This paper studies the resilient containment control of the second-order multi-agent systems under the Byzantine agents and the finite bit-rate communication. The Byzantine agents do not obey the desired control law, and due to the finite bit-rate communication, it is complicated for the cooperative agents to transmit the exact state values. Therefore, a novel control method, called the finite-level quantized mean-subsequence-reduced method, is designed. First, for the second-order cooperative agents, a state transformation is designed. By virtue of the above state transformation, a novel non-uniform quantizer and the associated scaling function are constructed, and a set of encoders and decoders are carefully designed. Then, based on the decoded output signals, the resilient containment controller is constructed. To avoid the influence of the Byzantine agents, every cooperative agent discards the most suspicious decoded output signals from the other agents. By means of the proposed method, the multi-agent systems can achieve the containment consensus, satisfy the limited communication data rates and withstand the Byzantine agents simultaneously. Additionally, in the single cooperative leader case, the designed method can realize the Byzantine-resilient exact leader-follower consensus under the limited bit-rate communication. The effectiveness of the proposed method is illustrated by the numerical simulations.