# gr.group theory – any subgroup is isomorphic to a normal subgroup

Let $$G$$ be a group and $$H$$ is a subgroup of $$G$$, then there exists a normal subgroup $$K$$ of $$G$$, such that $$H$$ is isomorphic to $$K$$. Under such conditions, can we determine the structure of $$G$$ ?

This question comes from the discussion of Dedekind group in group theory, where a Dedekind group means that all of its subgroups are normal, in this case, we know that a Dedekind group is an Abelian
group or a direct product of Quaternion group $$Q_8$$ and an Abelian group $$A$$, where $$A$$ has no elements with order $$4$$.

So my problem can be regarded as a promotion of Dedekind group, it only requires isomorphic, I don’t know how to deal with this case.