In his paper, ’On torsion in the cohomology of locally symmetric varieties’, Peter Scholze has introduced a new, purely topological method to construct classes on arithmetic quotients spaces reductive groups over $$\mathbb {Q}$$ originating from similar Levi subgroups maximal parabolic subgroups. We extend this construction beyond cases he considers, and, complex case, local systems.

For a finite group G, subgroup P of G is 2-minimal if B<P, where B=NG(S) for some Sylow 2-subgroup S and B contained in unique maximal P. fields odd characteristic, this paper contains detailed explicit description all the subgroups general orthogonal groups, certain their subgroups.