Let $G$ be a finite group. I will say that a set of a subgroups $H_1,\ldots ,H_k$ defines a basis for a group $G$ if any subgroup $H$ of $G$ there exists $S\subset [k]$ such that $H=\cap_{i\in S}H_i$.

My question is does it possible to give any upper bounds on the size of the minimal basis for $G$. For example does it possible to prove that for any $G$ there exists a basis of size at most $|G|^{10}$?

For example for an Abalian group the number of maximal subgroups is at most $|G|$ and they form a basis for $G$.