Subgroups — C₃⋊((C₃⋊(C₃⋊C₈))⋊C₃) #10619721

Order 648.

149 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊(C₃⋊C₈))	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊(C₃⋊C₈)	—	3	1	no	maximal
4	(C₃⋊(C₃⋊C₈))⋊C₃	—	3	3	no	maximal
5	C₂×C₃×H(3,𝔽₃)	C₄	1	1	yes	characteristic
6	C₃³×C₄	—	2	1	no	—
7	C₃³×C₄	D₃	1	1	yes	characteristic
8	C₃³×C₄	C₂×C₃	1	1	yes	characteristic
9	C₄×H(3,𝔽₃)	—	2	3	no	—
10	C₄×H(3,𝔽₃)	D₃	1	3	yes	—
11	C₃×H(3,𝔽₃)	C₈	1	1	yes	characteristic, complemented, Sylow
12	C₃⋊(C₃⋊C₈)	—	3	1	no	—
13	C₃⋊(C₃⋊C₈)	—	3	3	no	—
14	C₃⋊(C₃⋊C₈)	—	9	1	no	—
15	C₃⋊(C₃⋊C₈)	—	9	2	no	—
16	C₃×C₃⋊C₈	—	9	1	no	—
17	C₃×C₃⋊C₈	—	9	2	no	—
18	C₃×C₃⋊C₈	—	9	1	no	—
19	C₂×C₃³	—	2	1	no	—
20	C₂×C₃³	Dic₃	1	1	yes	characteristic
21	C₂×C₃³	C₃×C₄	1	1	yes	characteristic
22	C₂×H(3,𝔽₃)	—	2	3	no	—
23	C₂×H(3,𝔽₃)	Dic₃	1	3	yes	—
24	C₃²×C₄	—	2	1	no	—
25	C₃²×C₄	—	2	3	no	—
26	C₃²×C₄	—	3	1	no	—
27	C₃²×C₄	—	3	2	no	—
28	C₃²×C₄	—	3	1	no	—
29	C₃²×C₄	—	6	1	no	—
30	C₃²×C₄	—	6	2	no	—
31	C₃²×C₄	—	6	2	no	—
32	C₃²×C₄	C₃×D₃	1	3	yes	—
33	C₃²×C₄	C₃×D₃	1	1	yes	characteristic
34	C₃²×C₄	C₃⋊D₃	1	1	yes	characteristic
35	C₃³	—	2	1	no	—
36	C₃³	C₃⋊C₈	1	1	yes	characteristic, complemented
37	C₃³	C₃×C₈	1	1	yes	characteristic, complemented
38	H(3,𝔽₃)	—	2	3	no	—
39	H(3,𝔽₃)	C₃⋊C₈	1	3	yes	complemented
40	C₃×C₈	—	27	1	no	—
41	C₃⋊C₈	—	9	1	no	—
42	C₃⋊C₈	—	9	2	no	—
43	C₃⋊C₈	—	9	1	no	—
44	C₃⋊C₈	—	27	3	no	—
45	C₂×C₃²	—	2	1	no	—
46	C₂×C₃²	—	2	3	no	—
47	C₂×C₃²	—	3	1	no	—
48	C₂×C₃²	—	3	2	no	—
49	C₂×C₃²	—	3	1	no	—
50	C₂×C₃²	—	6	1	no	—
51	C₂×C₃²	—	6	2	no	—
52	C₂×C₃²	—	6	2	no	—
53	C₂×C₃²	C₃×Dic₃	1	3	yes	—
54	C₂×C₃²	C₃×Dic₃	1	1	yes	characteristic
55	C₂×C₃²	C₃⋊Dic₃	1	1	yes	characteristic
56	C₃×C₄	—	3	3	no	—
57	C₃×C₄	—	6	1	no	—
58	C₃×C₄	—	6	1	no	—
59	C₃×C₄	—	6	3	no	—
60	C₃×C₄	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
61	C₃×C₄	(C₃⋊D₃)⋊C₃	1	2	yes	—
62	C₃×C₄	C₃×C₃⋊D₃	1	1	yes	characteristic
63	C₃²	—	2	1	no	—
64	C₃²	—	2	3	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	3	2	no	—
67	C₃²	—	3	1	no	—
68	C₃²	—	6	1	no	—
69	C₃²	—	6	2	no	—
70	C₃²	—	6	2	no	—
71	C₃²	C₃×C₃⋊C₈	1	1	yes	characteristic
72	C₃²	C₃×C₃⋊C₈	1	3	yes	complemented
73	C₃²	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic, complemented
74	C₈	—	27	1	no	Sylow
75	C₂×C₃	—	3	3	no	—
76	C₂×C₃	—	6	1	no	—
77	C₂×C₃	—	6	3	no	—
78	C₂×C₃	—	6	1	no	—
79	C₂×C₃	(C₃⋊Dic₃)⋊C₃	1	1	yes	characteristic
80	C₂×C₃	(C₃⋊Dic₃)⋊C₃	1	2	yes	—
81	C₂×C₃	C₃×C₃⋊Dic₃	1	1	yes	characteristic
82	C₄	C₃⋊((C₃⋊D₃)⋊C₃)	1	1	yes	characteristic
83	C₃	—	3	3	no	—
84	C₃	—	6	1	no	—
85	C₃	—	6	3	no	—
86	C₃	—	6	1	no	—
87	C₃	(C₃⋊(C₃⋊C₈))⋊C₃	1	1	yes	characteristic, complemented
88	C₃	(C₃⋊(C₃⋊C₈))⋊C₃	1	2	yes	complemented
89	C₃	C₃×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
90	C₂	C₃⋊((C₃⋊Dic₃)⋊C₃)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	C₃⋊((C₃⋊D₃)⋊C₃)
commutator, hypocenter	C₃³	C₃×C₈
Frattini	C₃×C₄	C₃×C₃⋊D₃
Fitting	C₃×C₄×H(3,𝔽₃)	C₂
socle	C₂×C₃²	C₃×Dic₃
prime radical	C₂×C₃×H(3,𝔽₃)	C₄
cosocle	C₃³×C₄	C₂×C₃
LCS	G ▷ C₃³
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊(C₃⋊C₈))	C₃	semidirect
11. C₃×H(3,𝔽₃)	C₈	semidirect
36. C₃³	C₃⋊C₈	semidirect
37. C₃³	C₃×C₈	semidirect
39. H(3,𝔽₃)	C₃⋊C₈	semidirect
72. C₃²	C₃×C₃⋊C₈	semidirect
73. C₃²	C₃⋊(C₃⋊C₈)	semidirect
87. C₃	(C₃⋊(C₃⋊C₈))⋊C₃	semidirect
88. C₃	(C₃⋊(C₃⋊C₈))⋊C₃	semidirect

Maximal subgroups

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

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

Subgroups — C3⋊((C3⋊(C3⋊C8))⋊C3) #10619721

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃⋊((C₃⋊(C₃⋊C₈))⋊C₃) #10619721