Question

example of non cyclic group that have a cyclic subgroup

Answer 1

Theorem. If G is a cyclic group, then all the subgroups of G are cyclic. ... The groups D3 and Q8 are both non-abelian and hence non-cyclic, but each have 5 subgroups, all of which are cyclic. The group V4 happens to be abelian, but is non-cyclic.