If \(S\) is a subgroup of \(G\) such that \(SZ(G)/Z(G)\) is cyclic, then \(S\) is abelian. The author classifies those finite \(p\)-groups in which all subgroups are of this sort; he shows that in this case \(G/Z(G)\) is either elementary abelian or dihedral or non-abelian of order \(p^3\) and of exponent \(p\). For the two latter cases a detailed description of the isoclinism families is given. Furthermore groups \(G\) with one of the following properties are described: (i) \(G\) has a cyclic Thompson subgroup \(J(G)\), (ii) \(J(G)/Z(G)\) is cyclic, (iii) \(A/Z(G)\) is cyclic for some selfcentralizing normal subgroup \(A\) of \(G\).