Subgroups — #3292

Order 128.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#576	C₂	1	4	yes	maximal, complemented
2	#493	C₂	1	1	yes	maximal, characteristic, complemented
3	#472	C₂	1	4	yes	maximal, complemented
4	#453	C₂	1	4	yes	maximal, complemented
5	#443	C₂	1	2	yes	maximal, complemented
6	H(5,𝔽₂)	C₂²	1	4	yes	complemented
7	AGL(1,ℤ/8)	—	2	8	no	—
8	AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
9	#135	—	2	4	no	—
10	#135	C₂²	1	2	yes	—
11	C₂×D₈	—	2	4	no	—
12	C₂×D₈	C₂²	1	4	yes	complemented
13	#131	C₂²	1	4	yes	complemented
14	#127	C₂²	1	2	yes	complemented
15	#108	C₂²	1	2	yes	complemented
16	#104	C₂²	1	4	yes	complemented
17	#100	C₂²	1	4	yes	complemented
18	C₄×C₈	C₂²	1	1	yes	characteristic, complemented
19	#41	—	2	8	no	—
20	#41	C₂³	1	4	yes	—
21	C₂×D₄	—	2	8	no	—
22	C₂×D₄	—	4	2	no	—
23	C₂×D₄	—	4	4	no	—
24	C₂×D₄	C₂³	1	4	yes	—
25	Q₁₆	D₄	1	2	yes	complemented
26	SD₁₆	—	2	8	no	—
27	SD₁₆	D₄	1	4	yes	complemented
28	D₈	—	2	8	no	—
29	D₈	—	2	4	no	—
30	D₈	—	2	4	no	—
31	D₈	D₄	1	2	yes	complemented
32	M₁₆	—	2	4	no	—
33	M₁₆	C₂³	1	4	yes	—
34	C₂×C₈	—	2	4	no	—
35	C₂×C₈	C₂³	1	2	yes	—
36	C₄²	C₂³	1	1	yes	characteristic
37	C₂³	—	2	4	no	—
38	C₂³	—	4	4	no	—
39	Q₈	C₂×D₄	1	4	yes	—
40	D₄	—	2	4	no	—
41	D₄	—	2	8	no	—
42	D₄	—	4	8	no	—
43	D₄	—	8	4	no	—
44	D₄	C₂×D₄	1	4	yes	—
45	C₂×C₄	—	2	1	no	—
46	C₂×C₄	—	2	4	no	—
47	C₂×C₄	—	4	4	no	—
48	C₂×C₄	C₂⁴	1	1	yes	characteristic
49	C₈	—	2	4	no	—
50	C₈	C₂×D₄	1	2	yes	—
51	C₈	C₂×D₄	1	2	yes	—
52	C₂²	—	2	4	no	—
53	C₂²	—	4	4	no	—
54	C₂²	—	8	4	no	—
55	C₂²	—	16	4	no	—
56	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	2	4	no	—
59	C₄	C₂²×D₄	1	2	yes	—
60	C₂	—	2	1	no	—
61	C₂	—	4	4	no	—
62	C₂	—	8	4	no	—
63	C₂	D₄²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	D₄²
commutator, Frattini, cosocle	C₂×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #576	C₂	semidirect
2. #493	C₂	semidirect
3. #472	C₂	semidirect
4. #453	C₂	semidirect
5. #443	C₂	semidirect
6. H(5,𝔽₂)	C₂²	semidirect
8. AGL(1,ℤ/8)	C₂²	semidirect
12. C₂×D₈	C₂²	semidirect
13. #131	C₂²	semidirect
14. #127	C₂²	semidirect
15. #108	C₂²	semidirect
16. #104	C₂²	semidirect
17. #100	C₂²	semidirect
18. C₄×C₈	C₂²	semidirect
25. Q₁₆	D₄	semidirect
27. SD₁₆	D₄	semidirect
31. D₈	D₄	semidirect

Maximal subgroups

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

subgroup	index
#576	2
#493	2
#472	2
#453	2
#443	2