  We prove the following results: (1) Every group is a maximal subgroup of some
free idempotent generated semigroup. (2) Every finitely presented group is a
maximal subgroup of some free idempotent generated semigroup arising from a
finite semigroup. (3) Every group is a maximal subgroup of some free regular
idempotent generated semigroup. (4) Every finite group is a maximal subgroup of
some free regular idempotent generated semigroup arising from a finite regular
semigroup.
