Subgroups — C₂×C₄×AGL(1,ℤ/9) #88706

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	1	yes	maximal, characteristic
2	C₂×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×AGL(1,ℤ/9)	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₃×C₄×D₃	—	3	1	no	maximal
6	C₂×C₄×D₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₂²×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂×AGL(1,ℤ/9)	C₄	1	4	yes	direct product factor
9	C₂×AGL(1,ℤ/9)	C₂²	1	2	yes	—
10	C₄×3₋¹⁺²	C₂²	1	2	yes	complemented
11	3₋¹⁺²⋊C₄	C₂²	1	2	yes	complemented
12	C₂²×C₃×D₃	—	3	1	no	—
13	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
14	C₂×C₃×Dic₃	—	3	1	no	—
15	C₃×C₄×D₃	—	3	4	no	—
16	C₂²×D₉	C₂×C₃	1	1	yes	characteristic
17	C₂×C₄×C₉	—	2	1	no	—
18	C₂×C₄×C₉	C₂×C₃	1	1	yes	characteristic, complemented
19	C₂×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
20	C₄×D₉	C₂×C₃	1	4	yes	complemented
21	C₂×3₋¹⁺²	C₂×C₄	1	2	yes	complemented
22	C₂×3₋¹⁺²	C₂³	1	1	yes	characteristic
23	AGL(1,ℤ/9)	C₂×C₄	1	4	yes	direct product factor
24	C₂²×C₃×C₄	—	9	1	no	—
25	C₂×C₄×D₃	—	3	1	no	—
26	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
27	C₂×C₃×D₃	—	3	4	no	—
28	C₂×C₃×D₃	—	3	2	no	—
29	C₃²×C₄	C₂×D₃	1	2	yes	—
30	C₃×Dic₃	—	3	2	no	—
31	C₂²×C₉	—	2	1	no	—
32	C₂²×C₉	C₂²×C₃	1	1	yes	characteristic
33	C₂×D₉	C₃×C₄	1	4	yes	complemented
34	C₂×D₉	C₂²×C₃	1	2	yes	—
35	C₄×C₉	—	2	2	no	—
36	C₄×C₉	C₂²×C₃	1	2	yes	complemented
37	Dic₉	C₂²×C₃	1	2	yes	complemented
38	3₋¹⁺²	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
39	C₂³×C₃	—	9	1	no	—
40	C₂²×D₃	—	3	1	no	—
41	C₂×C₃×C₄	—	3	1	no	—
42	C₂×C₃×C₄	—	9	4	no	—
43	C₂×C₃×C₄	—	9	1	no	—
44	C₂×C₃×C₄	C₃×D₃	1	1	yes	characteristic
45	C₂×Dic₃	—	3	1	no	—
46	C₄×D₃	—	3	4	no	—
47	C₂×C₃²	C₄×D₃	1	2	yes	—
48	C₂×C₃²	C₂²×D₃	1	1	yes	characteristic
49	C₃×D₃	—	3	4	no	—
50	C₂×C₉	—	2	2	no	—
51	C₂×C₉	—	2	1	no	—
52	C₂×C₉	C₂×C₃×C₄	1	2	yes	complemented
53	C₂×C₉	C₂³×C₃	1	1	yes	characteristic
54	D₉	C₂×C₃×C₄	1	4	yes	complemented
55	C₂²×C₄	—	9	1	no	Sylow
56	C₂²×C₃	—	3	1	no	—
57	C₂²×C₃	—	9	2	no	—
58	C₂²×C₃	—	9	4	no	—
59	C₂²×C₃	C₂×C₃×D₃	1	1	yes	characteristic
60	C₂×D₃	—	3	4	no	—
61	C₂×D₃	—	3	2	no	—
62	C₃×C₄	—	3	2	no	—
63	C₃×C₄	—	9	2	no	—
64	C₃×C₄	C₂×C₃×D₃	1	2	yes	—
65	Dic₃	—	3	2	no	—
66	C₃²	C₂×C₄×D₃	1	1	yes	characteristic
67	C₉	—	2	1	no	—
68	C₉	C₂²×C₃×C₄	1	1	yes	characteristic, complemented
69	C₂³	—	9	1	no	—
70	C₂×C₄	—	9	1	no	—
71	C₂×C₄	—	9	4	no	—
72	C₂×C₄	AGL(1,ℤ/9)	1	1	yes	characteristic, direct product factor
73	C₂×C₃	—	3	2	no	—
74	C₂×C₃	—	3	1	no	—
75	C₂×C₃	—	9	4	no	—
76	C₂×C₃	C₃×C₄×D₃	1	2	yes	—
77	C₂×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
78	D₃	—	3	4	no	—
79	C₂²	—	9	4	no	—
80	C₂²	—	9	2	no	—
81	C₂²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
82	C₄	—	9	2	no	—
83	C₄	C₂×AGL(1,ℤ/9)	1	2	yes	direct product factor
84	C₃	—	3	1	no	—
85	C₃	C₂×C₃×C₄×D₃	1	1	yes	characteristic
86	C₂	—	9	4	no	—
87	C₂	C₄×AGL(1,ℤ/9)	1	2	yes	direct product factor
88	C₂	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×3₋¹⁺²	C₂	semidirect
3. C₂×3₋¹⁺²⋊C₄	C₂	semidirect
4. C₄×AGL(1,ℤ/9)	C₂	direct (N × H)
6. C₂×C₄×D₉	C₃	semidirect
8. C₂×AGL(1,ℤ/9)	C₄	direct (N × H)
10. C₄×3₋¹⁺²	C₂²	semidirect
11. 3₋¹⁺²⋊C₄	C₂²	semidirect
18. C₂×C₄×C₉	C₂×C₃	semidirect
19. C₂×Dic₉	C₂×C₃	semidirect
20. C₄×D₉	C₂×C₃	semidirect
21. C₂×3₋¹⁺²	C₂×C₄	semidirect
23. AGL(1,ℤ/9)	C₂×C₄	direct (N × H)
33. C₂×D₉	C₃×C₄	semidirect
36. C₄×C₉	C₂²×C₃	semidirect
37. Dic₉	C₂²×C₃	semidirect
38. 3₋¹⁺²	C₂²×C₄	semidirect
52. C₂×C₉	C₂×C₃×C₄	semidirect
54. D₉	C₂×C₃×C₄	semidirect
68. C₉	C₂²×C₃×C₄	semidirect
72. C₂×C₄	AGL(1,ℤ/9)	direct (N × H)
83. C₄	C₂×AGL(1,ℤ/9)	direct (N × H)
87. 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
C₂×3₋¹⁺²⋊C₄	2
C₄×AGL(1,ℤ/9)	2
C₂×C₃×C₄×D₃	3
C₂×C₄×D₉	3

Subgroups — C2×C4×AGL(1,ℤ/9) #88706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×AGL(1,ℤ/9) #88706