Subgroups — C₂×(C₃⋊(C₃⋊C₈))⋊C₃ #88489

Order 432.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, hypercenter	C₂×C₄	(C₃⋊D₃)⋊C₃
commutator, hypocenter	C₃²	C₂×C₃×C₈
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₂×C₄×H(3,𝔽₃)	C₂
socle	C₂²×C₃	C₃×Dic₃
prime radical	C₂²×H(3,𝔽₃)	C₄
cosocle	C₃²×C₄	C₂²×C₃
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. (C₃⋊(C₃⋊C₈))⋊C₃	C₂	direct (N × H)
3. C₂×C₃⋊(C₃⋊C₈)	C₃	semidirect
11. C₃⋊(C₃⋊C₈)	C₂×C₃	semidirect
13. C₂×H(3,𝔽₃)	C₈	semidirect
27. H(3,𝔽₃)	C₂×C₈	semidirect
37. C₂×C₃²	C₃⋊C₈	semidirect
38. C₂×C₃²	C₃×C₈	semidirect
55. C₃²	C₂×C₃⋊C₈	semidirect
56. C₃²	C₂×C₃×C₈	semidirect
74. C₂	(C₃⋊(C₃⋊C₈))⋊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
C₂×C₄×H(3,𝔽₃)	2
(C₃⋊(C₃⋊C₈))⋊C₃	2
C₂×C₃⋊(C₃⋊C₈)	3
C₂×C₃×C₃⋊C₈	3

Subgroups — C2×(C3⋊(C3⋊C8))⋊C3 #88489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₃⋊(C₃⋊C₈))⋊C₃ #88489