  We show that idempotent states on finite quantum groups correspond to
pre-subgroups in the sense of Baaj, Blanchard, and Skandalis. It follows that
the lattices formed by the idempotent states on a finite quantum group and by
its coidalgebras are isomorphic. We show furthermore that these lattices are
also isomorphic for compact quantum groups, if one restricts to expected
coidalgebras.
