Subgroups — #3416

Order 128.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#578	C₂	1	1	yes	maximal, characteristic, complemented
2	#576	C₂	1	1	yes	maximal, characteristic, complemented
3	#510	C₂	1	3	yes	maximal, complemented
4	#509	C₂	1	3	yes	maximal, complemented
5	#508	C₂	1	3	yes	maximal, complemented
6	C₂×SD₃₂	C₂	1	3	yes	maximal, complemented
7	#504	C₂	1	1	yes	maximal, characteristic, complemented
8	2₋¹⁺⁴	—	2	1	no	—
9	H(5,𝔽₂)	—	2	1	no	—
10	#137	—	2	3	no	—
11	AGL(1,ℤ/8)	—	2	3	no	—
12	#135	C₂²	1	3	yes	—
13	#135	C₂²	1	3	yes	complemented
14	C₂×Q₁₆	C₂²	1	3	yes	complemented
15	C₂×D₈	C₂²	1	3	yes	—
16	#131	C₂²	1	1	yes	characteristic
17	Q₃₂	C₂²	1	3	yes	complemented
18	SD₃₂	C₂²	1	3	yes	complemented
19	SD₃₂	C₂²	1	6	yes	complemented
20	SD₃₂	C₂²	1	1	yes	characteristic, complemented
21	D₁₆	C₂²	1	3	yes	complemented
22	M₃₂	C₂²	1	3	yes	complemented
23	C₂×C₁₆	C₂²	1	3	yes	complemented
24	#41	—	2	3	no	—
25	#41	—	2	3	no	—
26	#41	—	4	1	no	—
27	#41	—	4	3	no	—
28	#41	D₄	1	1	yes	characteristic
29	C₂×Q₈	—	2	3	no	—
30	C₂×Q₈	—	4	1	no	—
31	C₂×D₄	—	2	3	no	—
32	C₂×D₄	—	4	3	no	—
33	Q₁₆	—	2	3	no	—
34	Q₁₆	C₂³	1	3	yes	—
35	Q₁₆	C₂³	1	1	yes	characteristic
36	SD₁₆	—	2	3	no	—
37	SD₁₆	—	2	3	no	—
38	D₈	—	2	3	no	—
39	D₈	C₂³	1	1	yes	characteristic
40	D₈	C₂³	1	3	yes	—
41	M₁₆	D₄	1	3	yes	—
42	C₂×C₈	C₂³	1	3	yes	—
43	C₁₆	C₂³	1	1	yes	characteristic
44	C₁₆	C₂³	1	3	yes	—
45	C₂³	—	4	3	no	—
46	Q₈	—	2	1	no	—
47	Q₈	—	2	3	no	—
48	Q₈	—	4	3	no	—
49	Q₈	D₈	1	1	yes	characteristic
50	D₄	—	2	3	no	—
51	D₄	—	2	1	no	—
52	D₄	—	4	3	no	—
53	D₄	—	4	3	no	—
54	D₄	—	4	3	no	—
55	D₄	D₈	1	3	yes	—
56	C₂×C₄	—	4	3	no	—
57	C₂×C₄	—	4	3	no	—
58	C₂×C₄	—	4	3	no	—
59	C₂×C₄	C₂×D₄	1	3	yes	—
60	C₈	C₂×D₄	1	3	yes	—
61	C₈	C₂⁴	1	1	yes	characteristic
62	C₂²	—	4	1	no	—
63	C₂²	—	4	3	no	—
64	C₂²	—	16	3	no	—
65	C₂²	C₂×D₈	1	3	yes	—
66	C₄	—	4	1	no	—
67	C₄	—	4	3	no	—
68	C₄	C₂×D₈	1	3	yes	—
69	C₄	C₂²×D₄	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	8	1	no	—
72	C₂	—	8	3	no	—
73	C₂	C₂²×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #578	C₂	semidirect
2. #576	C₂	semidirect
3. #510	C₂	semidirect
4. #509	C₂	semidirect
5. #508	C₂	semidirect
6. C₂×SD₃₂	C₂	semidirect
7. #504	C₂	semidirect
13. #135	C₂²	semidirect
14. C₂×Q₁₆	C₂²	semidirect
17. Q₃₂	C₂²	semidirect
18. SD₃₂	C₂²	semidirect
19. SD₃₂	C₂²	semidirect
20. SD₃₂	C₂²	semidirect
21. D₁₆	C₂²	semidirect
22. M₃₂	C₂²	semidirect
23. C₂×C₁₆	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#578	2
#576	2
#510	2
#509	2
#508	2
C₂×SD₃₂	2
#504	2