TITLE: Groups with all subgroups normal
QUESTION [28 upvotes]: Is there any sort of classification of (say finite) groups with the property that every subgroup is normal?
Of course, any abelian group has this property, but the quaternions show commutativity isn't necessary.
If there isn't a classification, can we at least say the group must be of prime power order, or even a power of two?

REPLY [4 votes]: If we replace “(say finite)” by “(say connected Lie)”, then the fact that such groups are abelian may be first observed in:
Ahrens, W., Über Transformationsgruppen, deren sämtliche Untergruppen invariant sind, Hamb. Mitt. 4, 72-78 (1902). ZBL33.0162.02.