  When equipped with the natural topology first defined by Chabauty, the closed
subgroups of a locally compact group $G$ form a compact space $\Cal C(G)$. We
analyse the structure of $\Cal C(G)$ for some low-dimensional Lie groups,
concentrating mostly on the 3-dimensional Heisenberg group $H$. We prove that
$\Cal C(H)$ is a 6-dimensional space that is path--connected but not locally
connected. The lattices in $H$ form a dense open subset $\Cal L(H) \subset \Cal
C(H)$ that is the disjoint union of an infinite sequence of
pairwise--homeomorphic aspherical manifolds of dimension six, each a torus
bundle over $(\bold S^3 \smallsetminus T) \times \bold R$, where $T$ denotes a
trefoil knot. The complement of $\Cal L(H)$ in $\Cal C(H)$ is also described
explicitly. The subspace of $\Cal C(H)$ consisting of subgroups that contain
the centre $Z(H)$ is homeomorphic to the 4--sphere, and we prove that this is a
weak retract of $\Cal C(H)$.
