Subgroups — C₂×C₄⋊AGL(1,ℤ/9) #88707

Order 432.

170 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/9)	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊AGL(1,ℤ/9)	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃×D₁₂	—	3	1	no	maximal
5	C₂×D₃₆	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂²×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₂×AGL(1,ℤ/9)	—	2	4	no	—
8	C₂×AGL(1,ℤ/9)	C₂²	1	4	yes	complemented
9	C₄×3₋¹⁺²	C₂²	1	2	yes	complemented
10	C₂²×C₃×D₃	—	3	2	no	—
11	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
12	C₃×D₁₂	—	3	4	no	—
13	C₂²×D₉	C₂×C₃	1	2	yes	complemented
14	C₂×C₄×C₉	—	2	1	no	—
15	C₂×C₄×C₉	C₂×C₃	1	1	yes	characteristic, complemented
16	D₃₆	C₂×C₃	1	4	yes	complemented
17	C₂×3₋¹⁺²	D₄	1	2	yes	complemented
18	C₂×3₋¹⁺²	C₂³	1	1	yes	characteristic
19	AGL(1,ℤ/9)	—	2	4	no	—
20	C₂×C₃×D₄	—	9	1	no	—
21	C₂×D₁₂	—	3	1	no	—
22	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
23	C₂×C₃×D₃	—	3	4	no	—
24	C₂×C₃×D₃	—	6	4	no	—
25	C₃²×C₄	C₂×D₃	1	2	yes	—
26	C₂²×C₉	—	2	1	no	—
27	C₂²×C₉	C₂²×C₃	1	1	yes	characteristic
28	C₂×D₉	—	2	4	no	—
29	C₂×D₉	C₂²×C₃	1	4	yes	complemented
30	C₄×C₉	—	2	2	no	—
31	C₄×C₉	C₂²×C₃	1	2	yes	complemented
32	3₋¹⁺²	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
33	C₂³×C₃	—	9	2	no	—
34	C₂²×D₃	—	3	2	no	—
35	C₃×D₄	—	9	4	no	—
36	C₂×C₃×C₄	—	3	1	no	—
37	C₂×C₃×C₄	C₃×D₃	1	1	yes	characteristic
38	D₁₂	—	3	4	no	—
39	C₂×C₃²	D₁₂	1	2	yes	—
40	C₂×C₃²	C₂²×D₃	1	1	yes	characteristic
41	C₃×D₃	—	6	4	no	—
42	C₂×C₉	—	2	2	no	—
43	C₂×C₉	—	2	1	no	—
44	C₂×C₉	C₃×D₄	1	2	yes	complemented
45	C₂×C₉	C₂³×C₃	1	1	yes	characteristic
46	D₉	—	2	4	no	—
47	C₂×D₄	—	9	1	no	Sylow
48	C₂²×C₃	—	3	1	no	—
49	C₂²×C₃	—	9	4	no	—
50	C₂²×C₃	—	18	4	no	—
51	C₂²×C₃	C₂×C₃×D₃	1	1	yes	characteristic
52	C₂×D₃	—	3	4	no	—
53	C₂×D₃	—	6	4	no	—
54	C₃×C₄	—	3	2	no	—
55	C₃×C₄	C₂×C₃×D₃	1	2	yes	—
56	C₃²	C₂×D₁₂	1	1	yes	characteristic
57	C₉	—	2	1	no	—
58	C₉	C₂×C₃×D₄	1	1	yes	characteristic, complemented
59	C₂³	—	9	2	no	—
60	D₄	—	9	4	no	—
61	C₂×C₄	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
62	C₂×C₃	—	3	2	no	—
63	C₂×C₃	—	3	1	no	—
64	C₂×C₃	—	18	4	no	—
65	C₂×C₃	C₃×D₁₂	1	2	yes	—
66	C₂×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
67	D₃	—	6	4	no	—
68	C₂²	—	9	4	no	—
69	C₂²	—	18	4	no	—
70	C₂²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
71	C₄	C₂×AGL(1,ℤ/9)	1	2	yes	complemented
72	C₃	—	3	1	no	—
73	C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
74	C₂	—	18	4	no	—
75	C₂	C₄⋊AGL(1,ℤ/9)	1	2	yes	direct product factor
76	C₂	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₉	C₂³×C₃
Frattini	C₂×C₃	C₂²×C₃×D₃
Fitting	C₂×C₄×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×D₃
hypercenter	C₂×C₄	AGL(1,ℤ/9)
hypocenter	C₉	C₂×C₃×D₄
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×AGL(1,ℤ/9)	C₂	semidirect
2. C₂×C₄×3₋¹⁺²	C₂	semidirect
3. C₄⋊AGL(1,ℤ/9)	C₂	direct (N × H)
5. C₂×D₃₆	C₃	semidirect
8. C₂×AGL(1,ℤ/9)	C₂²	semidirect
9. C₄×3₋¹⁺²	C₂²	semidirect
13. C₂²×D₉	C₂×C₃	semidirect
15. C₂×C₄×C₉	C₂×C₃	semidirect
16. D₃₆	C₂×C₃	semidirect
17. C₂×3₋¹⁺²	D₄	semidirect
29. C₂×D₉	C₂²×C₃	semidirect
31. C₄×C₉	C₂²×C₃	semidirect
32. 3₋¹⁺²	C₂×D₄	semidirect
44. C₂×C₉	C₃×D₄	semidirect
58. C₉	C₂×C₃×D₄	semidirect
61. C₂×C₄	AGL(1,ℤ/9)	semidirect
71. C₄	C₂×AGL(1,ℤ/9)	semidirect
75. C₂	C₄⋊AGL(1,ℤ/9)	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×AGL(1,ℤ/9)	2
C₂×C₄×3₋¹⁺²	2
C₄⋊AGL(1,ℤ/9)	2
C₂×C₃×D₁₂	3
C₂×D₃₆	3

Subgroups — C2×C4⋊AGL(1,ℤ/9) #88707

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄⋊AGL(1,ℤ/9) #88707