A relevant theorem of B. H. Neumann states that in a group G each subgroup has finite index in its normal closure if and only if the commutator subgroup G' of G is finite, i.e. if and only if G is finite-by-abelian. In this article we prove that in a periodic group G each subgroup has finite index in a permutable subgroup if and only if G contains a finite normal subgroup N such that G/N is a quasihamiltonian group.