To provide a diagnostic device which suppresses a deviation of initial delay time in an output signal of a catalyst downstream sensor with respect to an appropriate value, and finds the initial delay time in a short period of time.

The presence or absence of abnormalities of an oxygen sensor 18 related to an initial timing delay of the output signal VO of the oxygen sensor 18 with respect to an oxygen concentration change in exhaust gas is determined using the initial delay time T. An oxygen storage capacity C1max and an intake air amount Ga1, and an oxygen storage capacity C2max and an intake air amount Ga2 used for finding the initial delay time T are respectively obtained in two engine operation areas which are considerably different from each other, such as an engine operation area where active air-fuel ratio control is executable, and an engine operation area where fuel cut control is executed. The discontinuation of fuel injection under the fuel cut control is executed at a higher frequency than the execution of the active air-fuel ratio control on the establishment of execution requirements thereof.