A subgroup A of a finite group G is called a local covering subgroup of G if AG = AB for all maximal G-invariant subgroup B of AG = 〈Ag , g ∈ G〉. Let p be a prime and d be a positive integer. Assume that all subgroups of pd, and all cyclic subgroups of order 4 when pd=2 and a Sylow 2-subgroup of G is nonabelian, of G are local covering subgroups. Then G is p-supersolvable whenever pd=p or pd≤ √|G|p or pd≤|Op'p(G)|p/p.