Another extended comment. The best bound on the number of subgroups is $|G|^{(1/4+o(1))\log_2 |G|}$ proved in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.100.913&rep=rep1&type=pdf by Borovik, Pyber and Shalev.

For the number of maximal solvable subgroups they get $|G|^c$ for some constant. They don't estimate c but conjecture it is 1.

Added. The survey László Pyber, Asymptotic results for simple groups and some applications. Groups and computation, II (New Brunswick, NJ, 1995), 309–327, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 28, Amer. Math. Soc., Providence, RI, 1997 claims there is a bound of the form $|G|^c$ for any group G on maximal subgroups but c is not described. It is conjectured c=1 will do.

Added. Jesse in the comments below cites a more recent and better bound of $c|G|^{3/2}$. I am making this CW to not get credit for his answer.

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Another extended comment. The best bound on the number of subgroups is $|G|(1/4+o(1)) \log_2 |G|^{(1/4+o(1))\log_2 |G|$ G|}$proved in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.100.913&rep=rep1&type=pdf by Borovik, Pyber and Shalev. For the number of maximal solvable subgroups they get$|G|^c$for some constant. They don't estimate c but conjecture it is 1. Added. The survey László Pyber, Asymptotic results for simple groups and some applications. Groups and computation, II (New Brunswick, NJ, 1995), 309–327, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 28, Amer. Math. Soc., Providence, RI, 1997 claims there is a bound of the form$|G|^c$for any group G on maximal subgroups but c is not described. It is conjectured c=1 will do. 2 added 72 characters in body Another extended comment. The best bound on the number of subgroups is$|G|(1/4+o(1)) \log_2 |G|$proved in http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.100.913&rep=rep1&type=pdf by Borovik, Pyber and Shalev. For the number of maximal solvable subgroups they get$|G|^c\$ for some constant. They don't estimate c but conjecture it is 1.

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