  Finding the number of maximal subgroups of infinite index of a finitely
generated group is a natural problem that has been solved for several classes
of `geometric' groups (linear groups, hyperbolic groups, mapping class groups,
etc). Here we provide a solution for a family of groups with a different
geometric origin: groups of intermediate growth that act on rooted binary
trees. In particular, we show that the non-torsion iterated monodromy groups of
the tent map (a special case of some groups first introduced by {\v{S}}uni{\'c}
in \cite{Sunic} as `siblings of the Grigorchuk group') have exactly countably
many maximal subgroups of infinite index, and describe them up to conjugacy.
This is in contrast to the torsion case (e.g. Grigorchuk group) where there are
no maximal subgroups of infinite index. It is also in contrast to the
above-mentioned geometric groups, where there are either none or uncountably
many such subgroups.
  Along the way we show that all the groups defined by {\v{S}}uni{\'c} have the
congruence subgroup property and are just infinite.
