Subgroups — #3664⋊C₃ #88475

Order 432.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#204	—	3	1	no	maximal
5	#3664	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×AGL(1,ℤ/9)	—	2	1	no	—
7	C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	3₋¹⁺²⋊C₄	—	2	1	no	—
9	C₃×D₁₂	—	3	1	no	—
10	C₃×Dic₆	—	3	1	no	—
11	C₃²×C₈	D₃	1	1	yes	characteristic
12	D₃₆	C₂×C₃	1	1	yes	characteristic
13	Dic₁₈	C₂×C₃	1	1	yes	characteristic, complemented
14	C₈×C₉	—	2	1	no	—
15	C₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
16	C₂×3₋¹⁺²	D₄	1	1	yes	characteristic
17	AGL(1,ℤ/9)	—	4	1	no	—
18	C₃×SD₁₆	—	9	1	no	—
19	#204	—	3	1	no	—
20	C₂×C₃×D₃	—	6	1	no	—
21	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
22	C₃×Dic₃	—	6	1	no	—
23	C₂×D₉	—	2	1	no	—
24	C₄×C₉	—	2	1	no	—
25	C₄×C₉	C₂²×C₃	1	1	yes	characteristic
26	Dic₉	—	2	1	no	—
27	3₋¹⁺²	SD₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	9	1	no	—
29	C₃×D₄	—	9	1	no	—
30	D₁₂	—	3	1	no	—
31	Dic₆	—	3	1	no	—
32	C₃×C₈	—	6	1	no	—
33	C₃×C₈	C₃×D₃	1	1	yes	characteristic
34	C₂×C₃²	D₁₂	1	1	yes	characteristic
35	C₃×D₃	—	12	1	no	—
36	C₂×C₉	—	2	1	no	—
37	C₂×C₉	C₃×D₄	1	1	yes	characteristic
38	D₉	—	4	1	no	—
39	SD₁₆	—	9	1	no	Sylow
40	C₂²×C₃	—	18	1	no	—
41	C₂×D₃	—	6	1	no	—
42	C₃×C₄	—	6	1	no	—
43	C₃×C₄	—	18	1	no	—
44	C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
45	Dic₃	—	6	1	no	—
46	C₃²	#204	1	1	yes	characteristic
47	C₉	—	2	1	no	—
48	C₉	C₃×SD₁₆	1	1	yes	characteristic, complemented
49	Q₈	—	9	1	no	—
50	D₄	—	9	1	no	—
51	C₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
52	C₂×C₃	—	6	1	no	—
53	C₂×C₃	—	36	1	no	—
54	C₂×C₃	C₃×D₁₂	1	1	yes	characteristic
55	D₃	—	12	1	no	—
56	C₂²	—	18	1	no	—
57	C₄	—	18	1	no	—
58	C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
59	C₃	—	6	1	no	—
60	C₃	C₃×#204	1	1	yes	characteristic
61	C₂	—	36	1	no	—
62	C₂	C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₄⋊AGL(1,ℤ/9)
commutator, cosocle	C₄×C₉	C₂²×C₃
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₈×3₋¹⁺²	C₂
socle	C₂×C₃	C₃×D₁₂
prime radical	C₄⋊AGL(1,ℤ/9)	C₂
hypercenter	C₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×SD₁₆
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. 3₋¹⁺²⋊Q₈	C₂	semidirect
3. C₈×3₋¹⁺²	C₂	semidirect
5. #3664	C₃	semidirect
13. Dic₁₈	C₂×C₃	semidirect
15. C₈×C₉	C₂×C₃	semidirect
27. 3₋¹⁺²	SD₁₆	semidirect
48. C₉	C₃×SD₁₆	semidirect
51. C₈	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
C₄⋊AGL(1,ℤ/9)	2
3₋¹⁺²⋊Q₈	2
C₈×3₋¹⁺²	2
C₃×#204	3
#3664	3

Subgroups — #3664⋊C3 #88475

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3664⋊C₃ #88475