A group G is called an " -QC-group" if for any element x of order 2 or 4 of G, x y = y x for all y in G. In this article, we investigate the structure of the groups G of the title. Suppose that G is non-2-closed and non-2-nilpotent, then it turns out that such a group is either S 4 or a uniquely determined group of order 48 or A 5 or SL 2 5 . The corresponding problem for G 2-closed or 2-nilpotent is open but very difficult.
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