Novel and efficient vibration control units such as inerter and grounded stiffness have contributed significantly to structural vibration reduction. However, as the performance of vibration suppression systems gradually improves, their structures become more complex. And the coupling effects among complex structures, as well as the effects on the system dynamics, are hazy. This aims to investigate the influence of combined structure of inerter and grounded stiffness on the bifurcation behaviors of nonlinear energy sink. The damping of the primary system, a parameter that has been neglected in the majority of studies, is also included in the model. The closed-form solutions of the system steady-state response are derived by the complexification-averaging method and verified numerically. Then, the control equations of stability judgment, saddle-node bifurcation, and Hopf bifurcation are calculated. Sensitivity analysis is performed for each parameter and is visualized in the form of two-dimensional and three-dimensional bifurcation diagrams. It is found that two kinds of bifurcation boundaries on the ( ξ 2 , f ) plane move slightly up and Hopf bifurcation boundaries become complicated both owing to the introduction of the damping in the primary system. The bifurcation zones on the ( ξ 2 , f ) and ( K 2 , f ) planes enlarge with the increase of inerter coefficient. Increasing grounded stiffness has the opposite influence on areas of the bifurcation boundary about ( ξ 2 , f ) and ( μ , f ), except for the special case that the area of saddle-node bifurcation on the ( μ , f ) plane increases accordingly. The amplitude of external force leading to bifurcations will increase when the inerter and grounded stiffness get greater jointly.