  A residually finite (profinite) group $G$ is just infinite if every
non-trivial (closed) normal subgroup of $G$ is of finite index. This paper
considers the problem of determining whether a (closed) subgroup $H$ of a just
infinite group is itself just infinite. If $G$ is not virtually abelian, we
give a description of the just infinite property for normal subgroups in terms
of maximal subgroups. In particular, we show that such a group $G$ is
hereditarily just infinite if and only if all maximal subgroups of finite index
are just infinite. We also obtain results for certain families of virtually
abelian groups, including all virtually abelian pro-$p$ groups and their
discrete analogues.
