Subgroups — C₃²×D₃×Q₈ #89059

Order 432.

276 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₃³×Q₈	C₂	semidirect
3. C₃²×Dic₆	C₂	semidirect
5. C₃×D₃×Q₈	C₃	direct (N × H)
11. C₃²×Q₈	D₃	direct (N × H)
12. C₃²×Q₈	C₂×C₃	semidirect
15. C₃×Dic₆	C₂×C₃	semidirect
17. C₃²×D₃	Q₈	direct (N × H)
19. D₃×Q₈	C₃²	direct (N × H)
27. C₃³	C₂×Q₈	semidirect
30. C₃×Q₈	C₃×D₃	direct (N × H)
31. C₃×Q₈	C₂×C₃²	semidirect
34. Dic₆	C₂×C₃²	semidirect
39. C₃×D₃	C₃×Q₈	direct (N × H)
49. C₃²	D₃×Q₈	direct (N × H)
50. C₃²	C₂×C₃×Q₈	semidirect
52. Q₈	C₃²×D₃	direct (N × H)
58. D₃	C₃²×Q₈	direct (N × H)
63. C₃	C₃×D₃×Q₈	direct (N × H)
64. C₃	C₂×C₃²×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₃³×Q₈	2
C₃²×C₄×D₃	2
C₃²×Dic₆	2
C₂×C₃²×Q₈	3
C₃×D₃×Q₈	3

Subgroups — C32×D3×Q8 #89059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×D₃×Q₈ #89059