O2n+2(2) is the group of a non-singular quadric in PG(2n+1,2). The related finite geometry is used to give a simple and systematic determination of the classes and characters of the maximal subgroup fixing a point on the quadric, of the intersection of this stabiliser with the simple subgroup ofO2n+2+(2) of index2, and of other subgroups. Explicit results are tabulated for groups of orders64, 128, 576, 960, 1152, 1920, 46080, 92160, 1290240, 2580480.
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