Subgroups — D₃×AGL(1,ℤ/9) #66426

Order 324.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₃²	—	3	1	no	maximal
5	C₂×AGL(1,ℤ/9)	—	3	1	no	maximal
6	D₃×D₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×3₋¹⁺²	C₂²	1	1	yes	characteristic, complemented, Sylow
8	C₃×C₃⋊D₃	—	3	1	no	—
9	C₃²×D₃	—	3	1	no	—
10	C₃²×D₃	D₃	1	1	yes	characteristic
11	C₂×3₋¹⁺²	—	3	1	no	—
12	C₃⋊D₉	C₂×C₃	1	1	yes	characteristic, complemented
13	AGL(1,ℤ/9)	—	2	1	no	—
14	AGL(1,ℤ/9)	—	3	1	no	—
15	AGL(1,ℤ/9)	—	6	1	no	—
16	AGL(1,ℤ/9)	D₃	1	1	yes	characteristic, direct product factor
17	D₃×C₉	—	2	1	no	—
18	D₃×C₉	C₂×C₃	1	1	yes	characteristic, complemented
19	C₃×D₉	C₂×C₃	1	1	yes	characteristic, complemented
20	C₂×C₃×D₃	—	9	1	no	—
21	C₂×C₃×D₃	—	9	1	no	—
22	D₃²	—	3	1	no	—
23	C₂×D₉	—	3	1	no	—
24	C₃³	C₂×D₃	1	1	yes	characteristic
25	3₋¹⁺²	—	2	1	no	—
26	3₋¹⁺²	—	2	1	no	—
27	3₋¹⁺²	—	4	1	no	—
28	3₋¹⁺²	C₂×D₃	1	1	yes	characteristic, complemented
29	C₃×C₉	—	2	1	no	—
30	C₃×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
31	C₂×C₃²	—	3	1	no	—
32	C₂×C₃²	—	9	1	no	—
33	C₃⋊D₃	—	3	1	no	—
34	C₃×D₃	—	3	1	no	—
35	C₃×D₃	—	3	1	no	—
36	C₃×D₃	—	3	1	no	—
37	C₃×D₃	—	6	1	no	—
38	C₃×D₃	—	9	1	no	—
39	C₃×D₃	—	9	1	no	—
40	C₃×D₃	—	18	1	no	—
41	C₃×D₃	C₃×D₃	1	1	yes	characteristic
42	C₂×C₉	—	3	1	no	—
43	C₂×C₉	—	6	1	no	—
44	D₉	—	3	1	no	—
45	D₉	—	6	1	no	—
46	D₉	C₃×D₃	1	1	yes	characteristic, complemented
47	C₂²×C₃	—	27	1	no	—
48	C₂×D₃	—	9	1	no	—
49	C₂×D₃	—	9	1	no	—
50	C₃²	—	2	1	no	—
51	C₃²	—	3	1	no	—
52	C₃²	—	6	1	no	—
53	C₃²	D₃²	1	1	yes	characteristic
54	C₃²	C₂×C₃×D₃	1	1	yes	characteristic
55	C₉	—	2	1	no	—
56	C₉	—	2	1	no	—
57	C₉	—	4	1	no	—
58	C₉	C₂×C₃×D₃	1	1	yes	characteristic, complemented
59	C₂×C₃	—	3	1	no	—
60	C₂×C₃	—	9	1	no	—
61	C₂×C₃	—	9	1	no	—
62	C₂×C₃	—	9	1	no	—
63	C₂×C₃	—	18	1	no	—
64	C₂×C₃	—	27	1	no	—
65	D₃	—	3	1	no	—
66	D₃	—	9	1	no	—
67	D₃	—	9	1	no	—
68	D₃	—	18	1	no	—
69	D₃	AGL(1,ℤ/9)	1	1	yes	characteristic, direct product factor
70	C₂²	—	27	1	no	Sylow
71	C₃	—	2	1	no	—
72	C₃	—	3	1	no	—
73	C₃	—	6	1	no	—
74	C₃	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
75	C₃	C₃×D₃²	1	1	yes	characteristic
76	C₂	—	3	1	no	—
77	C₂	—	9	1	no	—
78	C₂	—	27	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃×C₉	C₂²×C₃
Frattini	C₃	C₃×D₃²
Fitting	C₃×3₋¹⁺²	C₂²
socle	C₃²	C₂×C₃×D₃
LCS	G ▷ C₃×C₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₃⋊AGL(1,ℤ/9)	C₂	semidirect
2. D₃×3₋¹⁺²	C₂	semidirect
3. C₃×AGL(1,ℤ/9)	C₂	semidirect
6. D₃×D₉	C₃	semidirect
7. C₃×3₋¹⁺²	C₂²	semidirect
12. C₃⋊D₉	C₂×C₃	semidirect
16. AGL(1,ℤ/9)	D₃	direct (N × H)
18. D₃×C₉	C₂×C₃	semidirect
19. C₃×D₉	C₂×C₃	semidirect
28. 3₋¹⁺²	C₂×D₃	semidirect
30. C₃×C₉	C₂²×C₃	semidirect
46. D₉	C₃×D₃	semidirect
58. C₉	C₂×C₃×D₃	semidirect
69. D₃	AGL(1,ℤ/9)	direct (N × H)
74. C₃	C₂×AGL(1,ℤ/9)	semidirect

Maximal subgroups

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

subgroup	index
C₃⋊AGL(1,ℤ/9)	2
D₃×3₋¹⁺²	2
C₃×AGL(1,ℤ/9)	2
C₃×D₃²	3
C₂×AGL(1,ℤ/9)	3
D₃×D₉	3

Subgroups — D3×AGL(1,ℤ/9) #66426

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×AGL(1,ℤ/9) #66426