Subgroups — C₂²×AGL(1,ℤ/8) #3578

Order 128.

860 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂³	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#41	C₂	semidirect
2. C₂³×D₄	C₂	semidirect
3. C₂×AGL(1,ℤ/8)	C₂	direct (N × H)
4. C₂²×SD₁₆	C₂	semidirect
5. C₂²×D₈	C₂	semidirect
6. C₂²×M₁₆	C₂	semidirect
9. C₂×#41	C₂²	semidirect
10. C₂²×Q₈	C₂²	semidirect
15. C₂²×D₄	C₂²	semidirect
16. C₂²×D₄	C₂²	semidirect
19. AGL(1,ℤ/8)	C₂²	direct (N × H)
20. C₂×SD₁₆	C₂²	semidirect
21. C₂×D₈	C₂²	semidirect
22. C₂×M₁₆	C₂²	semidirect
23. C₂²×C₈	C₂²	semidirect
30. #41	C₂³	semidirect
32. C₂×Q₈	C₂³	semidirect
39. C₂×D₄	C₂³	semidirect
40. C₂×D₄	C₂³	semidirect
47. SD₁₆	C₂³	semidirect
48. D₈	C₂³	semidirect
49. M₁₆	C₂³	semidirect
50. C₂×C₈	C₂³	semidirect
61. Q₈	C₂⁴	semidirect
67. D₄	C₂⁴	semidirect
75. C₈	C₂⁴	semidirect
82. C₂²	AGL(1,ℤ/8)	direct (N × H)
92. C₂	C₂×AGL(1,ℤ/8)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#41	2
C₂³×D₄	2
C₂×AGL(1,ℤ/8)	2
C₂²×SD₁₆	2
C₂²×D₈	2
C₂²×M₁₆	2

Subgroups — C22×AGL(1,ℤ/8) #3578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×AGL(1,ℤ/8) #3578