File:D16latticeofsubgroups.png

This is the lattice of subgroups of dihedral group:D16. For more information, see subgroup structure of dihedral group:D16.

This picture was generated using Graphviz. The generating code is as follows:

digraph G{       rankdir = "BT"; Wholegroup[peripheries=3]; Trivialsubgroup[peripheries=3]; Center[peripheries=2]; Involutionone; Involutiontwo; Involutionthree; Involutionfour; Involutionfive; Involutionsix; Involutionseven; Involutioneight; Cyclicfour[peripheries=2]; Cycliceight; ElementaryAbelianone; ElementaryAbeliantwo; ElementaryAbelianthree; ElementaryAbelianfour; Dihedraleightone; Dihedraleighttwo; Trivialsubgroup -> Center; Trivialsubgroup -> Involutionone; Trivialsubgroup -> Involutiontwo; Trivialsubgroup -> Involutionthree; Trivialsubgroup -> Involutionfour; Trivialsubgroup -> Involutionfive; Trivialsubgroup -> Involutionsix; Trivialsubgroup -> Involutionseven; Trivialsubgroup -> Involutioneight; Involutionone -> ElementaryAbelianone; Involutionfive -> ElementaryAbelianone; Involutiontwo -> ElementaryAbeliantwo; Involutionsix -> ElementaryAbeliantwo; Involutionthree -> ElementaryAbelianthree; Involutionseven -> ElementaryAbelianthree; Involutionfour -> ElementaryAbelianfour; Involutioneight -> ElementaryAbelianfour; Center -> ElementaryAbelianone; Center -> Cyclicfour; Center -> ElementaryAbeliantwo; Center -> ElementaryAbelianthree; Center -> ElementaryAbelianfour; ElementaryAbelianone -> Dihedraleightone; ElementaryAbelianthree -> Dihedraleightone; ElementaryAbeliantwo -> Dihedraleighttwo; ElementaryAbelianfour -> Dihedraleighttwo; Cyclicfour -> Cycliceight; Cyclicfour -> Dihedraleightone; Cyclicfour -> Dihedraleighttwo; Cycliceight -> Wholegroup; Dihedraleightone -> Wholegroup; Dihedraleighttwo -> Wholegroup; Wholegroup[label="Whole group {e, a, a^2, a^3, a^4, a^5, a^6, a^7, ax, a^2x, a^3x, a^4x, a^5x, a^6x, a^7x }"]; Center[label="Center {e, a^4}, Order two, characteristic"]; Trivialsubgroup[label="Trivial subgroup {e}"]; Involutionone[label = "{e, x}, Order two, 3-subnormal"]; Involutiontwo[label = "{e, ax}, Order two, 3-subnormal"]; Involutionthree[label = "{e, a^2x}, Order two, 3-subnormal"]; Involutionfour[label = "{e, a^3x}, Order two, 3-subnormal"]; Involutionfive[label = "{e, a^4x}, Order two, 3-subnormal"]; Involutionsix[label = "{e, a^5x}, Order two, 3-subnormal"]; Involutionseven[label = "{e, a^6x}, Order two, 3-subnormal"]; Involutioneight[label = "{e, a^7x}, Order two, 3-subnormal"]; ElementaryAbelianone[label = "{e, x, a^4, a^4x}, Order four, 2-subnormal"]; ElementaryAbeliantwo[label = "{e, ax, a^5, a^5x}, Order four, 2-subnormal"]; ElementaryAbelianthree[label = "{e, a^2x, a^6, a^6x}, Order four, 2-subnormal"]; ElementaryAbelianfour[label = "{e, a^3x, a^7, a^7x}, Order four, 2-subnormal"]; Cyclicfour[label = "{e, a^2, a^4, a^6}, Order four, characteristic"]; Cycliceight[label = "{e, a, a^2, a^3, a^4, a^5, a^6, a^7}, Order eight, characteristic"]; Dihedraleightone[label = "{e, a^2, a^4, a^6, x, a^2x, a^4x, a^6x}, Order eight, normal"]; Dihedraleighttwo[label = "{e, a^2, a^4, a^6, ax, a^3x, a^5x, a^7x}, Order eight, normal"]; }