The Enhanced Hasse Diagrams for D_4 and the Quaterion group Q_8

The two non-abelian groups of order 8, D4 and Q8, do not need Simplified Enhanced Hasse diagrams. There are so few subgroups, simplification seems unnecessary. So instead, we get Enhanced Hasse Diagrams, which show the differences in their structures.

 

D4 has a total of ten subgroups, six of them are normal and the other four subnormal. There are two different copies of the Klein-4 group, {(1), (12)(34), (13)(24), (14)(23)} and {(1), (13), (24), (13)(24)}. They are isomorphic, but the second version has two odd permutations and two even, while the first version is all even permutations.

 

All subgroups, other than the Klein-4 subgroups and D4 itself,  are defined by a single generator.

The Q8 diagram much less cluttered, with only six subgroups total and all of them normal. In abelian groups, all subgroups are normal, but it is rare when this is true in a non-abelian group. Only Q8 itself is not define by a single generator.