We classify the combinations of parameters which lead three generations of quarks and leptons in the framework of magnetized twisted orbifolds on \(T^2/Z_2\), \(T^2/Z_3\), \(T^2/Z_4\) and \(T^2/Z_6\) with allowing nonzero discretized Wilson line phases and Scherk-Schwarz phases. We also analyze two actual examples with nonzero phases leading to one-pair Higgs and five-pair Higgses and discuss the difference from the results without nonzero phases studied previously.