  We discuss whether finiteness properties of a profinite group $G$ can be
deduced from the probabilistic zeta function $P_G(s)$. In particular we prove
that if $P_G(s)$ is rational and all but finitely many nonabelian composition
factors of $G$ are groups of Lie type in a fixed characteristic, then $G$
contains only finitely many maximal subgroups.
