This paper presents a new genetic algorithm for job-shop scheduling problems. When we design a genetic algorithm for difficult ordering problems such as job-shop scheduling problems, it is important to design encoding/crossover that is excellent in characteristic preservation. We regard a sub-sequence on each machine as a characteristic to be preserved between parents and their children. The proposed method uses a job sequence for encoding. This paper proposes a new crossover, the sub-sequence exchange crossover (SXX), that can preserve the characteristic very well. Since the children generated by SXX are not always feasible, we propose a technique to transform them into active schedules by using the GT method with a few modifications. Maintaining a diversity of population is important for preventing premature convergence. We present a mutation based on the shift change operator for efficiently introducing a diversity. Furthermore, we design a generation alternation model that is excellent in diversity maintaining. By applying the proposed method to Fisher's and Thompson's 10×10 and 20×5 problems, we show its effectiveness.