It is well known that the set of commutators in a group does not form a subgroup and numerous examples with this property are in the literature. A similar phenomenon should occur for the set of autocommutators. But so far, no example appears in the literature. We show that for a finite abelian group the set of autocommutators forms a subgroup and that there exists a group of order 64 in which not every element of the autocommutator subgroup is an autocommutator. With the help of GAP we show that this group is of minimal order with this property.