Fano plane duality group ($G{L}_3(2)⋊⟨\tau ⟩,|\overline{G}|=336$)

Adjoin the inverse-transpose automorphism to the collineation group of the Fano plane. Its maximal subgroups other than the collineation group are unordered incident-pair stabilizers, unordered nonincident-pair stabilizers and Sylow 7-normalizers, of orders 16, 12 and 42.