Subgroups — C₂×#576 #3581

Order 128.

756 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×H(5,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	#576	C₂	1	16	yes	maximal, direct product factor
3	C₂×AGL(1,ℤ/8)	C₂	1	6	yes	maximal, complemented
4	C₂×#135	C₂	1	3	yes	maximal, complemented
5	C₂²×D₈	C₂	1	3	yes	maximal, complemented
6	C₂×#131	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)	—	2	8	no	—
8	H(5,𝔽₂)	C₂²	1	16	yes	complemented
9	C₂×#41	—	2	2	no	—
10	C₂×#41	C₂²	1	1	yes	characteristic
11	C₂×#41	C₂²	1	6	yes	complemented
12	C₂²×D₄	—	2	6	no	—
13	C₂²×D₄	C₂²	1	6	yes	complemented
14	AGL(1,ℤ/8)	C₂²	1	48	yes	complemented
15	#135	C₂²	1	24	yes	complemented
16	C₂×Q₁₆	C₂²	1	1	yes	characteristic, complemented
17	C₂×SD₁₆	C₂²	1	6	yes	complemented
18	C₂×D₈	C₂²	1	3	yes	complemented
19	C₂×D₈	C₂²	1	6	yes	complemented
20	C₂×D₈	C₂²	1	24	yes	complemented
21	#131	C₂²	1	8	yes	complemented
22	C₂×M₁₆	C₂²	1	3	yes	complemented
23	C₂²×C₈	C₂²	1	3	yes	complemented
24	C₂⁴	—	2	6	no	—
25	#41	—	2	16	no	—
26	#41	—	2	12	no	—
27	#41	D₄	1	4	yes	—
28	#41	C₂³	1	4	yes	—
29	#41	C₂³	1	24	yes	complemented
30	C₂×Q₈	D₄	1	1	yes	characteristic
31	C₂×Q₈	C₂³	1	2	yes	—
32	C₂×D₄	—	2	6	no	—
33	C₂×D₄	—	2	48	no	—
34	C₂×D₄	—	2	6	no	—
35	C₂×D₄	—	2	12	no	—
36	C₂×D₄	D₄	1	3	yes	—
37	C₂×D₄	C₂³	1	6	yes	—
38	C₂×D₄	C₂³	1	24	yes	complemented
39	C₂²×C₄	—	2	6	no	—
40	C₂²×C₄	C₂³	1	3	yes	—
41	Q₁₆	C₂³	1	4	yes	complemented
42	SD₁₆	C₂³	1	24	yes	complemented
43	D₈	C₂³	1	12	yes	complemented
44	D₈	C₂³	1	24	yes	complemented
45	M₁₆	C₂³	1	12	yes	complemented
46	C₂×C₈	C₂³	1	1	yes	characteristic
47	C₂×C₈	C₂³	1	3	yes	—
48	C₂×C₈	C₂³	1	12	yes	complemented
49	C₂³	—	2	6	no	—
50	C₂³	—	2	24	no	—
51	C₂³	—	8	12	no	—
52	C₂³	C₂×D₄	1	3	yes	—
53	Q₈	—	2	2	no	—
54	Q₈	C₂×D₄	1	4	yes	—
55	Q₈	C₂⁴	1	4	yes	—
56	D₄	—	2	24	no	—
57	D₄	—	2	6	no	—
58	D₄	—	2	24	no	—
59	D₄	C₂×D₄	1	12	yes	—
60	D₄	C₂⁴	1	12	yes	—
61	C₂×C₄	—	2	2	no	—
62	C₂×C₄	—	2	24	no	—
63	C₂×C₄	C₂×D₄	1	6	yes	—
64	C₂×C₄	C₂×D₄	1	3	yes	—
65	C₂×C₄	C₂⁴	1	6	yes	—
66	C₂×C₄	C₂⁴	1	1	yes	characteristic
67	C₈	C₂⁴	1	2	yes	—
68	C₈	C₂⁴	1	6	yes	—
69	C₂²	—	2	6	no	—
70	C₂²	—	2	12	no	—
71	C₂²	—	4	12	no	—
72	C₂²	—	8	24	no	—
73	C₂²	C₂²×D₄	1	6	yes	—
74	C₂²	C₂²×D₄	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	C₂²×D₄	1	6	yes	—
77	C₄	C₂²×D₄	1	1	yes	characteristic
78	C₄	C₂⁵	1	1	yes	characteristic
79	C₂	—	2	6	no	—
80	C₂	—	4	12	no	—
81	C₂	#576	1	2	yes	direct product factor
82	C₂	C₂³×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁵
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂×H(5,𝔽₂)	C₂	semidirect
2. #576	C₂	direct (N × H)
3. C₂×AGL(1,ℤ/8)	C₂	semidirect
4. C₂×#135	C₂	semidirect
5. C₂²×D₈	C₂	semidirect
6. C₂×#131	C₂	semidirect
8. H(5,𝔽₂)	C₂²	semidirect
11. C₂×#41	C₂²	semidirect
13. C₂²×D₄	C₂²	semidirect
14. AGL(1,ℤ/8)	C₂²	semidirect
15. #135	C₂²	semidirect
16. C₂×Q₁₆	C₂²	semidirect
17. C₂×SD₁₆	C₂²	semidirect
18. C₂×D₈	C₂²	semidirect
19. C₂×D₈	C₂²	semidirect
20. C₂×D₈	C₂²	semidirect
21. #131	C₂²	semidirect
22. C₂×M₁₆	C₂²	semidirect
23. C₂²×C₈	C₂²	semidirect
29. #41	C₂³	semidirect
38. C₂×D₄	C₂³	semidirect
41. Q₁₆	C₂³	semidirect
42. SD₁₆	C₂³	semidirect
43. D₈	C₂³	semidirect
44. D₈	C₂³	semidirect
45. M₁₆	C₂³	semidirect
48. C₂×C₈	C₂³	semidirect
81. C₂	#576	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×H(5,𝔽₂)	2
#576	2
C₂×AGL(1,ℤ/8)	2
C₂×#135	2
C₂²×D₈	2
C₂×#131	2

Subgroups — C2×#576 #3581

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#576 #3581