Subgroups — Dic₃×3₋¹⁺² #66403

Order 324.

70 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	C₃	1	1	yes	maximal, characteristic
3	C₄×3₋¹⁺²	—	3	1	no	maximal
4	C₉×Dic₃	C₃	1	3	yes	maximal, complemented
5	C₃×3₋¹⁺²	C₄	1	1	yes	characteristic, complemented, Sylow
6	C₂×C₃³	C₂×C₃	1	1	yes	characteristic
7	C₂×3₋¹⁺²	—	2	1	no	—
8	C₂×3₋¹⁺²	—	2	3	no	—
9	C₂×3₋¹⁺²	D₃	1	1	yes	characteristic
10	C₂×C₃×C₉	C₂×C₃	1	3	yes	—
11	C₃²×C₄	—	3	1	no	—
12	C₃×Dic₃	—	3	1	no	—
13	C₃×Dic₃	C₃²	1	1	yes	characteristic
14	C₄×C₉	—	3	3	no	—
15	C₃³	C₃×C₄	1	1	yes	characteristic
16	3₋¹⁺²	—	2	1	no	—
17	3₋¹⁺²	—	2	3	no	—
18	3₋¹⁺²	Dic₃	1	1	yes	characteristic, direct product factor
19	C₃×C₉	C₃×C₄	1	3	yes	complemented
20	C₂×C₃²	—	2	1	no	—
21	C₂×C₃²	—	3	1	no	—
22	C₂×C₃²	—	6	1	no	—
23	C₂×C₃²	C₃×D₃	1	1	yes	characteristic
24	C₂×C₃²	C₂×C₃²	1	1	yes	characteristic
25	C₂×C₉	—	2	3	no	—
26	C₂×C₉	C₃×D₃	1	3	yes	—
27	C₃×C₄	—	3	1	no	—
28	C₃×C₄	—	9	1	no	—
29	Dic₃	3₋¹⁺²	1	1	yes	characteristic, direct product factor
30	C₃²	—	2	1	no	—
31	C₃²	—	3	1	no	—
32	C₃²	—	6	1	no	—
33	C₃²	C₃×Dic₃	1	1	yes	characteristic
34	C₃²	C₃²×C₄	1	1	yes	characteristic
35	C₉	—	2	3	no	—
36	C₉	C₃×Dic₃	1	3	yes	complemented
37	C₂×C₃	—	2	1	no	—
38	C₂×C₃	—	3	1	no	—
39	C₂×C₃	—	6	1	no	—
40	C₂×C₃	C₂×3₋¹⁺²	1	1	yes	characteristic
41	C₂×C₃	C₃²×D₃	1	1	yes	characteristic
42	C₄	—	3	1	no	Sylow
43	C₃	—	2	1	no	—
44	C₃	—	3	1	no	—
45	C₃	—	6	1	no	—
46	C₃	C₄×3₋¹⁺²	1	1	yes	characteristic, complemented
47	C₃	C₃²×Dic₃	1	1	yes	characteristic
48	C₂	D₃×3₋¹⁺²	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₉×Dic₃	C₃	semidirect
5. C₃×3₋¹⁺²	C₄	semidirect
18. 3₋¹⁺²	Dic₃	direct (N × H)
19. C₃×C₉	C₃×C₄	semidirect
29. Dic₃	3₋¹⁺²	direct (N × H)
36. C₉	C₃×Dic₃	semidirect
46. C₃	C₄×3₋¹⁺²	semidirect

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
C₄×3₋¹⁺²	3
C₉×Dic₃	3

Subgroups — Dic3×3−1+2 #66403

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃×3₋¹⁺² #66403