  Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the
$n$-dimensional $\mathbb{F}_q$-vector space $V=\mathbb{F}_q^n\,$. In this paper
we define a closure operator on the subgroup lattice of the group $G =
\mathrm{PGL}(V)$. Let $\mu$ denote the M\"obius function of this lattice. The
aim is to use this closure operator to characterize subgroups $H$ of $G$ for
which $\mu(H,G)\neq 0$. Moreover, we establish a polynomial bound on the number
$c(m)$ of closed subgroups $H$ of index $m$ in $G$ for which the lattice of
$H$-invariant subspaces of $V$ is isomorphic to a product of chains. This bound
depends only on $m$ and not on the choice of $n$ and $q$. It is achieved by
considering a similar closure operator for the subgroup lattice of
$\mathrm{GL}(V)$ and the same results proven for this group.
