@tao@mathstodon.xyz I followed the extension structure of 24T10478 a bit further. The minimal obstruction to splitting occurs over Q₈ < GL(2,3). Surprisingly, its exceptional defect geometry has a clean projective description: the 64 local complements partition canonically into four 16-element blocks, equivariantly indexed by the four Sylow-3 subgroups of GL(2,3), i.e. P¹(F₃). More explicitly, an ordered generating pair (a,b) of Q₈ selects the unique order-3 generator p satisfying aᵖ = b, bᵖ = ab. The 24 pairs map equivariantly onto the four projective points, with fibres of size 6. GAP certificate available.