Summary: Let \(D\) be a division ring with the center \(F\) and suppose that \(G\) is a subnormal subgroup of the multiplicative group \(D^*\) of \(D\). In this paper, we show that if \(D\) is locally centrally finite and \(G\) is radical over \(F\), then \(G\) is central. Also, for an arbitrary \(D\), if \(G\) is either locally finite or an FC-group, then \(G\) is central. Further, we consider the question of what subgroups of the group \(D^*\) can occur as multiplicative groups of some maximal subfields of \(D\).