ABSTRACT The kinematic morphology–density relation of galaxies is normally attributed to a changing distribution of galaxy stellar masses with the local environment. However, earlier studies were largely focused on slow rotators; the dynamical properties of the overall population in relation to environment have received less attention. We use the SAMI Galaxy Survey to investigate the dynamical properties of ∼1800 early and late-type galaxies with log (M⋆/M⊙) > 9.5 as a function of mean environmental overdensity (Σ5) and their rank within a group or cluster. By classifying galaxies into fast and slow rotators, at fixed stellar mass above log (M⋆/M⊙) > 10.5, we detect a higher fraction (∼3.4σ) of slow rotators for group and cluster centrals and satellites as compared to isolated-central galaxies. We find similar results when using Σ5 as a tracer for environment. Focusing on the fast-rotator population, we also detect a significant correlation between galaxy kinematics and their stellar mass as well as the environment they are in. Specifically, by using inclination-corrected or intrinsic $\lambda _{R_{\rm {e}}}$ values, we find that, at fixed mass, satellite galaxies on average have the lowest $\lambda _{\, R_{\rm {e}},\rm {intr}}$, isolated-central galaxies have the highest $\lambda _{\, R_{\rm {e}},\rm {intr}}$, and group and cluster centrals lie in between. Similarly, galaxies in high-density environments have lower mean $\lambda _{\, R_{\rm {e}},\rm {intr}}$ values as compared to galaxies at low environmental density. However, at fixed Σ5, the mean $\lambda _{\, R_{\rm {e}},\rm {intr}}$ differences for low and high-mass galaxies are of similar magnitude as when varying Σ5 ($\Delta \lambda _{\, R_{\rm {e}},\rm {intr}} \sim 0.05$, with σrandom = 0.025, and σsyst < 0.03). Our results demonstrate that after stellar mass, environment plays a significant role in the creation of slow rotators, while for fast rotators we also detect an independent, albeit smaller, impact of mass and environment on their kinematic properties.