Let p be a prime number. We prove that if G is a compact Lie group with a non-trivial p-subgroup, then the orbit space $\left(B_p\left(G\right)\right)/G$ of the classifying space of the category associated to the G-poset ${}_p\left(G\right)$ of all non-trivial elementary abelian p-subgroups of G is contractible. This gives, for every G-CW-complex X each of whose isotropy groups contains a non-trivial p-subgroup, a decomposition of X/G as a homotopy colimit of the functor $X^{Eₙ}/(NE₀\cap ...\cap NEₙ)$ defined over the poset $\left(s{d}_p\left(G\right)\right)/G$, where sd is the barycentric subdivision. We also...