We compute the rate of supernovae (SNe) of different types along the Hubble sequence normalized to the near-infrared luminosity and to the stellar mass of the parent galaxies. This is made possible by the new complete catalog of near-infrared galaxy magnitudes obtained by 2MASS. We find that the rates of all SN types, including Ia, Ib/c and II, show a sharp dependence on both the morphology and the (B-K) colors of the parent galaxies and, therefore, on the star formation activity. In particular we find, with a high statistical significance, that the type Ia rate in late type galaxies is a factor ~20 higher than in E/S0. Similarly, the type Ia rate in the galaxies bluer than B-K=2.6 is about a factor of 30 larger than in galaxies with B-K>4.1. These findings can be explained by assuming that a significant fraction of Ia events in late spirals/irregulars originates in a relatively young stellar component.