The group \(G\) is a \(CC\)-group if \(G/C_ G(x^ G)\) is Chernikov for all \(x\in G\). If \(G\) is not a \(CC\)-group but all its proper subgroups are \(CC\)-groups \(G\) is said to be a minimal non \(CC\)-group. \textit{J. Otal} and \textit{J. M. Peña} [Commun. Algebra 16, 1231-1242 (1988; Zbl 0644.20025)] have shown that a locally graded minimal non-\(CC\)-group is locally finite, countable and has no nontrivial finite or abelian images. From this starting point the authors adapt the methods of \textit{M. Kuzucuoglu} and \textit{R. E. Phillips} [Math. Proc. Camb. Philos. Soc. 105, 417-420 (1989; Zbl 0686.20034)] to show that a locally graded minimal non-\(CC\)- group must be a \(p\)-group. The existence of such a group remains an open question.