Let G be one of the lamplighter groups \({({\Bbb Z}/p{\Bbb Z})^n} \wr {\Bbb Z}\) and Sub(G) the space of all subgroups of G. We determine the perfect kernel and Cantor-Bendixson rank of Sub(G). The space of all conjugation-invariant Borel probability measures on Sub(G) is a simplex. We show that this simplex has a canonical Poulsen subsimplex whose complement has only a countable number of extreme points. If F is a finite group and Γ an infinite group which does not have property (T), then the conjugation-invariant probability measures on Sub(\(F \wr \Gamma \)) supported on \(_{ \oplus \Gamma }F\) also form a Poulsen simplex.