Subgroups — C₂×D₃×3₋¹⁺² #66449

Order 324.

102 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, complemented
2	D₃×3₋¹⁺²	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃²×D₃	C₃	1	1	yes	maximal, characteristic
4	C₂²×3₋¹⁺²	—	3	1	no	maximal
5	C₂×D₃×C₉	C₃	1	3	yes	maximal, complemented
6	C₃×3₋¹⁺²	C₂²	1	1	yes	characteristic, complemented, Sylow
7	C₂×C₃³	C₂×C₃	1	1	yes	characteristic
8	C₃²×D₃	C₂×C₃	1	2	yes	—
9	C₂×3₋¹⁺²	—	2	1	no	—
10	C₂×3₋¹⁺²	—	2	3	no	—
11	C₂×3₋¹⁺²	—	3	2	no	—
12	C₂×3₋¹⁺²	D₃	1	1	yes	characteristic, direct product factor
13	C₂×C₃×C₉	C₂×C₃	1	3	yes	complemented
14	D₃×C₉	C₂×C₃	1	6	yes	complemented
15	C₂²×C₃²	—	3	1	no	—
16	C₂×C₃×D₃	—	3	1	no	—
17	C₂×C₃×D₃	C₃²	1	1	yes	characteristic
18	C₂²×C₉	—	3	3	no	—
19	C₃³	C₂²×C₃	1	1	yes	characteristic
20	3₋¹⁺²	—	2	1	no	—
21	3₋¹⁺²	—	2	3	no	—
22	3₋¹⁺²	C₂×D₃	1	1	yes	characteristic, direct product factor
23	C₃×C₉	C₂²×C₃	1	3	yes	complemented
24	C₂×C₃²	—	2	1	no	—
25	C₂×C₃²	—	3	1	no	—
26	C₂×C₃²	—	3	2	no	—
27	C₂×C₃²	—	6	1	no	—
28	C₂×C₃²	C₃×D₃	1	1	yes	characteristic
29	C₂×C₃²	C₂×C₃²	1	1	yes	characteristic
30	C₃×D₃	—	3	2	no	—
31	C₃×D₃	C₂×C₃²	1	2	yes	—
32	C₂×C₉	—	2	3	no	—
33	C₂×C₉	—	3	6	no	—
34	C₂×C₉	C₃×D₃	1	3	yes	complemented
35	C₂²×C₃	—	3	1	no	—
36	C₂²×C₃	—	9	1	no	—
37	C₂×D₃	3₋¹⁺²	1	1	yes	characteristic, direct product factor
38	C₃²	—	2	1	no	—
39	C₃²	—	3	1	no	—
40	C₃²	—	6	1	no	—
41	C₃²	C₂×C₃×D₃	1	1	yes	characteristic
42	C₃²	C₂²×C₃²	1	1	yes	characteristic
43	C₉	—	2	3	no	—
44	C₉	C₂×C₃×D₃	1	3	yes	complemented
45	C₂×C₃	—	2	1	no	—
46	C₂×C₃	—	3	2	no	—
47	C₂×C₃	—	3	1	no	—
48	C₂×C₃	—	6	1	no	—
49	C₂×C₃	—	9	2	no	—
50	C₂×C₃	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
51	C₂×C₃	C₃²×D₃	1	1	yes	characteristic
52	D₃	C₂×3₋¹⁺²	1	2	yes	direct product factor
53	C₂²	—	3	1	no	Sylow
54	C₃	—	2	1	no	—
55	C₃	—	3	1	no	—
56	C₃	—	6	1	no	—
57	C₃	C₂²×3₋¹⁺²	1	1	yes	characteristic, complemented
58	C₃	C₂×C₃²×D₃	1	1	yes	characteristic
59	C₂	—	3	2	no	—
60	C₂	D₃×3₋¹⁺²	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center	C₂×C₃	C₃²×D₃
commutator, cosocle	C₃²	C₂²×C₃²
Frattini	C₃	C₂×C₃²×D₃
Fitting	C₂×C₃×3₋¹⁺²	C₂
socle	C₂×C₃²	C₂×C₃²
prime radical	C₂×C₃²×D₃	C₃
hypercenter	C₂×3₋¹⁺²	D₃
hypocenter	C₃	C₂²×3₋¹⁺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
1. C₂×C₃×3₋¹⁺²	C₂	semidirect
2. D₃×3₋¹⁺²	C₂	direct (N × H)
5. C₂×D₃×C₉	C₃	semidirect
6. C₃×3₋¹⁺²	C₂²	semidirect
12. C₂×3₋¹⁺²	D₃	direct (N × H)
13. C₂×C₃×C₉	C₂×C₃	semidirect
14. D₃×C₉	C₂×C₃	semidirect
22. 3₋¹⁺²	C₂×D₃	direct (N × H)
23. C₃×C₉	C₂²×C₃	semidirect
34. C₂×C₉	C₃×D₃	semidirect
37. C₂×D₃	3₋¹⁺²	direct (N × H)
44. C₉	C₂×C₃×D₃	semidirect
50. C₂×C₃	C₂×3₋¹⁺²	semidirect
52. D₃	C₂×3₋¹⁺²	direct (N × H)
57. C₃	C₂²×3₋¹⁺²	semidirect
60. C₂	D₃×3₋¹⁺²	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
D₃×3₋¹⁺²	2
C₂×C₃²×D₃	3
C₂²×3₋¹⁺²	3
C₂×D₃×C₉	3

Subgroups — C2×D3×3−1+2 #66449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₃×3₋¹⁺² #66449