Subgroups — D₄² #545

Order 64.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄	C₂	1	4	yes	maximal, complemented
2	#127	C₂	1	1	yes	maximal, characteristic, complemented
3	#121	C₂	1	4	yes	maximal, complemented
4	#120	C₂	1	4	yes	maximal, complemented
5	C₄×D₄	C₂	1	2	yes	maximal, complemented
6	C₂⁴	C₂²	1	4	yes	complemented
7	C₂×D₄	—	2	16	no	—
8	C₂×D₄	C₂²	1	2	yes	—
9	C₂×D₄	C₂²	1	8	yes	complemented
10	C₂×D₄	C₂²	1	4	yes	complemented
11	C₂×D₄	C₂²	1	2	yes	complemented
12	C₂²×C₄	C₂²	1	4	yes	complemented
13	C₄⋊C₄	C₂²	1	2	yes	complemented
14	C₂²⋊C₄	C₂²	1	4	yes	complemented
15	C₂²⋊C₄	C₂²	1	4	yes	complemented
16	C₄²	C₂²	1	1	yes	characteristic, complemented
17	C₂³	—	2	16	no	—
18	C₂³	—	4	4	no	—
19	C₂³	C₂³	1	4	yes	—
20	C₂³	C₂³	1	4	yes	—
21	D₄	—	2	16	no	—
22	D₄	—	2	8	no	—
23	D₄	—	4	2	no	—
24	D₄	D₄	1	8	yes	direct product factor
25	C₂×C₄	—	2	8	no	—
26	C₂×C₄	C₂³	1	1	yes	characteristic
27	C₂×C₄	C₂³	1	4	yes	—
28	C₂×C₄	C₂³	1	2	yes	—
29	C₂²	—	2	4	no	—
30	C₂²	—	2	4	no	—
31	C₂²	—	2	8	no	—
32	C₂²	—	2	4	no	—
33	C₂²	—	4	16	no	—
34	C₂²	C₂×D₄	1	8	yes	complemented
35	C₂²	C₂⁴	1	1	yes	characteristic
36	C₄	—	2	4	no	—
37	C₄	—	2	1	no	—
38	C₄	C₂×D₄	1	4	yes	complemented
39	C₂	—	2	8	no	—
40	C₂	—	4	4	no	—
41	C₂	C₂²×D₄	1	2	yes	—
42	C₂	H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.