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→Subgroup-defining functions and associated quotient-defining functions

group = dihedral group:D8|

connective = of}}

{{fblike}}

The '''dihedral group''' <math>D_8</math>, sometimes called <math>D_4</math>, also called the {{dihedral group}} of order eight or the dihedral group acting on four elements, is defined by the following presentation:

<math>a = (1,2,3,4), \qquad x = (1,3)</math>.

|-

| ~~trivial subgroup ~~[[center of dihedral group:D8|center]] || <math>\{ e,a^2 \}</math> || [[~~trivial ~~cyclic group:Z2]] || 2 || 4 || 1 || 1 || 1 || [[~~dihedral ~~Klein four-group~~:D8~~]] || 1 || ~~0~~ 1

|-

| [[~~center ~~non-normal subgroups of dihedral group:D8|~~center~~other subgroups of order two]] || <math>\{e,x \}, \{ e,a^2x \}</math> <br> <math>\{ e,ax \}, \{ e,a^3x \}</math> || [[cyclic group:Z2]] || ~~1 ~~2 || 4 || 2 || 2 || ~~1 ~~4 || ~~[[Klein four~~-~~group]] ~~- || ~~1 ~~2 || 1

|-

| [[~~2~~Klein four-~~subnormal ~~subgroups of dihedral group:D8|~~other ~~Klein four-subgroups ~~of order two~~]] || <math>\{ e,x,a^2,a^2x \}</math>, <math>\{ e,ax,a^2,a^3x \}</math> || [[~~cyclic ~~Klein four-group~~:Z2~~]] || 4 || 2 || 2 || ~~-- ~~1 || 2 || [[cyclic group:Z2]] || 1 || 1

|-

| [[~~Klein four-subgroups ~~cyclic maximal subgroup of dihedral group:D8|~~Klein four-subgroups~~cyclic maximal subgroup]] || <math>\{ e,a,a^2,a^3 \}</math> || [[~~Klein four-~~cyclic group:Z4]] || 4 || 2 || 1 || 1 || 1 || [[cyclic group:Z2]] || 1 || 1

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| whole group || <math>\{ e,a,a^2,a^3,x,ax,a^2x,a^3x \}</math> || [[~~cyclic maximal subgroup of ~~dihedral group:D8~~|cyclic maximal subgroup~~]] || ~~[[cyclic group:Z4]] ~~8 || 1 || 1 || 1 || 1 || [[~~cyclic ~~trivial group~~:Z2~~]] || ~~1 ~~0 || ~~1~~2

|-

|}

<section end="summary"/>

===Table classifying isomorphism types of subgroups===

The first part of the GAP ID is the order of the group. {| class="~~wikitable~~sortable" border="1"! Group name !! GAP ID !! Index !! Occurrences as subgroup (=1 iff [[isomorph-free subgroup]]) !! Conjugacy classes of occurrence as subgroup (=1 iff [[isomorph-conjugate subgroup]]) !! Automorphism classes of occurrence as subgroup (=1 iff [[isomorph-automorphic subgroup]]) !! Occurrences as normal subgroup (=1 iff [[normal-isomorph-free subgroup]]; =Occurrences as subgroup iff [[isomorph-normal subgroup]]) !! Occurrences as characteristic subgroup(=1 iff [[characteristic-isomorph-free subgroup]]; =Occurences as subgroup iff [[isomorph-characteristic subgroup]])

|-

| [[Trivial group]] || <math>(1,1)</math> || 8 || 1 || 1 || 1 || 1 || 1

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| [[Cyclic group:Z2]] || <math>(2,1)</math> || 4 || 5 || 3 || 2 || 1 || 1

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| [[Cyclic group:Z4]] || <math>(4,1)</math> || 2 || 1 || 1 || 1 || 1 || 1

|-

| [[Klein four-group]] || <math>(4,2)</math> || 2 || 2 || 2 || 1 || 2 || 0

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| [[Dihedral group:D8]] || <math>(8,3)</math> || 1 || 1 || 1 || 1 ||1 || 1

|-

| Total || -- || -- || 10 || 8 || 6 || 6 || 4

|}

===Table listing number of subgroups by order===

Note that these orders satisfy the [[congruence condition on number of subgroups of given prime power order]]: the number of subgroups of order <math>p^r</math> is congruent to <math>1</math> modulo <math>p</math>. Here, <math>p = 2</math>, so this means that the number of subgroups of any given order is odd. {| class="~~wikitable~~sortable" border="1"! Group order !! Index !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Automorphism classes of occurrences as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup

|-

| ~~<math>~~1~~</math>~~|| 8 || 1 || 1 || 1 || 1 || 1

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| ~~<math>~~2~~</math> ~~|| 4 || 5 || 3 || 2 || 1 || 1

|-

| ~~<math>~~4~~</math> ~~|| 2 || 3 || 3 || 2 || 3 || 1

|-

| ~~<math>~~8~~</math> ~~|| 1 || 1 || 1 || 1 || 1 || 1

|-

| Total || -- || 10 || 8 || 6 || 6 || 4

|}

==~~The center (type (2))~~Defining functions==

<section end=~~==Subgroup-defining functions yielding this subgroup===~~"sdf summary"/>

==Lattice of subgroups==

[[Image:D8latticeofsubgroups.png|1000px]]

===The entire lattice===

===The sublattice of normal subgroups===

[[File:D8latticeofnormalsubgroups.png|1000px]]

For the sublattice of normal subgroups, we delete the four 2-subnormal subgroups of order two, leaving only the center. The center is the unique minimal normal subgroup (i.e., a [[monolith]]) and is contained in three maximal normal subgroups of order four. Note that this lattice is isomorphic to the lattice of normal subgroups of the [[quaternion group]], but the quaternion group has no non-normal subgroups.

Some aspects of this generalize to arbitrary <math>p</math>-groups: if the center is of order <math>p</math> it is the unique minimal normal subgroup. In fact, more general results include: [[prime power order implies center is normality-large]], [[minimal normal implies central in nilpotent]], [[omega-1 of center is normality-large in nilpotent p-group]]. The upshot of all these results is that the minimal normal subgroups of a finite <math>p</math>-group are ''precisely'' the subgroups of order <math>p</math> in the center.

The lattice of normal subgroups is isomorphic to that for the [[quaternion group]]. In fact, these are both [[groups having the same Hall-Senior genus]], namely <math>8\Gamma_2a</math>. The picture is given by:

[[File:8Gamma2a.png|1000px]]

===The sublattice of characteristic subgroups===

The sublattice of characteristic subgroups is totally ordered, comprising the identity, the center, the four-element cyclic subgroup, and the whole group. Note that there is one characteristic subgroup of every order dividing the group order. This differs from the quaternion group case, which has no characteristic subgroup of order four.

===The sublattice of fully ~~characteristic ~~invariant subgroups===

The sublattice of fully ~~characteristic ~~invariant subgroups comprises the identity element, the center, and the whole group. It is totally ordered.

===Abelian subgroups and elementary abelian subgroups===

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