Problem 5

Suppose $E$ is the splitting field of some polynomial over $\mathrm{GF}(p)$, where $p$ is a prime. If $|\mathrm{Gal}(E/\mathrm{GF}(p))|=q^{6}$ where $q$ is also a prime, how many fields are strictly between $E$ and $\mathrm{GF}(p)$? That is, how many fields $F$ are there such that $\mathrm{GF}(p)\subset F\subset E$?