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## Subgroup structure of dihedral group:D8

, 16:19, 22 June 2011
Tables for quick information
{| class="sortable" border="1"
! Automorphism class of subgroups !! List of subgroups !! Isomorphism class !! Order of subgroups !! Index of subgroups !! Number of conjugacy classes !! Size of each conjugacy class !! Total number of subgroups !! Isomorphism class of quotient (if exists) !! [[Subnormal depth]] (if proper and normal, this equals 1) !! [[Nilpotency class]]
|-
| trivial subgroup || [itex]\{ e \}[/itex] || [[trivial group]] || 1 || 8 || 1 || 1 || 1 || [[dihedral group:D8]] || 1 || 0
|-
| [[center of dihedral group:D8|center]] || [itex]\{ e,a^2, e \}[/itex] || [[cyclic group:Z2]] || 2 || 4 || 1 || 1 || 1 || [[Klein four-group]] || 1 || 1
|-
| [[2-subnormal subgroups of dihedral group:D8|other subgroups of order two]] || [itex]\{e,x ,e \}, \{ axe, e a^2x \}, [/itex] <br> [itex]\{ a^2xe, e ax \}, \{ e,a^3x, e \}[/itex] || [[cyclic group:Z2]] || 2 || 4 || 2 || 2 || 4 || -- || 2 || 1
|-
| [[Klein four-subgroups of dihedral group:D8|Klein four-subgroups]] || [itex]\{ e,x,a^2,a^2x \}[/itex], [itex]\{ e,ax,a^2,a^3x \}[/itex] || [[Klein four-group]] || 4 || 2 || 2 || 1 || 2 || [[cyclic group:Z2]] || 1 || 1
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