Every countable group G can be embedded in a finitely generated group $G^*$ that is hopfian and complete , that is, $G^*$ has trivial centre and every epimorphism $G^*\to G^*$ is an inner automorphism. Every finite subgroup of $G^*$ is conjugate to a finite subgroup of G . If G has a finite presentation (respectively, a finite classifying space), then so does $G^*$ . Our construction of $G^*$ relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken.