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Subgroup structure of dihedral group:D8

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# The trivial subgroup (1)
# The [[center]], which is the unique minimal normal subgroup, and is a two-element subgroup generated by <math>a^2</math>. Isomorphic to [[cyclic group:Z2]]. {{further|[[center of dihedral group:D8]]}}(1)
# The two-element subgroups generated by <math>x</math>, <math>ax</math>, <math>a^2x</math> and <math>a^3x</math>. Isomorphic to [[cyclic group:Z2]]. These come in two conjugacy classes: the subgroups generated by <math>x</math> and by <math>a^2x</math> are conjugate, and the subgroups generated by <math>ax</math> and by <math>a^3x</math> are conjugate. (4){{further|[[non-normal subgroups of dihedral group:D8]]}}
# The four-element subgroup generated by <math>a^2</math> and <math>x</math>. This comprises elements <math>e,a^2,x,a^2x</math>. It is isomorphic to the [[Klein four-group]]. A similar four-element subgroup is obtained as that generated by <math>a^2</math> and <math>ax</math>. These are both normal. (2) {{further|[[Klein four-subgroups of dihedral group:D8]]}}
# The four-element subgroup generated by <math>a</math>. Isomorphic to [[cyclic group:Z4]]. (1) {{further|[[Cyclic maximal subgroup of dihedral group:D8]]}}
==The non-characteristic four-element subgroups (type (4))==
 
{{further|[[Klein four-subgroups of dihedral group:D8]]}}
These two subgroups are related by an outer automorphism, but are ''not'' conjugate (in fact, both are normal subgroups). Since they're [[automorph]]s, they in particular satisfy and dissatisfy the same subgroup properties.
==The two-element non-normal subgroups (type (3))==
 
{{further|[[non-normal subgroups of dihedral group:D8]]}}
There are four of these: <math>\{ e, x \}</math>, <math>\{ e, ax \}</math>, <math>\{ e,a^2x \}</math>, and <math>\{ e,a^3x \}</math>. In terms of permutations, these are the subgroups <math>\{ (), (1,3) \}</math>, <math>\{ (), (1,2)(3,4) \}</math>, <math>\{ (), (2,4) \}</math> and <math>\{ (), (1,4)(2,3)</math>.
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