  We consider the question: When do two finite abelian groups have isomorphic
lattices of characteristic subgroups? An explicit description of the
characteristic subgroups of such groups enables us to give a complete answer to
this question in the case where at least one of the groups has odd order. An
"exceptional" isomorphism, which occurs between the lattice of characteristic
subgroups of $Z_p\times Z_{p^2}\times Z_{p^4}$ and $Z_{p^2} \times Z_{p^5}$,
for any prime $p$, is noteworthy.
