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2 Changed a false statement.

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. More generally, the group

$\langle \rlap{////////////////////////////////////////////////}\langle a, x_1,\ldots,x_n|\text{ } ax_i=x_ia, \text{ }a^2=1, \text{ }x_i^2=a, \text{ }x_i^{-1}x_jx_i=x_j^{-1}\rangle$

will have this property.(See answer below). 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?

1

Groups with all subgroups normal

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. More generally, the group

$\langle a, x_1,\ldots,x_n|\text{ } ax_i=x_ia, \text{ }a^2=1, \text{ }x_i^2=a, \text{ }x_i^{-1}x_jx_i=x_j^{-1}\rangle$

will have this property. 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?