  The classification of the maximal subgroups of the Monster $\mathbf{M}$ is
believed to be complete subject to an unpublished result of Holmes and Wilson
asserting that $\mathbf{M}$ has no maximal subgroups that are almost simple
with socle isomorphic to $\text{PSL}_2(8)$, $\text{PSL}_2(16)$, or
$\text{PSU}_3(4)$. We prove this result for $\text{PSL}_2(16)$, with the
intention that the other two cases will be dealt with in an expanded version of
this paper. Our proof is supported by reproducible computations carried out
using Seysen's publicly available Python package mmgroup for computing with
$\mathbf{M}$.
