  Let $(T(t))_{t\geq0}$ be a $C_0$ semigroups and $A$ be its infinitesimal
generator. In this work, we prove that the spectral inclusion for
$(T(t))_{t\geq0}$ remains true for the Drazin invertible and Quasi-Fredholm
spectra. Also, we will give conditions under which facts $A$ is quasi-Fredholm,
$A$ is Drazin invertible and $A$ is B-Fredholm are equivalent.
