Maximal intersections in finite groups

For a finite group $G$, we investigate the behaviour of four invariants, $\text{MaxDim}(G),$ $\text{MinDim}(G),$ $\text{MaxInt}(G)$ and $\text{MinInt}(G),$ measuring in some way the width and the height of the lattice $\mathcal{M}(G)$ consisting of the intersections of the maximal subgroups of $G\mathrm{.}$