  The paper is concerned with maximal subgroups of the ample (better known as
topological full) groups of homeomorphisms of totally disconnected compact
metrizable topological spaces. We describe all maximal subgroups that are
stabilizers of finite sets. Under certain assumptions on the ample group
(including minimality), we describe all maximal subgroups that are stabilizers
of closed sets or stabilizers of partitions into clopen sets. In particular,
our results apply to the ample groups associated with Cantor minimal systems.
