Subgroups — C₄×AGL(1,ℤ/9) #6216

Order 216.

51 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×AGL(1,ℤ/9)	C₂	1	1	yes	maximal, characteristic
2	C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	3₋¹⁺²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₄×D₃	—	3	1	no	maximal
5	C₄×D₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	AGL(1,ℤ/9)	C₄	1	2	yes	direct product factor
8	C₂×C₃×D₃	—	3	1	no	—
9	C₃²×C₄	D₃	1	1	yes	characteristic
10	C₃×Dic₃	—	3	1	no	—
11	C₂×D₉	C₂×C₃	1	1	yes	characteristic
12	C₄×C₉	—	2	1	no	—
13	C₄×C₉	C₂×C₃	1	1	yes	characteristic, complemented
14	Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
15	3₋¹⁺²	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₃×C₄	—	9	1	no	—
17	C₄×D₃	—	3	1	no	—
18	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
19	C₃×D₃	—	3	2	no	—
20	C₂×C₉	—	2	1	no	—
21	C₂×C₉	C₂²×C₃	1	1	yes	characteristic
22	D₉	C₃×C₄	1	2	yes	complemented
23	C₂²×C₃	—	9	1	no	—
24	C₂×D₃	—	3	1	no	—
25	C₃×C₄	—	3	1	no	—
26	C₃×C₄	—	9	1	no	—
27	C₃×C₄	C₃×D₃	1	1	yes	characteristic
28	Dic₃	—	3	1	no	—
29	C₃²	C₄×D₃	1	1	yes	characteristic
30	C₉	—	2	1	no	—
31	C₉	C₂×C₃×C₄	1	1	yes	characteristic, complemented
32	C₂×C₄	—	9	1	no	Sylow
33	C₂×C₃	—	3	1	no	—
34	C₂×C₃	—	9	2	no	—
35	C₂×C₃	C₂×C₃×D₃	1	1	yes	characteristic
36	D₃	—	3	2	no	—
37	C₂²	—	9	1	no	—
38	C₄	—	9	1	no	—
39	C₄	AGL(1,ℤ/9)	1	1	yes	characteristic, direct product factor
40	C₃	—	3	1	no	—
41	C₃	C₃×C₄×D₃	1	1	yes	characteristic
42	C₂	—	9	2	no	—
43	C₂	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₄×3₋¹⁺²	C₂	semidirect
3. 3₋¹⁺²⋊C₄	C₂	semidirect
5. C₄×D₉	C₃	semidirect
7. AGL(1,ℤ/9)	C₄	direct (N × H)
13. C₄×C₉	C₂×C₃	semidirect
14. Dic₉	C₂×C₃	semidirect
15. 3₋¹⁺²	C₂×C₄	semidirect
22. D₉	C₃×C₄	semidirect
31. C₉	C₂×C₃×C₄	semidirect
39. 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
C₂×AGL(1,ℤ/9)	2
C₄×3₋¹⁺²	2
3₋¹⁺²⋊C₄	2
C₃×C₄×D₃	3
C₄×D₉	3

Subgroups — C4×AGL(1,ℤ/9) #6216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×AGL(1,ℤ/9) #6216