The Alperin–Brauer–Gorenstein theorem classifies the simple groups, and to a degree the finite groups, with quasidihedral Sylow 2-subgroups. Examples The Sylow 2-subgroups of the following groups are quasidihedral: PSL3(Fq) for q ≡ 3 mod 4, PSU3(Fq) for q ≡ 1 mod 4, the Mathieu group M11, GL2(Fq) for q ≡ 3 mod 4. References Finite groups