LARGE SUBGROUPS OF A FINITE GROUP OF EVEN ORDER

It is shown that if G is a group of even order with trivial center such that vertical bar G vertical bar > 2 vertical bar C(G)(t)vertical bar(3) for some involution t is an element of G, then there exists a proper subgroup H of G such that vertical bar G vertical bar < vertical bar H vertical bar(2). If vertical bar G vertical bar > vertical bar C(G)(t)vertical bar(3) and k(G) is the class number of G, then vertical bar G vertical bar <= k(G)(3).