Existence of intermediate fields with order $9$ and $3$ of a Galois field extension of order $27$.

Question

Here is the question again:

Let $F$ be a Galois extension of $K$ such that $[F:K] = 27$. Show that there are intermediate fields of order $9$ and $3$.

I am stumped. By the Galois correspondence, the Galois group of the extension has order 27 and it suffices to find subgroups of order $9$ and $3$. I know by Lagrange's Theorem that intermediate fields of order $9$ and $3$ are allowed.

Answer 1

The Galois group of $F/K$ is a $p$-group.

A $p$-group of order $p^n$ has subgroups of order $p^m$ for all $m \le n$. This follows from the Sylow theorems.

Therefore, the Galois group has subgroups of order $3$ and $9$. The fixed fields of these subgroups have degree $9$ and $3$, respectively.