We study the entanglement dynamics of the three-qubit Dicke model by means of the adiabatic approximation and the exact diagonalization in the parameter regime where the qubit transition frequencies are far off-resonance with the radiation field and the interaction strengths reach the ultrastrong-coupling regime. The single-mode field is prepared in the coherent state and two typical states GHZ and W are chosen as the initial three-qubit states. In the process of evolution, the interaction between the quantized field and three-qubit system leads to the generation of entanglement between the field and qubits, as well as between different parties in the three-qubit system, i.e. the pairwise entanglement of two qubits and the tripartite entanglement, which are of ongoing interest in quantum information process. The generalized concurrence and negativity are adopted to quantify different kinds of entanglement. The qubit-field entanglement never reaches the maximum and no sudden death occurs in the the tripartite entanglement for GHZ state, but it is exactly the opposite for W state. This reflects that the tripartite entanglement of the GHZ state is more robust than W sate, which is the same as in the rotating wave approximation. The results beyond the rotating wave approximation show that the pairwise entanglement gradually decreases and vanishes in the evolution of both initial states, with the tripartite entanglement periodically reaching relatively high level. This means that the interaction in system supports the tripartite entanglement at the cost of pairwise entanglement. The conclusions provide theoretical reference for the robustness of entanglement state and quantum information processing using Dicke model.