Subgroups — C₈×(C₃⋊D₃)⋊C₃ #88468

Order 432.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×(C₃⋊D₃)⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₃⋊(C₃⋊C₈))⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃×D₃×C₈	—	3	1	no	maximal
6	C₂×(C₃⋊D₃)⋊C₃	C₄	1	1	yes	characteristic
7	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	(C₃⋊Dic₃)⋊C₃	C₄	1	1	yes	characteristic
9	C₄×C₃⋊D₃	C₂×C₃	1	1	yes	characteristic
10	C₃×C₄×D₃	—	3	1	no	—
11	C₃²×C₈	—	2	1	no	—
12	C₃²×C₈	D₃	1	1	yes	characteristic, complemented
13	C₃²×C₈	C₂×C₃	1	1	yes	characteristic, complemented
14	C₃⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃×C₃⋊C₈	—	3	1	no	—
16	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
17	(C₃⋊D₃)⋊C₃	C₈	1	2	yes	direct product factor
18	C₂×C₃×C₈	—	9	1	no	—
19	D₃×C₈	—	3	1	no	—
20	D₃×C₈	—	9	1	no	—
21	C₂×C₃⋊D₃	C₃×C₄	1	1	yes	characteristic
22	C₂×C₃×D₃	—	3	1	no	—
23	C₃²×C₄	—	2	1	no	—
24	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
25	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
26	C₃⋊Dic₃	C₃×C₄	1	1	yes	characteristic
27	C₃×Dic₃	—	3	1	no	—
28	H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic, complemented, Sylow
29	C₂×C₃×C₄	—	9	1	no	—
30	C₄×D₃	—	3	1	no	—
31	C₄×D₃	—	9	1	no	—
32	C₃×C₈	—	3	1	no	—
33	C₃×C₈	—	6	1	no	—
34	C₃×C₈	—	6	1	no	—
35	C₃×C₈	—	9	1	no	—
36	C₃×C₈	C₃×D₃	1	1	yes	characteristic
37	C₃⋊C₈	—	3	1	no	—
38	C₃⋊C₈	—	9	1	no	—
39	C₂×C₃²	—	2	1	no	—
40	C₂×C₃²	C₄×D₃	1	1	yes	characteristic
41	C₂×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
42	C₃⋊D₃	C₃×C₈	1	2	yes	complemented
43	C₃×D₃	—	3	2	no	—
44	C₂×C₈	—	9	1	no	Sylow
45	C₂²×C₃	—	9	1	no	—
46	C₂×D₃	—	3	1	no	—
47	C₂×D₃	—	9	1	no	—
48	C₃×C₄	—	3	1	no	—
49	C₃×C₄	—	6	1	no	—
50	C₃×C₄	—	6	1	no	—
51	C₃×C₄	—	9	1	no	—
52	C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
53	Dic₃	—	3	1	no	—
54	Dic₃	—	9	1	no	—
55	C₃²	—	2	1	no	—
56	C₃²	D₃×C₈	1	1	yes	characteristic, complemented
57	C₃²	C₂×C₃×C₈	1	1	yes	characteristic, complemented
58	C₂×C₄	—	9	1	no	—
59	C₈	—	9	1	no	—
60	C₈	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, direct product factor
61	C₂×C₃	—	3	1	no	—
62	C₂×C₃	—	6	1	no	—
63	C₂×C₃	—	6	1	no	—
64	C₂×C₃	—	9	2	no	—
65	C₂×C₃	C₃×C₄×D₃	1	1	yes	characteristic
66	D₃	—	3	2	no	—
67	D₃	—	9	2	no	—
68	C₂²	—	9	1	no	—
69	C₄	—	9	1	no	—
70	C₄	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
71	C₃	—	3	1	no	—
72	C₃	—	6	1	no	—
73	C₃	—	6	1	no	—
74	C₃	C₃×D₃×C₈	1	1	yes	characteristic
75	C₂	—	9	2	no	—
76	C₂	C₄×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₈	(C₃⋊D₃)⋊C₃
commutator, hypocenter	C₃²	C₂×C₃×C₈
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₈×H(3,𝔽₃)	C₂
socle	C₂×C₃	C₃×C₄×D₃
prime radical	C₂×(C₃⋊D₃)⋊C₃	C₄
cosocle	C₃²×C₄	C₂²×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₈×H(3,𝔽₃)	C₂	semidirect
3. (C₃⋊(C₃⋊C₈))⋊C₃	C₂	semidirect
4. C₈×C₃⋊D₃	C₃	semidirect
12. C₃²×C₈	D₃	semidirect
13. C₃²×C₈	C₂×C₃	semidirect
14. C₃⋊(C₃⋊C₈)	C₂×C₃	semidirect
17. (C₃⋊D₃)⋊C₃	C₈	direct (N × H)
28. H(3,𝔽₃)	C₂×C₈	semidirect
42. C₃⋊D₃	C₃×C₈	semidirect
56. C₃²	D₃×C₈	semidirect
57. C₃²	C₂×C₃×C₈	semidirect
60. C₈	(C₃⋊D₃)⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×(C₃⋊D₃)⋊C₃	2
C₈×H(3,𝔽₃)	2
(C₃⋊(C₃⋊C₈))⋊C₃	2
C₈×C₃⋊D₃	3
C₃×D₃×C₈	3

Subgroups — C8×(C3⋊D3)⋊C3 #88468

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×(C₃⋊D₃)⋊C₃ #88468