Subgroups — C₂²×C₃²×Dic₃ #89061

Order 432.

452 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×Dic₃	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃²×C₄	—	3	1	no	maximal
4	C₂²×C₃×Dic₃	C₃	1	4	yes	maximal, direct product factor
5	C₂²×C₃³	C₄	1	4	yes	complemented
6	C₂²×C₃³	C₂²	1	3	yes	—
7	C₃²×Dic₃	C₂²	1	4	yes	direct product factor
8	C₂³×C₃²	—	2	4	no	—
9	C₂³×C₃²	D₃	1	1	yes	characteristic
10	C₂³×C₃²	C₂×C₃	1	4	yes	—
11	C₂×C₃²×C₄	—	3	6	no	—
12	C₂×C₃×Dic₃	C₂×C₃	1	24	yes	direct product factor
13	C₂×C₃³	C₂×C₄	1	6	yes	complemented
14	C₂×C₃³	C₂³	1	1	yes	characteristic
15	C₂²×C₃×C₄	—	3	4	no	—
16	C₂²×Dic₃	C₃²	1	1	yes	characteristic, direct product factor
17	C₂²×C₃²	—	2	16	no	—
18	C₂²×C₃²	—	2	12	no	—
19	C₂²×C₃²	Dic₃	1	4	yes	direct product factor
20	C₂²×C₃²	C₃×C₄	1	16	yes	complemented
21	C₂²×C₃²	C₂×D₃	1	3	yes	—
22	C₂²×C₃²	C₂²×C₃	1	12	yes	—
23	C₃²×C₄	—	3	4	no	—
24	C₃×Dic₃	C₂²×C₃	1	16	yes	direct product factor
25	C₃³	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
26	C₂³×C₃	—	2	4	no	—
27	C₂³×C₃	C₃×D₃	1	4	yes	—
28	C₂³×C₃	C₂×C₃²	1	1	yes	characteristic
29	C₂×C₃×C₄	—	3	24	no	—
30	C₂×Dic₃	C₂×C₃²	1	6	yes	direct product factor
31	C₂×C₃²	—	2	24	no	—
32	C₂×C₃²	—	2	4	no	—
33	C₂×C₃²	C₂×Dic₃	1	6	yes	direct product factor
34	C₂×C₃²	C₂×C₃×C₄	1	24	yes	complemented
35	C₂×C₃²	C₂²×D₃	1	1	yes	characteristic
36	C₂×C₃²	C₂³×C₃	1	4	yes	—
37	C₂²×C₄	—	3	1	no	Sylow
38	C₂²×C₃	—	2	16	no	—
39	C₂²×C₃	—	2	12	no	—
40	C₂²×C₃	C₃×Dic₃	1	16	yes	direct product factor
41	C₂²×C₃	C₃²×C₄	1	4	yes	complemented
42	C₂²×C₃	C₂×C₃×D₃	1	12	yes	—
43	C₂²×C₃	C₂²×C₃²	1	3	yes	—
44	C₃×C₄	—	3	16	no	—
45	Dic₃	C₂²×C₃²	1	4	yes	direct product factor
46	C₃²	—	2	4	no	—
47	C₃²	C₂²×Dic₃	1	1	yes	characteristic, direct product factor
48	C₃²	C₂²×C₃×C₄	1	4	yes	complemented
49	C₂³	C₃²×D₃	1	1	yes	characteristic
50	C₂×C₄	—	3	6	no	—
51	C₂×C₃	—	2	24	no	—
52	C₂×C₃	—	2	4	no	—
53	C₂×C₃	C₂×C₃×Dic₃	1	24	yes	direct product factor
54	C₂×C₃	C₂×C₃²×C₄	1	6	yes	complemented
55	C₂×C₃	C₂²×C₃×D₃	1	4	yes	—
56	C₂×C₃	C₂³×C₃²	1	1	yes	characteristic
57	C₂²	C₃²×Dic₃	1	4	yes	direct product factor
58	C₂²	C₂×C₃²×D₃	1	3	yes	—
59	C₄	—	3	4	no	—
60	C₃	—	2	4	no	—
61	C₃	C₂²×C₃×Dic₃	1	4	yes	direct product factor
62	C₃	C₂²×C₃²×C₄	1	1	yes	characteristic, complemented
63	C₂	C₂×C₃²×Dic₃	1	6	yes	direct product factor
64	C₂	C₂²×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₃²	D₃
commutator, hypocenter	C₃	C₂²×C₃²×C₄
Frattini	C₂	C₂²×C₃²×D₃
Fitting, socle, prime radical	C₂³×C₃³	C₂
cosocle	C₂×C₃	C₂³×C₃²
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₃²

Complements

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

N	H ≅ G/N	type
2. C₂×C₃²×Dic₃	C₂	direct (N × H)
4. C₂²×C₃×Dic₃	C₃	direct (N × H)
5. C₂²×C₃³	C₄	semidirect
7. C₃²×Dic₃	C₂²	direct (N × H)
12. C₂×C₃×Dic₃	C₂×C₃	direct (N × H)
13. C₂×C₃³	C₂×C₄	semidirect
16. C₂²×Dic₃	C₃²	direct (N × H)
19. C₂²×C₃²	Dic₃	direct (N × H)
20. C₂²×C₃²	C₃×C₄	semidirect
24. C₃×Dic₃	C₂²×C₃	direct (N × H)
25. C₃³	C₂²×C₄	semidirect
30. C₂×Dic₃	C₂×C₃²	direct (N × H)
33. C₂×C₃²	C₂×Dic₃	direct (N × H)
34. C₂×C₃²	C₂×C₃×C₄	semidirect
40. C₂²×C₃	C₃×Dic₃	direct (N × H)
41. C₂²×C₃	C₃²×C₄	semidirect
45. Dic₃	C₂²×C₃²	direct (N × H)
47. C₃²	C₂²×Dic₃	direct (N × H)
48. C₃²	C₂²×C₃×C₄	semidirect
53. C₂×C₃	C₂×C₃×Dic₃	direct (N × H)
54. C₂×C₃	C₂×C₃²×C₄	semidirect
57. C₂²	C₃²×Dic₃	direct (N × H)
61. C₃	C₂²×C₃×Dic₃	direct (N × H)
62. C₃	C₂²×C₃²×C₄	semidirect
63. C₂	C₂×C₃²×Dic₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₃³	2
C₂×C₃²×Dic₃	2
C₂²×C₃²×C₄	3
C₂²×C₃×Dic₃	3

Subgroups — C22×C32×Dic3 #89061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×Dic₃ #89061