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In this paper we give a bound to the number of conjugacy classes of maximalsubgroups of any almost simple group whose socle is a classical group of Lietype. The bound is $2n^{5.2}+n\log_2\log_2 q$, where $n$ is the dimension ofthe classical socle and $q$ is the size of the defining field. To obtain thebound, we first bound the number of projective cross-characteristicrepresentations of simple groups of Lie type as a function of therepresentation degree. These bounds are computed for different families ofgroups separately. In the computation, we use information on class numbers,minimal character degrees and gaps between character degrees.

Growth of cross-characteristic representations of finite quasisimple groups of Lie type