The semigroup of partial symmetries of a polygon P is the collection of all distance-preserving bijections between subpolygons of P, with composition as the operation. Around every idempotent of the semigroup there is a maximal subgroup that is the group of symmetries of a subpolygon of P. In this paper we construct all of the maximal subgroups that can occur for any regular polygon P, and determine for which P there exist nontrivial cyclic maximal subgroups, and for which there are only dihedral maximal subgroups.