Finite groups with large Chermak-Delgado lattices

Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by $\mathcal{C}\mathcal{D}(G)$ the Chermak-Delgado lattice of $G$. In this note, we determine the finite groups $G$ such that $\mathrm{\mid }\mathcal{C}\mathcal{D}(G)\mathrm{\mid }=\mathrm{\mid }L(G)\mathrm{\mid }-k$, $k=1,2$.