Can there be a homomophism f, from Z16 onto Z2+Z2?

To figure this out I tried to use the properties of subgroups of homomorphisms. If H is a subgroup Z16, then f(H) is cyclic. But all of the subgroups of Z2+Z2 are also cyclic so I am having a hard time getting a contradiction. Can anyone explain to me were I am going wrong on this?

Thanks