Subgroups — #3291

Order 128.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#577	C₂	1	4	yes	maximal, complemented
2	#492	C₂	1	1	yes	maximal, characteristic, complemented
3	#471	C₂	1	4	yes	maximal, complemented
4	#456	C₂	1	2	yes	maximal, complemented
5	#453	C₂	1	2	yes	maximal, complemented
6	#443	C₂	1	2	yes	maximal, complemented
7	2₋¹⁺⁴	C₂²	1	2	yes	—
8	H(5,𝔽₂)	C₂²	1	2	yes	complemented
9	#137	—	2	4	no	—
10	#137	C₂²	1	4	yes	complemented
11	AGL(1,ℤ/8)	—	2	4	no	—
12	AGL(1,ℤ/8)	C₂²	1	4	yes	—
13	#135	—	2	4	no	—
14	#135	C₂²	1	2	yes	—
15	C₂×SD₁₆	—	2	4	no	—
16	C₂×SD₁₆	C₂²	1	4	yes	complemented
17	#131	C₂²	1	4	yes	complemented
18	C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	#127	C₂²	1	1	yes	characteristic
20	#108	C₂²	1	2	yes	complemented
21	#104	C₂²	1	4	yes	complemented
22	#101	C₂²	1	2	yes	complemented
23	#100	C₂²	1	2	yes	complemented
24	C₄×C₈	C₂²	1	1	yes	characteristic, complemented
25	#41	—	2	4	no	—
26	#41	—	2	4	no	—
27	#41	—	4	2	no	—
28	#41	C₂³	1	4	yes	—
29	C₂×Q₈	—	2	4	no	—
30	C₂×Q₈	C₂³	1	2	yes	—
31	C₂×D₄	—	2	4	no	—
32	C₂×D₄	—	4	1	no	—
33	C₂×D₄	—	4	2	no	—
34	C₂×D₄	C₂³	1	2	yes	—
35	Q₁₆	—	2	4	no	—
36	Q₁₆	D₄	1	2	yes	complemented
37	SD₁₆	—	2	4	no	—
38	SD₁₆	—	2	4	no	—
39	SD₁₆	—	2	4	no	—
40	SD₁₆	—	2	4	no	—
41	SD₁₆	D₄	1	4	yes	complemented
42	D₈	—	2	4	no	—
43	D₈	D₄	1	2	yes	complemented
44	M₁₆	—	2	4	no	—
45	M₁₆	C₂³	1	4	yes	—
46	C₂×C₈	—	2	4	no	—
47	C₂×C₈	C₂³	1	2	yes	—
48	C₄⋊C₄	—	4	1	no	—
49	C₄²	C₂³	1	1	yes	characteristic
50	C₂³	—	2	2	no	—
51	C₂³	—	4	2	no	—
52	Q₈	—	2	4	no	—
53	Q₈	—	4	2	no	—
54	Q₈	C₂×D₄	1	4	yes	—
55	D₄	—	2	4	no	—
56	D₄	—	2	4	no	—
57	D₄	—	4	4	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	2	no	—
60	D₄	C₂×D₄	1	4	yes	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	4	4	no	—
66	C₂×C₄	C₂⁴	1	1	yes	characteristic
67	C₈	—	2	4	no	—
68	C₈	C₂×D₄	1	2	yes	—
69	C₈	C₂×D₄	1	2	yes	—
70	C₂²	—	2	4	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	8	2	no	—
73	C₂²	—	16	2	no	—
74	C₂²	H(5,𝔽₂)	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	2	4	no	—
77	C₄	—	4	2	no	—
78	C₄	C₂²×D₄	1	2	yes	—
79	C₂	—	2	1	no	—
80	C₂	—	4	4	no	—
81	C₂	—	8	2	no	—
82	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. #577	C₂	semidirect
2. #492	C₂	semidirect
3. #471	C₂	semidirect
4. #456	C₂	semidirect
5. #453	C₂	semidirect
6. #443	C₂	semidirect
8. H(5,𝔽₂)	C₂²	semidirect
10. #137	C₂²	semidirect
16. C₂×SD₁₆	C₂²	semidirect
17. #131	C₂²	semidirect
18. C₄⋊Q₈	C₂²	semidirect
20. #108	C₂²	semidirect
21. #104	C₂²	semidirect
22. #101	C₂²	semidirect
23. #100	C₂²	semidirect
24. C₄×C₈	C₂²	semidirect
36. Q₁₆	D₄	semidirect
41. SD₁₆	D₄	semidirect
43. D₈	D₄	semidirect

Maximal subgroups

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

subgroup	index
#577	2
#492	2
#471	2
#456	2
#453	2
#443	2