$ be a prime and $n$ be a positive integer. Is it possible to find a finite solvable group $G$ with a maximal subgroup $M$ such that $|G:M|=p^n$?
If $n=1$, we can surely find it taking a group of order $pq$ with a normal Sylow $p$-subgroup. What if $n\geq 2$?
I can't even find examples of solvable groups with a maximal subgroup of cube prime index. Are there?