We consider the algorithmic task of computing a maximal autarky for a clause-set F, i.e., a partial assignment which satisfies every clause of F it touches, and where this property is destroyed by adding any non-empty set of further assignments. We employ SAT solvers as oracles, using various capabilities. Using the standard SAT oracle, log_2(n(F)) oracle calls suffice, where n(F) is the number of variables, but the drawback is that (translated) cardinality constraints are employed, which makes this approach less efficient in practice. Using an extended SAT oracle, motivated by the capabilities of modern SAT solvers, we show how to compute maximal autarkies with 2 n(F)^{1/2} simpler oracle calls. This novel algorithm combines the previous two main approaches, based on the autarky-resolution duality and on SAT translations.