This paper proves a combinatorial rule giving all maximal and minimal partitions \lambda such that the Schur function s_\lambda appears in a plethysm of two arbitrary Schur functions. Determining the decomposition of these plethysms
has been identified by Stanley as a key open problem in algebraic combinatorics. As corollaries we prove three conjectures of Agaoka on the partitions labelling the lexicographically greatest and least Schur functions appearing in an arbitrary plethysm. We also show that the multiplicity of the Schur function labelled by the lexicographically least constituent may be arbitrarily large. The proof is carried out in the symmetric group and gives an explicit non-zero homomorphism corresponding to each maximal or minimal partition.