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Motivated by the search for new gravity duals to M2 branes with N > 4 supersymmetry — equivalently, M-theory backgrounds with Killing superalgebra osposp(N |4) for N > 4 — we classify (except for a small gap) homogeneous M-theory backgrounds with symmetry Lie algebra soso(n) ⊕ soso(3, 2) for n = 5, 6, 7. We find that there are no new backgrounds with n = 6, 7 but we do find a number of new (to us) backgrounds with n = 5. All backgrounds are metrically products of the form AdS4 × P 7, with P riemannian and homogeneous under the action of SO(5), or S 4 × Q 7 with Q lorentzian and homogeneous under the action of SO(3,2). At least one of the new backgrounds is supersymmetric (albeit with only N = 2) and we show that it can be constructed from a supersymmetric Freund-Rubin background via a Wick rotation. Two of the new backgrounds have only been approximated numerically.
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