Subgroups — C₃×3₋¹⁺²⋊C₄ #66402

Order 324.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₃²×Dic₃	—	3	1	no	maximal
3	3₋¹⁺²⋊C₄	C₃	1	3	yes	maximal, direct product factor
4	C₃×Dic₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×3₋¹⁺²	C₄	1	1	yes	characteristic, complemented, Sylow
6	C₂×C₃³	D₃	1	1	yes	characteristic
7	C₂×3₋¹⁺²	—	2	3	no	—
8	C₂×3₋¹⁺²	C₂×C₃	1	3	yes	—
9	C₂×C₃×C₉	—	2	1	no	—
10	C₂×C₃×C₉	C₂×C₃	1	1	yes	characteristic
11	C₃²×C₄	—	9	1	no	—
12	C₃×Dic₃	—	3	1	no	—
13	C₃×Dic₃	—	3	3	no	—
14	Dic₉	C₃²	1	1	yes	characteristic, complemented
15	C₃³	Dic₃	1	1	yes	characteristic
16	3₋¹⁺²	—	2	3	no	—
17	3₋¹⁺²	C₃×C₄	1	3	yes	complemented
18	C₃×C₉	—	2	1	no	—
19	C₃×C₉	C₃×C₄	1	1	yes	characteristic, complemented
20	C₂×C₃²	—	3	1	no	—
21	C₂×C₃²	—	6	1	no	—
22	C₂×C₃²	C₃×D₃	1	1	yes	characteristic
23	C₂×C₃²	C₃×D₃	1	3	yes	—
24	C₂×C₉	—	2	3	no	—
25	C₂×C₉	—	2	1	no	—
26	C₂×C₉	C₂×C₃²	1	1	yes	characteristic
27	C₃×C₄	—	9	3	no	—
28	C₃×C₄	—	9	1	no	—
29	Dic₃	—	3	1	no	—
30	C₃²	—	3	1	no	—
31	C₃²	—	6	1	no	—
32	C₃²	C₃×Dic₃	1	1	yes	characteristic
33	C₃²	C₃×Dic₃	1	3	yes	—
34	C₉	—	2	3	no	—
35	C₉	—	2	1	no	—
36	C₉	C₃²×C₄	1	1	yes	characteristic, complemented
37	C₂×C₃	—	2	1	no	—
38	C₂×C₃	—	3	3	no	—
39	C₂×C₃	AGL(1,ℤ/9)	1	1	yes	characteristic
40	C₂×C₃	C₃²×D₃	1	1	yes	characteristic
41	C₄	—	9	1	no	Sylow
42	C₃	—	2	1	no	—
43	C₃	—	3	3	no	—
44	C₃	3₋¹⁺²⋊C₄	1	1	yes	characteristic, direct product factor
45	C₃	C₃²×Dic₃	1	1	yes	characteristic
46	C₂	C₃×AGL(1,ℤ/9)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	AGL(1,ℤ/9)
commutator, hypocenter	C₉	C₃²×C₄
Frattini	C₂×C₃	C₃²×D₃
Fitting	C₂×C₃×3₋¹⁺²	C₂
socle	C₂×C₃²	C₃×D₃
prime radical	C₂×C₃³	D₃
cosocle	C₂×C₉	C₂×C₃²
LCS	G ▷ C₉
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
3. 3₋¹⁺²⋊C₄	C₃	direct (N × H)
4. C₃×Dic₉	C₃	semidirect
5. C₃×3₋¹⁺²	C₄	semidirect
14. Dic₉	C₃²	semidirect
17. 3₋¹⁺²	C₃×C₄	semidirect
19. C₃×C₉	C₃×C₄	semidirect
36. C₉	C₃²×C₄	semidirect
44. C₃	3₋¹⁺²⋊C₄	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×C₃×3₋¹⁺²	2
C₃²×Dic₃	3
3₋¹⁺²⋊C₄	3
C₃×Dic₉	3

Subgroups — C3×3−1+2⋊C4 #66402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×3₋¹⁺²⋊C₄ #66402