In earlier papers the author has shown that normality of certain subgroups insures solvability or supersolvability. In this paper he shows that normality can be replaced by quasinormality. Let G be a finite group and define \(A_ 1=\{H\leq G:\) H has prime order or is cyclic of order \(4\}\), \(A_ 2=\{H\leq G:\) H has order 2p, p an odd prime\(\}\) and \(A_ 3=\{H\leq G:\) H has order 2pq, p and q primes not necessarily odd or distinct\(\}\). Theorem 3.1. If each element of \(A_ 1\) is quasinormal in G then G is supersolvable. Theorem 3.2. If G has even order and each element of \(A_ 2\) is quasinormal in G then G is solvable. Theorem 3.3. If G has even order and each element of \(A_ 3\) is quasinormal in G then G is solable. (Note: The proof of Theorem 3.3. does not seem to allow \(A_ 3\) to contain a subgroup of order \(2p^ 2\), p an odd prime.)