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The Maximal Subgroups of the Low-Dimensional Finite Classical Groups by Colva M. Roney-Dougal, Derek F. Holt, John N. Bray

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Proposition 4.5.18  (i)  If p ≡ 1 or 19 (mod 24), then there are exactly six conjugacy classes of image-subgroups of SL6(p) isomorphic to image. The subgroups are scalar-normalising in GL6(p), and the classes are fused under the diagonal automorphisms of SL6(p).

(ii)  If p ≡ 7 or 13 (mod 24), then there are exactly three conjugacy classes of image-subgroups of SL6(p) isomorphic to 6 ̇ L3(4). The normalisers of these subgroups G in GL6(p) are generated by G, scalars, ...

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