In this short paper, counterexamples are constructed to prove that the maximum clique problem (MCP) is not in the class P. First of all, several new concepts are defined to describe how to solve the MCP. And then based on these new concepts, it is proved that the MCP is not in the class P in 3 steps. The first step, a key work which any algorithm solving the MCP has to do is found. The second step, for any n > 9, a special sample graph with n vertices is constructed which contains an exponential number of maximum cliques. The third step, it is proved that the time complexity of any algorithm solving the MCP is exponential when the input is the special sample graph. Because it is well-known that the MCP is in the class NP, and now it is proved that the MCP is not in the class P, P != NP is proved.