A group \(G\) satisfies the weak maximal condition on non-Abelian subgroups (max-\(\infty\)-\(\overline{\text{ab}}\)) if for every ascending chain \(H_1\leq H_2\leq\cdots\leq H_n\leq\cdots\) of non-Abelian subgroups there exists a positive integer \(n_0\) such that the indices \(|H_{n+1}:H_n|\) are finite for all \(n\geq n_0\). In this paper the authors prove that a locally finite group satisfies max-\(\infty\)-\(\overline{\text{ab}}\) if and only if it is either Abelian or a Chernikov group. Moreover they characterize non-Abelian groups in max-\(\infty\)-\(\overline{\text{ab}}\), provided that they have an ascending normal series whose factors are either locally nilpotent or locally finite.