Proposition 3.7.3 Let *n* = 8 and let *H* be a -subgroup of Ω, preserving a decomposition into four subspaces. Then *H* is maximal amongst the geometric subgroups of Ω if and only if *q* > 2. If *q* = 2 then *H* does not extend to a novel maximal subgroup.

Proof The statement for *q* = 2 is immediate from Proposition 2.3.6, so we assume that *q* > 2. Assume, by way of contradiction, that , where *K* is maximal amongst the geometric subgroups of Ω and is not of the same type as *H*. Note that if *q* > 3 then *H*^{∞} ≅ SL_{2}(*q*)^{4}.

It is immediate from Lemma 2.3.7 (iv) that . Suppose ...

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