  Fix a finite semigroup $S$ and let $a_1,\ldots,a_k, b$ be tuples in a direct
power $S^n$. The subpower membership problem (SMP) for $S$ asks whether $b$ can
be generated by $a_1,\ldots,a_k$. For bands (idempotent semigroups), we provide
a dichotomy result: if a band $S$ belongs to a certain quasivariety, then
$SMP(S)$ is in P; otherwise it is NP-complete.
  Furthermore we determine the greatest variety of bands all of whose finite
members induce a tractable SMP. Finally we present the first example of two
finite algebras that generate the same variety and have tractable and
NP-complete SMPs, respectively.
