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Subgroup structure of dihedral group:D8

, 16:15, 29 June 2011
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==Defining functions==

===Subgroup-defining functions and associated quotient-defining functions===

{| class="sortable" border="1"
! [[Subgroup-defining function]] !! What it means !! Value as subgroup !! Value as group !! Order !! Associated quotient-defining function !! Value as group !! Order (= index of subgroup)
|-
| [[center]] || elements that commute with every group element || [[center of dihedral group:D8]]: [itex]\{ e, a^2 \}[/itex] || [[cyclic group:Z2]] || 2 || [[inner automorphism group]] || [[Klein four-group]] || 4
|-
| [[derived subgroup]] || subgroup generated by all [[commutator]]s || [[center of dihedral group:D8]]: [itex]\{ e, a^2 \}[/itex] || [[cyclic group:Z2]] || 2 || [[abelianization]] || [[Klein four-group]] || 4
|-
| [[Frattini subgroup]] || intersection of all [[maximal subgroup]]s || [[center of dihedral group:D8]]: [itex]\{ e, a^2 \}[/itex] || [[cyclic group:Z2]] || 2 || [[Frattini quotient]] || [[Klein four-group]] || 4
|-
| [[Jacobson radical]] || intersection of all [[maximal normal subgroup]]s || [[center of dihedral group:D8]]: [itex]\{ e, a^2 \}[/itex] || [[cyclic group:Z2]] || 2 || ? || [[Klein four-group]] || 4
|-
| [[socle]] || join of all [[minimal normal subgroup]]s || [[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
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| [[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
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| [[join of abelian subgroups of maximum rank]] || join of all abelian subgroups of maximum rank among abelian subgroups || whole group || [[dihedral group:D8]] || 8 || ? || [[trivial group]] || 1
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| [[join of elementary abelian subgroups of maximum order]] || join of all elementary abelian subgroups of maximum order among abelian subgroups || whole group || [[dihedral group:D8]] || 8 || ? || [[trivial group]] || 1
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| [[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
|}

===Subgroup series-defining functions===

{| class="sortable" border="1"
! Series-defining function !! Type !! Zeroth member !! First member !! Second member !! Third member !! Stable member
|-
| [[upper central series]] || ascending || trivial || [[center]]: [itex]\{ e, a^2 \}[/itex] -- [[center of dihedral group:D8]] || [[second center]]: whole group || whole group || whole group
|-
| [[lower central series]] || descending || -- || whole group || [[derived subgroup]]: [itex]\{ e, a^2 \}[/itex] -- [[center of dihedral group:D8]] || trivial || trivial || trivial
|-
| [[derived series]] || descending || whole group || [[derived subgroup]]: [itex]\{ e, a^2 \}[/itex] -- [[center of dihedral group:D8]] || trivial || trivial || trivial || trivial
|-
| [[Frattini series]] || descending || whole group || [[Frattini subgroup]]: [itex]\{ e, a^2 \}[/itex] -- [[center of dihedral group:D8]] || trivial || trivial || trivial || trivial
|-
| [[Fitting series]] || ascending || trivial || [[Fitting subgroup]]: whole group || whole group || whole group || whole group
|-
| [[socle series]] || ascending || trivial || [[socle]]: [itex]\{ e, a^2 \}[/itex] -- [[center of dihedral group:D8]] || whole group || whole group || whole group
|}
==Lattice of subgroups==