The class semigroup of a commutative integral domainRis the semi­groupS(R) of the isomorphism classes of the nonzero ideals ofRwith operation induced by multiplication. We consider Prufer domains of finite character, i.e. Prüfer domains in which every nonzero ideal is contained but in a finite number of maximal ideals. In [1] it is proved that, ifRis such a Prüfer domain, then the semigroupS(Ris a Clifford semigroup, namely it is the disjoint union of the subgroups associated to each idempotent element. In [2] we gave a description of a generating set for the A-semilattice of the idempotent elements ofS(R). In this paper we consider the constituent groups of the class semigroup. We prove that the groups associated to idempotent elements ofS(R) are extensions of class groups of overrings of (R) by means of direct products of archimedean groups of localizations of(R) at idempotent prime ideals.