Subgroups — D₄×C₃⋊(C₃⋊D₃) #89077

Order 432.

756 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
2	C₃³×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#6312	C₂	1	2	yes	maximal, complemented
4	C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₃⋊D₃	—	3	13	no	maximal
7	C₂²×C₃³	C₂²	1	2	yes	complemented
8	C₂×C₃⋊(C₃⋊D₃)	—	2	4	no	—
9	C₂×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	complemented
11	C₃³×C₄	C₂²	1	1	yes	characteristic, complemented
12	C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₃⋊D₃	—	3	26	no	—
14	C₃²×D₄	D₃	1	13	yes	complemented
15	#638	—	3	26	no	—
16	C₄⋊(C₃⋊D₃)	—	3	13	no	—
17	C₄×C₃⋊D₃	—	3	13	no	—
18	C₂×C₃³	—	2	2	no	—
19	C₂×C₃³	C₂³	1	1	yes	characteristic
20	C₃⋊(C₃⋊D₃)	—	2	2	no	—
21	C₃⋊(C₃⋊D₃)	D₄	1	2	yes	direct product factor
22	D₃×D₄	—	9	13	no	—
23	C₂²×C₃²	C₂×D₃	1	26	yes	complemented
24	C₂×C₃⋊D₃	—	3	13	no	—
25	C₂×C₃⋊D₃	—	3	26	no	—
26	C₂×C₃⋊D₃	—	6	52	no	—
27	C₃²×C₄	C₂×D₃	1	13	yes	complemented
28	C₃⋊Dic₃	—	3	13	no	—
29	C₃³	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
30	C₂²×D₃	—	9	26	no	—
31	C₃×D₄	C₃⋊D₃	1	13	yes	complemented
32	#67	—	9	26	no	—
33	D₁₂	—	9	13	no	—
34	C₄×D₃	—	9	13	no	—
35	C₂×C₃²	—	2	26	no	—
36	C₂×C₃²	C₂²×D₃	1	13	yes	—
37	C₃⋊D₃	—	3	26	no	—
38	C₃⋊D₃	—	6	26	no	—
39	C₂×D₄	—	27	1	no	Sylow
40	C₂²×C₃	C₂×C₃⋊D₃	1	26	yes	complemented
41	C₂×D₃	—	9	13	no	—
42	C₂×D₃	—	9	26	no	—
43	C₂×D₃	—	18	52	no	—
44	C₃×C₄	C₂×C₃⋊D₃	1	13	yes	complemented
45	Dic₃	—	9	13	no	—
46	C₃²	D₃×D₄	1	13	yes	complemented
47	C₂³	—	27	2	no	—
48	D₄	—	27	2	no	—
49	D₄	—	27	1	no	—
50	D₄	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
51	C₂×C₄	—	27	1	no	—
52	C₂×C₃	—	2	26	no	—
53	C₂×C₃	C₂²×C₃⋊D₃	1	13	yes	—
54	D₃	—	9	26	no	—
55	D₃	—	18	26	no	—
56	C₂²	—	27	1	no	—
57	C₂²	—	27	2	no	—
58	C₂²	—	54	4	no	—
59	C₂²	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	complemented
60	C₄	—	27	1	no	—
61	C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
62	C₃	D₄×C₃⋊D₃	1	13	yes	complemented
63	C₂	—	2	2	no	—
64	C₂	—	27	2	no	—
65	C₂	—	54	2	no	—
66	C₂	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂×C₃³	C₂³
Fitting	C₃³×D₄	C₂
hypercenter	D₄	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₂×D₄
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃⋊(C₃⋊D₃)	C₂	semidirect
2. C₃³×D₄	C₂	semidirect
3. #6312	C₂	semidirect
4. C₄⋊(C₃⋊(C₃⋊D₃))	C₂	semidirect
5. C₄×C₃⋊(C₃⋊D₃)	C₂	semidirect
7. C₂²×C₃³	C₂²	semidirect
10. C₂×C₃⋊(C₃⋊D₃)	C₂²	semidirect
11. C₃³×C₄	C₂²	semidirect
12. C₃⋊(C₃⋊Dic₃)	C₂²	semidirect
14. C₃²×D₄	D₃	semidirect
21. C₃⋊(C₃⋊D₃)	D₄	direct (N × H)
23. C₂²×C₃²	C₂×D₃	semidirect
27. C₃²×C₄	C₂×D₃	semidirect
29. C₃³	C₂×D₄	semidirect
31. C₃×D₄	C₃⋊D₃	semidirect
40. C₂²×C₃	C₂×C₃⋊D₃	semidirect
44. C₃×C₄	C₂×C₃⋊D₃	semidirect
46. C₃²	D₃×D₄	semidirect
50. D₄	C₃⋊(C₃⋊D₃)	direct (N × H)
59. C₂²	C₂×C₃⋊(C₃⋊D₃)	semidirect
61. C₄	C₂×C₃⋊(C₃⋊D₃)	semidirect
62. C₃	D₄×C₃⋊D₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₃⋊D₃)	2
C₃³×D₄	2
#6312	2
C₄⋊(C₃⋊(C₃⋊D₃))	2
C₄×C₃⋊(C₃⋊D₃)	2
D₄×C₃⋊D₃	3

Subgroups — D4×C3⋊(C3⋊D3) #89077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×C₃⋊(C₃⋊D₃) #89077