In the present paper, we study the relation between criticality and information transmission in the one-dimensional contact process with infection parameter $λ.$ To do this we define the {\it sensitivity} of the process to its initial condition. This sensitivity increases for values of $λ< λ_c, $ the value of the critical parameter. The main point of the present paper is that we show that actually it continues increasing even after $ λ_c $ and only starts decreasing for sufficiently large values of $λ.$ This provides a counterexample to the common belief that associates maximal information transmission to criticality.