  We give a necessary and sufficient condition for a locally compact group to
be isomorphic to a closed cocompact subgroup in the isometry group of a
Diestel-Leader graph. As a consequence of this condition, we see that every
cocompact lattice in the isometry group of a Diestel-Leader graph admits a
transitive, proper action on some other Diestel-Leader graph. We also give some
examples of lattices that are not virtually lamplighters. This implies the
class of discrete groups commensurable to lamplighter groups is not closed
under quasi-isometries and, combined with work of Eskin, Fisher and Whyte,
gives a characterization of their quasi-isometry class.
