  We initiate a study of maximal subgroups and maximal von Neumann subalgebras
which have the Haagerup property. We determine maximal Haagerup subgroups
inside $\mathbb{Z}^2 \rtimes SL_2(\mathbb{Z})$ and obtain several explicit
instances where maximal Haagerup subgroups yield maximal Haagerup subalgebras.
Our techniques are on one hand based on group-theoretic considerations, and on
the other on certain results on intermediate von Neumann algebras, in
particular these allowing us to deduce that all the intermediate algebras for
certain inclusions arise from groups or from group actions. Some remarks and
examples concerning maximal non-(T) subgroups and subalgebras are also
presented, and we answer two questions of Ge regarding maximal von Neumann
subalgebras.
