Homotopy types of group lattices

In this article, we study group lattices using the ideas by K. S. Brown and D. Quillen of associating a certain topological space to a partially ordered set. We determine the exact homotopy type for the subgroup lattice of $\operatorname{PSL}(2,7)$, find a connection between different group lattices, and obtain some estimates for the Betty numbers of these lattices using the spectral sequence method.