Based on the finite element method, the propagation of flexural vibration in a binary phononic crystal thick plate with a point defect is studied. The plate is composed of a square array of concrete cylinders embedded in the rubber matrix. Complete band structure and frequency response function of this perfect thick plate indicates the existence of low-frequency absolute band gap. Detailed investigations have been carried out to study the dependence of the width of absolute band gap on both structural and material parameters. For the point defect, the defect modes are localized around the defect, and the frequency and the number of the defect bands are significantly dependent on the filling fraction, the size and the mass density of the defect cylinder. To better support the statement of the defect band structures, we also represent the frequency response function of the propagation of flexural vibration in the thick plate with a point defect. Based on the detailed investigations, both the absolute band gap and the defect bands of a binary thick plate could be modulated with appropriate parameters. It may be useful to vibration control in engineering structure.