Cyclic irreducible subgroups of $GL(2,p)$

Question

Let $p$ be an odd prime. Is it possible to have an cyclic irreducible subgroup of $GL(2,p)$ of order $q^n$, say, with $n>0$ and some prime $q$, such that some proper subgroup is still irreducible?

Answer 1

Yes this can happen. For example ${\rm GL}(2,17)$ has an irreducible cyclic subgroup of order $9$, and its subgroup of order $3$ is also irreducible.

In fact I think it always happens. Given $q^n$ with $q$ and odd prime and $q>1$, choose any prime $p$ with $q^n$ dividing $p+1$. Then ${\rm GL}(2,p)$ has an irreducible cyclic subgroup of order $q^n$ and all of its nontrivial subgroups are also irreducible, because $q$ cannot divide $p-1$.