We prove that the number of conjugacy classes of maximal subgroups
of bounded order in a finite group of Lie type of bounded rank is
bounded. For exceptional groups this solves a long-standing open
problem. The proof uses, among other tools, some methods from
geometric invariant theory. Using this result, we provide a sharp
bound for the total number of conjugacy classes of maximal
subgroups of Lie-type groups of fixed rank, drawing conclusions
regarding the behaviour of the corresponding ``zeta function''
$\zeta_G(s) = \sum_{M \max G} |G:M|^{-s}$ , which appears in many
probabilistic applications. More specifically, we are able to show
that for simple groups $G$ and for any fixed real number $s>1$ ,
$\zeta_G(s) \rightarrow 0$ as $|G| \rightarrow \infty$ . This
confirms a conjecture made in [27, page 84].
We also apply these results to prove the conjecture made in
[28, Conjecture 1, page 343, that the symmetric group
$S_n$ has $n^{o(1)}$ conjugacy classes of primitive maximal
subgroups.