New collections of p-subgroups and homology decompositions for   classifying spaces of finite groups
  Let G be a finite group and p a prime dividing its order. We define new collections of p-subgroups of G. We study the homotopy relations among them and with the standard collections of p-subgroups. We determine their ampleness and sharpness properties. 
John Maginnis, Silvia Onofrei