Subgroups — C₃²×AGL(1,ℤ/8) #64185

Order 288.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×SD₁₆	C₂	1	2	yes	maximal, complemented
4	C₃²×D₈	C₂	1	2	yes	maximal, complemented
5	C₃²×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×AGL(1,ℤ/8)	C₃	1	4	yes	maximal, direct product factor
7	C₂³×C₃²	—	2	1	no	—
8	C₃²×Q₈	C₂²	1	1	yes	characteristic, complemented
9	C₃²×D₄	—	2	1	no	—
10	C₃²×D₄	—	2	1	no	—
11	C₃²×D₄	C₂²	1	2	yes	—
12	C₃²×D₄	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₃²×C₄	—	2	1	no	—
14	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
15	C₃²×C₈	C₂²	1	2	yes	complemented
16	C₃×#41	C₂×C₃	1	4	yes	complemented
17	C₂×C₃×D₄	C₂×C₃	1	4	yes	complemented
18	C₃×SD₁₆	C₂×C₃	1	8	yes	complemented
19	C₃×D₈	C₂×C₃	1	8	yes	complemented
20	C₃×M₁₆	C₂×C₃	1	4	yes	complemented
21	C₂²×C₃²	—	2	1	no	—
22	C₂²×C₃²	—	2	2	no	—
23	C₂²×C₃²	—	4	2	no	—
24	C₂²×C₃²	D₄	1	1	yes	characteristic
25	C₃²×C₄	—	2	1	no	—
26	C₃²×C₄	D₄	1	1	yes	characteristic
27	C₃²×C₄	C₂³	1	1	yes	characteristic
28	AGL(1,ℤ/8)	C₃²	1	1	yes	characteristic, direct product factor, Sylow
29	C₂³×C₃	—	2	4	no	—
30	C₃×Q₈	C₂²×C₃	1	4	yes	complemented
31	C₃×D₄	—	2	4	no	—
32	C₃×D₄	—	2	4	no	—
33	C₃×D₄	C₂²×C₃	1	8	yes	—
34	C₃×D₄	C₂²×C₃	1	4	yes	complemented
35	C₂×C₃×C₄	—	2	4	no	—
36	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
37	C₃×C₈	C₂²×C₃	1	8	yes	complemented
38	C₂×C₃²	—	2	1	no	—
39	C₂×C₃²	—	4	1	no	—
40	C₂×C₃²	—	4	2	no	—
41	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
42	#41	C₂×C₃²	1	1	yes	characteristic, complemented
43	C₂×D₄	C₂×C₃²	1	1	yes	characteristic, complemented
44	SD₁₆	C₂×C₃²	1	2	yes	complemented
45	D₈	C₂×C₃²	1	2	yes	complemented
46	M₁₆	C₂×C₃²	1	1	yes	characteristic, complemented
47	C₂²×C₃	—	2	4	no	—
48	C₂²×C₃	—	2	8	no	—
49	C₂²×C₃	—	4	8	no	—
50	C₂²×C₃	C₃×D₄	1	4	yes	—
51	C₃×C₄	—	2	4	no	—
52	C₃×C₄	C₃×D₄	1	4	yes	—
53	C₃×C₄	C₂³×C₃	1	4	yes	—
54	C₃²	AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³	—	2	1	no	—
56	Q₈	C₂²×C₃²	1	1	yes	characteristic, complemented
57	D₄	—	2	1	no	—
58	D₄	—	2	1	no	—
59	D₄	C₂²×C₃²	1	2	yes	—
60	D₄	C₂²×C₃²	1	1	yes	characteristic, complemented
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
63	C₈	C₂²×C₃²	1	2	yes	complemented
64	C₂×C₃	—	2	4	no	—
65	C₂×C₃	—	4	8	no	—
66	C₂×C₃	—	4	4	no	—
67	C₂×C₃	C₂×C₃×D₄	1	4	yes	—
68	C₂²	—	2	1	no	—
69	C₂²	—	2	2	no	—
70	C₂²	—	4	2	no	—
71	C₂²	C₃²×D₄	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	C₃²×D₄	1	1	yes	characteristic
74	C₄	C₂³×C₃²	1	1	yes	characteristic
75	C₃	C₃×AGL(1,ℤ/8)	1	4	yes	direct product factor
76	C₂	—	2	1	no	—
77	C₂	—	4	2	no	—
78	C₂	—	4	1	no	—
79	C₂	C₂×C₃²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃²	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂³×C₃²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃²×#41	C₂	semidirect
2. C₂×C₃²×D₄	C₂	semidirect
3. C₃²×SD₁₆	C₂	semidirect
4. C₃²×D₈	C₂	semidirect
5. C₃²×M₁₆	C₂	semidirect
6. C₃×AGL(1,ℤ/8)	C₃	direct (N × H)
8. C₃²×Q₈	C₂²	semidirect
12. C₃²×D₄	C₂²	semidirect
15. C₃²×C₈	C₂²	semidirect
16. C₃×#41	C₂×C₃	semidirect
17. C₂×C₃×D₄	C₂×C₃	semidirect
18. C₃×SD₁₆	C₂×C₃	semidirect
19. C₃×D₈	C₂×C₃	semidirect
20. C₃×M₁₆	C₂×C₃	semidirect
28. AGL(1,ℤ/8)	C₃²	direct (N × H)
30. C₃×Q₈	C₂²×C₃	semidirect
34. C₃×D₄	C₂²×C₃	semidirect
37. C₃×C₈	C₂²×C₃	semidirect
42. #41	C₂×C₃²	semidirect
43. C₂×D₄	C₂×C₃²	semidirect
44. SD₁₆	C₂×C₃²	semidirect
45. D₈	C₂×C₃²	semidirect
46. M₁₆	C₂×C₃²	semidirect
54. C₃²	AGL(1,ℤ/8)	direct (N × H)
56. Q₈	C₂²×C₃²	semidirect
60. D₄	C₂²×C₃²	semidirect
63. C₈	C₂²×C₃²	semidirect
75. C₃	C₃×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
C₃²×#41	2
C₂×C₃²×D₄	2
C₃²×SD₁₆	2
C₃²×D₈	2
C₃²×M₁₆	2
C₃×AGL(1,ℤ/8)	3

Subgroups — C32×AGL(1,ℤ/8) #64185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×AGL(1,ℤ/8) #64185