We consider   the   Neumann  problem  for    time-periodic  reaction-diffusion
equations. Our main concern is the  question whether a stable periodic solution
can be subharmonic,  that is, whether  its minimal period can be larger than the
period  of the equation. We present two theorems answering this question; one
dealing with spatially inhomogeneous, the other with spatially homogeneous
equations.   The results presented in this note have been obtained jointly with
E. Yanagida .