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Subgroup structure of dihedral group:D8

, 23:22, 4 January 2010
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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 [itex]a^2[/itex]. Isomorphic to [[cyclic group:Z2]]. {{further|[[center of dihedral group:D8]]}}(1)
# The two-element subgroups generated by [itex]x[/itex], [itex]ax[/itex], [itex]a^2x[/itex] and [itex]a^3x[/itex]. Isomorphic to [[cyclic group:Z2]]. These come in two conjugacy classes: the subgroups generated by [itex]x[/itex] and by [itex]a^2x[/itex] are conjugate, and the subgroups generated by [itex]ax[/itex] and by [itex]a^3x[/itex] are conjugate. (4){{further|[[non-normal subgroups of dihedral group:D8]]}}
# The four-element subgroup generated by [itex]a^2[/itex] and [itex]x[/itex]. This comprises elements [itex]e,a^2,x,a^2x[/itex]. It is isomorphic to the [[Klein four-group]]. A similar four-element subgroup is obtained as that generated by [itex]a^2[/itex] and [itex]ax[/itex]. These are both normal. (2) {{further|[[Klein four-subgroups of dihedral group:D8]]}}
# The four-element subgroup generated by [itex]a[/itex]. 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: [itex]\{ e, x \}[/itex], [itex]\{ e, ax \}[/itex], [itex]\{ e,a^2x \}[/itex], and [itex]\{ e,a^3x \}[/itex]. In terms of permutations, these are the subgroups [itex]\{ (), (1,3) \}[/itex], [itex]\{ (), (1,2)(3,4) \}[/itex], [itex]\{ (), (2,4) \}[/itex] and [itex]\{ (), (1,4)(2,3)[/itex].