We present a comprehensive computational study of some 900 possible lambda-lac mutants of the lysogeny maintenance switch in phage lambda, of which up to date only 19 have been studied experimentally (Atsumi & Little, PNAS 103: 4558-4563, (2006)). We clarify that these mutants realise regulatory schemes quite different from wild-type lambda, and can therefore be expected to behave differently, within the conventional mechanistic setting in which this problem has traditionally been framed. We verify that indeed, with reasonable modelling assumptions and across this wide selection of mutants, the lambda-lac mutants for the most part either have no stable lytic states, or should only be inducible with difficulty. In particular, the computational results contradicts the experimental finding that four lambda-lac mutants both show stable lysogeny and are inducible. This work hence suggests either that the four out of 900 mutants are special, or that lambda lysogeny and inducibility are holistic effects involving other molecular players or other mechanisms, or both. The approach illustrates the power and versatility of computational systems biology to systematically and quickly test a wide variety of examples and alternative hypotheses for future closer experimental study.