Finite groups with certain subgroups of Sylow subgroups complemented

Let G be a finite group and H a subgroup of G. We say that H is complemented in G if there exists a subgroup K of G such that G = HK and H boolean AND K = 1. For each prime p dividing the order of G let P be a Sylow p-subgroup of G. We fix in each P a subgroup D such that 1 <= vertical bar D vertical bar < vertical bar P vertical bar and study the structure of G under the assumption that each subgroup H of P with vertical bar H vertical bar = vertical bar D vertical bar and vertical bar H vertical bar = p vertical bar D vertical bar is complemented in G. (C) 2010 Elsevier Inc. All rights reserved.