Subgroups — C₂×C₃²×Dic₆ #89053

Order 432.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×Dic₃	C₂	1	2	yes	maximal
3	C₃²×Dic₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₃²×Q₈	—	3	1	no	maximal
5	C₂×C₃×Dic₆	C₃	1	4	yes	maximal, direct product factor
6	C₂²×C₃³	C₂²	1	1	yes	characteristic
7	C₃³×C₄	C₂²	1	2	yes	—
8	C₃²×Dic₃	C₂²	1	4	yes	—
9	C₃²×Q₈	—	3	4	no	—
10	C₂×C₃²×C₄	—	2	4	no	—
11	C₂×C₃²×C₄	—	3	2	no	—
12	C₂×C₃²×C₄	D₃	1	1	yes	characteristic
13	C₂×C₃²×C₄	C₂×C₃	1	4	yes	—
14	C₂×C₃×Dic₃	C₂×C₃	1	8	yes	—
15	C₃×Dic₆	C₂×C₃	1	16	yes	direct product factor
16	C₂×C₃³	Q₈	1	2	yes	complemented
17	C₂×C₃³	C₂³	1	1	yes	characteristic
18	C₂×C₃×Q₈	—	3	4	no	—
19	C₂×Dic₆	C₃²	1	1	yes	characteristic, direct product factor
20	C₂²×C₃²	—	2	4	no	—
21	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
22	C₂²×C₃²	C₂²×C₃	1	4	yes	—
23	C₃²×C₄	—	2	8	no	—
24	C₃²×C₄	—	3	4	no	—
25	C₃²×C₄	C₂×D₃	1	2	yes	—
26	C₃²×C₄	C₂²×C₃	1	8	yes	—
27	C₃×Dic₃	C₂²×C₃	1	16	yes	—
28	C₃³	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
29	C₃×Q₈	—	3	16	no	—
30	C₂×C₃×C₄	—	2	4	no	—
31	C₂×C₃×C₄	—	3	8	no	—
32	C₂×C₃×C₄	C₃×D₃	1	4	yes	—
33	C₂×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
34	C₂×Dic₃	C₂×C₃²	1	2	yes	—
35	Dic₆	C₂×C₃²	1	4	yes	direct product factor
36	C₂×C₃²	—	2	8	no	—
37	C₂×C₃²	—	2	4	no	—
38	C₂×C₃²	Dic₆	1	2	yes	direct product factor
39	C₂×C₃²	C₃×Q₈	1	8	yes	complemented
40	C₂×C₃²	C₂²×D₃	1	1	yes	characteristic
41	C₂×C₃²	C₂³×C₃	1	4	yes	—
42	C₂×Q₈	—	3	1	no	Sylow
43	C₂²×C₃	—	2	4	no	—
44	C₂²×C₃	C₂×C₃×D₃	1	4	yes	—
45	C₂²×C₃	C₂²×C₃²	1	1	yes	characteristic
46	C₃×C₄	—	2	8	no	—
47	C₃×C₄	—	3	16	no	—
48	C₃×C₄	C₂×C₃×D₃	1	8	yes	—
49	C₃×C₄	C₂²×C₃²	1	2	yes	—
50	Dic₃	C₂²×C₃²	1	4	yes	—
51	C₃²	—	2	4	no	—
52	C₃²	C₂×Dic₆	1	1	yes	characteristic, direct product factor
53	C₃²	C₂×C₃×Q₈	1	4	yes	complemented
54	Q₈	—	3	4	no	—
55	C₂×C₄	—	3	2	no	—
56	C₂×C₄	C₃²×D₃	1	1	yes	characteristic
57	C₂×C₃	—	2	8	no	—
58	C₂×C₃	—	2	4	no	—
59	C₂×C₃	C₃×Dic₆	1	8	yes	direct product factor
60	C₂×C₃	C₃²×Q₈	1	2	yes	complemented
61	C₂×C₃	C₂²×C₃×D₃	1	4	yes	—
62	C₂×C₃	C₂³×C₃²	1	1	yes	characteristic
63	C₂²	C₂×C₃²×D₃	1	1	yes	characteristic
64	C₄	—	3	4	no	—
65	C₄	C₂×C₃²×D₃	1	2	yes	—
66	C₃	—	2	4	no	—
67	C₃	C₂×C₃×Dic₆	1	4	yes	direct product factor
68	C₃	C₂×C₃²×Q₈	1	1	yes	characteristic, complemented
69	C₂	C₃²×Dic₆	1	2	yes	direct product factor
70	C₂	C₂²×C₃²×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃²	C₂×D₃
commutator, cosocle	C₂×C₃	C₂³×C₃²
Frattini	C₂	C₂²×C₃²×D₃
Fitting	C₂×C₃³×C₄	C₂
socle, prime radical	C₂²×C₃³	C₂²
hypercenter	C₂×C₃²×C₄	D₃
hypocenter	C₃	C₂×C₃²×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃² ◁ C₂×C₃²×C₄

Complements

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

N	H ≅ G/N	type
3. C₃²×Dic₆	C₂	direct (N × H)
5. C₂×C₃×Dic₆	C₃	direct (N × H)
15. C₃×Dic₆	C₂×C₃	direct (N × H)
16. C₂×C₃³	Q₈	semidirect
19. C₂×Dic₆	C₃²	direct (N × H)
28. C₃³	C₂×Q₈	semidirect
35. Dic₆	C₂×C₃²	direct (N × H)
38. C₂×C₃²	Dic₆	direct (N × H)
39. C₂×C₃²	C₃×Q₈	semidirect
52. C₃²	C₂×Dic₆	direct (N × H)
53. C₃²	C₂×C₃×Q₈	semidirect
59. C₂×C₃	C₃×Dic₆	direct (N × H)
60. C₂×C₃	C₃²×Q₈	semidirect
67. C₃	C₂×C₃×Dic₆	direct (N × H)
68. C₃	C₂×C₃²×Q₈	semidirect
69. C₂	C₃²×Dic₆	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×C₃³×C₄	2
C₂×C₃²×Dic₃	2
C₃²×Dic₆	2
C₂×C₃²×Q₈	3
C₂×C₃×Dic₆	3

Subgroups — C2×C32×Dic6 #89053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×Dic₆ #89053