No; see here.
RonL
1)If a group is abelian then all its subgroups are normal.
But turn the question around,
If all the subgroups of a group are normal then it is abelian?
I was not able to show that.
Since this is my own question I believe that is is wrong.
I think we should consider cases when the group is finite and infinite. Maybe that will help because if it is finite then it has a composition series (but I am not sure what that leads to).