Smooth Freund-Rubin backgrounds of eleven-dimensional supergravity of the
form AdS_4 x X^7 and preserving at least half of the supersymmetry have been
recently classified. Requiring that amount of supersymmetry forces X to be a
spherical space form, whence isometric to the quotient of the round 7-sphere by
a freely-acting finite subgroup of SO(8). The classification is given in terms
of ADE subgroups of the quaternions embedded in SO(8) as the graph of an
automorphism. In this paper we extend this classification by dropping the
requirement that the background be smooth, so that X is now allowed to be an
orbifold of the round 7-sphere. We find that if the background preserves more
than half of the supersymmetry, then it is automatically smooth in accordance
with the homogeneity conjecture, but that there are many half-BPS orbifolds,
most of them new. The classification is now given in terms of pairs of ADE
subgroups of quaternions fibred over the same finite group. We classify such
subgroups and then describe the resulting orbifolds in terms of iterated
quotients. In most cases the resulting orbifold can be described as a sequence
of cyclic quotients.Comment: 51 pages; v3: substantial revision (20% longer): we had missed some
cases, but the paper now includes a check of our results via comparison with
extant classification of finite subgroups of SO(4
