# Changes

## Subgroup structure of dihedral group:D8

, 15:12, 2 August 2011
Subgroup-defining functions and associated quotient-defining functions
| [[Baer norm]] || intersection of [[normalizer]]s of all subgroups || [[center of dihedral group:D8]]: [itex]\{ e, a^2 \}[/itex] || [[cyclic group:Z2]] || 2 || ? || [[Klein four-group]] || 4
|-
| [[Fitting subgroup]] || join of all [[nilpotent normal subgroup]]s || whole group || [[dihedral group:D8]] || 8 || [[Fitting quotient]] || [[trivial group]] || 1
|-
| [[join of abelian subgroups of maximum order]] || join of all abelian subgroups of maximum order among abelian subgroups || whole group || [[dihedral group:D8]] || 8 || ? || [[trivial group]] || 1
|-
| [[ZJ-subgroup]] || center of the [[join of abelian subgroups of maximum order]] || [[center of dihedral group:D8]]: [itex]\{ e, a^2 \}[/itex] || [[cyclic group:Z2]] || 2 || ? || [[Klein four-group]] || 4
|}

Some more notes:

* The following subgroup-defining functions are equal to the whole group on account of the group being a [[nilpotent group]]: [[Fitting subgroup]], [[hypercenter]], [[solvable radical]].
* The following subgroup-defining functions are equal to the trivial subgroup on account of the group being a [[solvable group]]: [[hypocenter]], [[nilpotent residual]], [[perfect core]], [[solvable residual]].

<section end="sdf summary"/>