Let $p$ be an odd prime number and let $\mathbf{F}_{p}$ be the finite field of order $p$. The group $H$ is a cyclic subgroup of $GL_{3}(\mathbf{F}_{p})$ such that there does not exist an abelian subgroup H' of $GL_{3}(\mathbf{F}_{p})$ with H $\subsetneq$ H'.
My question is that how to compute the number of such subgroups $H$ of $GL_{3}(\mathbf{F}_{p})$.
I think that we should first find some cyclic subgroups which satisfy this condition but I don't know how.
Thank you in advance.