Let $G$ be a finite group. I want to find an upper bound on the number of the maximal subgroups. My questions is does it possible to prove that the number of maximal subgroups of any finite group $G$ is at most $|G|^{100}$?


One can easily find that any subgroup is generated by at most $\log|G|$ elements thus the number of subgroups(in particular maximal subgroups) is at most $|G|^{\log|G|}$. 
Does it possible to improve this upper bound. 
For an abelian group the number of maximal subgroups is at most $|G|$ and in fact I do not know any example where the number of maximal subgroups is more than $|G|$.


I am almost sure that I am not the first who is asking this question I would like to know if the answer to this question is known or either this is a hard question.



Maximal number of maximal subgroups
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. 


