The existence of infinity of subharmonics for Duffing equations with convex and oscillatory nonlinearities is shown. This result is a corollary of two theorems. These theorems, one for a weak sub-quadratic potential and another for a geometric case, roughly speaking, are complementary. The approach of this paper is based on the phase-plane analysis for the time map and using the Poincaré-Birkhoff twist theorem.