Subgroup structure of dihedral group:D16

This article discusses the subgroup structure of dihedral group:D16, the dihedral group of order sixteen (and hence, degree eight). We use here the presentation:

$$G := \langle a,x \mid a^8 = x^2 = e, xax = a^{-1} \rangle$$.

$$G$$ has 16 elements:

$$\{ e, a, a^2, a^3, a^4, a^5, a^6, a^7, x, ax, a^2x, a^3x, a^4x, a^5x, a^6x, a^7x \}$$



Table classifying subgroups up to automorphisms
In case a single equivalence class of subgroups under automorphisms comprises multiple conjugacy classes of subgroups, outer curly braces are used to bucket the conjugacy classes.

Table classifying isomorphism types of subgroups
The first part of the GAP ID is the order of the subgroup.

Table listing number of subgroups by order
Note that due to the congruence condition on number of subgroups of given prime power order, the number of subgroups of any fixed order is congruent to 1 mod 2, i.e., is odd.