Let (Zn)n≥0 be a branching process in a random environment defined by a Markov chain (Xn)n≥0 with values in a finite state space X. Let Pi be the probability law generated by the trajectories of Xnn≥0 starting at X0=i∈X. We study the asymptotic behaviour of the joint survival probability PiZn>0,Xn=j, j∈X as n→+∞ in the critical and strongly, intermediate and weakly subcritical cases.