Group #14517867 — subgroups

Subgroups — C₃×C₉×C₃⋊(C₃⋊C₈) #14517867

Order 1944.

744 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₃⋊(C₃⋊C₈)	C₃	1	1	yes	maximal, characteristic
3	C₉×C₃⋊(C₃⋊C₈)	C₃	1	3	yes	maximal, direct product factor
4	C₃×C₉×C₃⋊C₈	—	3	4	no	maximal
5	C₂×C₃³×C₉	C₄	1	1	yes	characteristic
6	C₃⁴×C₄	C₂×C₃	1	1	yes	characteristic
7	C₃²×C₄×C₉	—	2	4	no	—
8	C₃²×C₄×C₉	—	2	12	no	—
9	C₃²×C₄×C₉	D₃	1	4	yes	—
10	C₃²×C₄×C₉	C₂×C₃	1	3	yes	—
11	C₃³×C₉	C₈	1	1	yes	characteristic, complemented, Sylow
12	C₃×C₃⋊(C₃⋊C₈)	C₉	1	3	yes	direct product factor
13	C₃×C₃⋊(C₃⋊C₈)	C₃²	1	1	yes	characteristic
14	C₃²×C₃⋊C₈	—	3	4	no	—
15	C₃×C₈×C₉	—	9	1	no	—
16	C₉×C₃⋊C₈	—	3	12	no	—
17	C₂×C₃⁴	C₃×C₄	1	1	yes	characteristic
18	C₂×C₃²×C₉	—	2	4	no	—
19	C₂×C₃²×C₉	—	2	12	no	—
20	C₂×C₃²×C₉	Dic₃	1	4	yes	—
21	C₂×C₃²×C₉	C₃×C₄	1	3	yes	—
22	C₃³×C₄	—	2	12	no	—
23	C₃³×C₄	—	2	4	no	—
24	C₃³×C₄	C₂×C₉	1	3	yes	—
25	C₃³×C₄	C₃×D₃	1	4	yes	—
26	C₃³×C₄	C₂×C₃²	1	1	yes	characteristic
27	C₃×C₄×C₉	—	2	24	no	—
28	C₃×C₄×C₉	—	2	12	no	—
29	C₃×C₄×C₉	—	2	4	no	—
30	C₃×C₄×C₉	—	2	12	no	—
31	C₃×C₄×C₉	C₃×D₃	1	12	yes	—
32	C₃×C₄×C₉	C₃⋊D₃	1	1	yes	characteristic
33	C₃⁴	C₃×C₈	1	1	yes	characteristic
34	C₃²×C₉	—	2	4	no	—
35	C₃²×C₉	—	2	12	no	—
36	C₃²×C₉	C₃⋊C₈	1	4	yes	complemented
37	C₃²×C₉	C₃×C₈	1	3	yes	complemented
38	C₃²×C₈	—	9	1	no	—
39	C₃⋊(C₃⋊C₈)	C₃×C₉	1	1	yes	characteristic, direct product factor
40	C₃×C₃⋊C₈	—	3	12	no	—
41	C₃×C₃⋊C₈	—	3	4	no	—
42	C₈×C₉	—	9	3	no	—
43	C₂×C₃³	—	2	12	no	—
44	C₂×C₃³	—	2	4	no	—
45	C₂×C₃³	C₄×C₉	1	3	yes	—
46	C₂×C₃³	C₃×Dic₃	1	4	yes	—
47	C₂×C₃³	C₃²×C₄	1	1	yes	characteristic
48	C₂×C₃×C₉	—	2	24	no	—
49	C₂×C₃×C₉	—	2	12	no	—
50	C₂×C₃×C₉	—	2	4	no	—
51	C₂×C₃×C₉	—	2	12	no	—
52	C₂×C₃×C₉	C₃×Dic₃	1	12	yes	—
53	C₂×C₃×C₉	C₃⋊Dic₃	1	1	yes	characteristic
54	C₃²×C₄	—	2	24	no	—
55	C₃²×C₄	—	2	12	no	—
56	C₃²×C₄	—	2	4	no	—
57	C₃²×C₄	—	2	12	no	—
58	C₃²×C₄	—	2	4	no	—
59	C₃²×C₄	D₃×C₉	1	12	yes	—
60	C₃²×C₄	C₂×C₃×C₉	1	1	yes	characteristic
61	C₃²×C₄	C₃²×D₃	1	4	yes	—
62	C₃²×C₄	C₃×C₃⋊D₃	1	1	yes	characteristic
63	C₄×C₉	—	2	12	no	—
64	C₄×C₉	C₃×C₃⋊D₃	1	3	yes	—
65	C₃³	—	2	12	no	—
66	C₃³	—	2	4	no	—
67	C₃³	C₈×C₉	1	3	yes	complemented
68	C₃³	C₃×C₃⋊C₈	1	4	yes	—
69	C₃³	C₃²×C₈	1	1	yes	characteristic
70	C₃×C₉	—	2	24	no	—
71	C₃×C₉	—	2	12	no	—
72	C₃×C₉	—	2	4	no	—
73	C₃×C₉	—	2	12	no	—
74	C₃×C₉	C₃×C₃⋊C₈	1	12	yes	complemented
75	C₃×C₉	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic, direct product factor
76	C₃×C₈	—	9	3	no	—
77	C₃×C₈	—	9	1	no	—
78	C₃⋊C₈	—	3	4	no	—
79	C₂×C₃²	—	2	24	no	—
80	C₂×C₃²	—	2	12	no	—
81	C₂×C₃²	—	2	4	no	—
82	C₂×C₃²	—	2	12	no	—
83	C₂×C₃²	—	2	4	no	—
84	C₂×C₃²	C₉×Dic₃	1	12	yes	—
85	C₂×C₃²	C₃×C₄×C₉	1	1	yes	characteristic
86	C₂×C₃²	C₃²×Dic₃	1	4	yes	—
87	C₂×C₃²	C₃×C₃⋊Dic₃	1	1	yes	characteristic
88	C₂×C₉	—	2	12	no	—
89	C₂×C₉	C₃×C₃⋊Dic₃	1	3	yes	—
90	C₃×C₄	—	2	12	no	—
91	C₃×C₄	—	2	4	no	—
92	C₃×C₄	C₃×D₃×C₉	1	4	yes	—
93	C₃×C₄	C₉×C₃⋊D₃	1	3	yes	—
94	C₃×C₄	C₃²×C₃⋊D₃	1	1	yes	characteristic
95	C₃²	—	2	24	no	—
96	C₃²	—	2	12	no	—
97	C₃²	—	2	4	no	—
98	C₃²	—	2	12	no	—
99	C₃²	—	2	4	no	—
100	C₃²	C₉×C₃⋊C₈	1	12	yes	complemented
101	C₃²	C₃×C₈×C₉	1	1	yes	characteristic, complemented
102	C₃²	C₃²×C₃⋊C₈	1	4	yes	—
103	C₃²	C₃×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
104	C₉	—	2	12	no	—
105	C₉	C₃×C₃⋊(C₃⋊C₈)	1	3	yes	direct product factor
106	C₈	—	9	1	no	Sylow
107	C₂×C₃	—	2	12	no	—
108	C₂×C₃	—	2	4	no	—
109	C₂×C₃	C₃×C₉×Dic₃	1	4	yes	—
110	C₂×C₃	C₉×C₃⋊Dic₃	1	3	yes	—
111	C₂×C₃	C₃²×C₃⋊Dic₃	1	1	yes	characteristic
112	C₄	C₃×C₉×C₃⋊D₃	1	1	yes	characteristic
113	C₃	—	2	12	no	—
114	C₃	—	2	4	no	—
115	C₃	C₃×C₉×C₃⋊C₈	1	4	yes	complemented
116	C₃	C₉×C₃⋊(C₃⋊C₈)	1	3	yes	direct product factor
117	C₃	C₃²×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
118	C₂	C₃×C₉×C₃⋊Dic₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃³×C₄×C₉

C₂

yes

maximal, characteristic

C₃²×C₃⋊(C₃⋊C₈)

C₃

yes

maximal, characteristic

C₉×C₃⋊(C₃⋊C₈)

C₃

yes

maximal, direct product factor

C₃×C₉×C₃⋊C₈

—

maximal

C₂×C₃³×C₉

C₄

yes

characteristic

C₃⁴×C₄

C₂×C₃

yes

characteristic

C₃²×C₄×C₉

—

C₃²×C₄×C₉

—

C₃²×C₄×C₉

D₃

yes

—

C₃²×C₄×C₉

C₂×C₃

yes

—

C₃³×C₉

C₈

yes

characteristic, complemented, Sylow

C₃×C₃⋊(C₃⋊C₈)

C₉

yes

direct product factor

C₃×C₃⋊(C₃⋊C₈)

C₃²

yes

characteristic

C₃²×C₃⋊C₈

—

C₃×C₈×C₉

—

C₉×C₃⋊C₈

—

C₂×C₃⁴

C₃×C₄

yes

characteristic

C₂×C₃²×C₉

—

C₂×C₃²×C₉

—

C₂×C₃²×C₉

Dic₃

yes

—

C₂×C₃²×C₉

C₃×C₄

yes

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

C₂×C₉

yes

—

C₃³×C₄

C₃×D₃

yes

—

C₃³×C₄

C₂×C₃²

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃×D₃

yes

—

C₃×C₄×C₉

C₃⋊D₃

yes

characteristic

C₃⁴

C₃×C₈

yes

characteristic

C₃²×C₉

—

C₃²×C₉

—

C₃²×C₉

C₃⋊C₈

yes

complemented

C₃²×C₉

C₃×C₈

yes

complemented

C₃²×C₈

—

C₃⋊(C₃⋊C₈)

C₃×C₉

yes

characteristic, direct product factor

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

C₈×C₉

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₄×C₉

yes

—

C₂×C₃³

C₃×Dic₃

yes

—

C₂×C₃³

C₃²×C₄

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×Dic₃

yes

—

C₂×C₃×C₉

C₃⋊Dic₃

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

D₃×C₉

yes

—

C₃²×C₄

C₂×C₃×C₉

yes

characteristic

C₃²×C₄

C₃²×D₃

yes

—

C₃²×C₄

C₃×C₃⋊D₃

yes

characteristic

C₄×C₉

—

C₄×C₉

C₃×C₃⋊D₃

yes

—

C₃³

—

C₃³

—

C₃³

C₈×C₉

yes

complemented

C₃³

C₃×C₃⋊C₈

yes

—

C₃³

C₃²×C₈

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×C₃⋊C₈

yes

complemented

C₃×C₉

C₃⋊(C₃⋊C₈)

yes

characteristic, direct product factor

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₉×Dic₃

yes

—

C₂×C₃²

C₃×C₄×C₉

yes

characteristic

C₂×C₃²

C₃²×Dic₃

yes

—

C₂×C₃²

C₃×C₃⋊Dic₃

yes

characteristic

C₂×C₉

—

C₂×C₉

C₃×C₃⋊Dic₃

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₃×D₃×C₉

yes

—

C₃×C₄

C₉×C₃⋊D₃

yes

—

C₃×C₄

C₃²×C₃⋊D₃

yes

characteristic

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

C₉×C₃⋊C₈

yes

complemented

101

C₃²

C₃×C₈×C₉

yes

characteristic, complemented

102

C₃²

C₃²×C₃⋊C₈

yes

—

103

C₃²

C₃×C₃⋊(C₃⋊C₈)

yes

characteristic

104

C₉

—

105

C₉

C₃×C₃⋊(C₃⋊C₈)

yes

direct product factor

106

C₈

—

Sylow

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

C₃×C₉×Dic₃

yes

—

110

C₂×C₃

C₉×C₃⋊Dic₃

yes

—

111

C₂×C₃

C₃²×C₃⋊Dic₃

yes

characteristic

112

C₄

C₃×C₉×C₃⋊D₃

yes

characteristic

113

C₃

—

114

C₃

—

115

C₃

C₃×C₉×C₃⋊C₈

yes

complemented

116

C₃

C₉×C₃⋊(C₃⋊C₈)

yes

direct product factor

117

C₃

C₃²×C₃⋊(C₃⋊C₈)

yes

characteristic

118

C₂

C₃×C₉×C₃⋊Dic₃

yes

characteristic

	subgroup	G/H
center, hypercenter	C₃×C₄×C₉	C₃⋊D₃
commutator, hypocenter	C₃²	C₃×C₈×C₉
Frattini	C₃×C₄	C₃²×C₃⋊D₃
Fitting	C₃³×C₄×C₉	C₂
socle, prime radical	C₂×C₃⁴	C₃×C₄
cosocle	C₃³×C₄	C₂×C₃²
LCS	G ▷ C₃²
UCS	C₁ ◁ C₃×C₄×C₉

subgroup

G/H

center, hypercenter

C₃×C₄×C₉

C₃⋊D₃

commutator, hypocenter

C₃²

C₃×C₈×C₉

Frattini

C₃×C₄

C₃²×C₃⋊D₃

Fitting

C₃³×C₄×C₉

C₂

socle, prime radical

C₂×C₃⁴

C₃×C₄

cosocle

C₃³×C₄

C₂×C₃²

LCS

G ▷ C₃²

UCS

C₁ ◁ C₃×C₄×C₉

N	H ≅ G/N	type
3. C₉×C₃⋊(C₃⋊C₈)	C₃	direct (N × H)
11. C₃³×C₉	C₈	semidirect
12. C₃×C₃⋊(C₃⋊C₈)	C₉	direct (N × H)
36. C₃²×C₉	C₃⋊C₈	semidirect
37. C₃²×C₉	C₃×C₈	semidirect
39. C₃⋊(C₃⋊C₈)	C₃×C₉	direct (N × H)
67. C₃³	C₈×C₉	semidirect
74. C₃×C₉	C₃×C₃⋊C₈	semidirect
75. C₃×C₉	C₃⋊(C₃⋊C₈)	direct (N × H)
100. C₃²	C₉×C₃⋊C₈	semidirect
101. C₃²	C₃×C₈×C₉	semidirect
105. C₉	C₃×C₃⋊(C₃⋊C₈)	direct (N × H)
115. C₃	C₃×C₉×C₃⋊C₈	semidirect
116. C₃	C₉×C₃⋊(C₃⋊C₈)	direct (N × H)

H ≅ G/N

type

3. C₉×C₃⋊(C₃⋊C₈)

C₃

direct (N × H)

11. C₃³×C₉

C₈

semidirect

12. C₃×C₃⋊(C₃⋊C₈)

C₉

direct (N × H)

36. C₃²×C₉

C₃⋊C₈

semidirect

37. C₃²×C₉

C₃×C₈

semidirect

39. C₃⋊(C₃⋊C₈)

C₃×C₉

direct (N × H)

67. C₃³

C₈×C₉

semidirect

74. C₃×C₉

C₃×C₃⋊C₈

semidirect

75. C₃×C₉

C₃⋊(C₃⋊C₈)

direct (N × H)

100. C₃²

C₉×C₃⋊C₈

semidirect

101. C₃²

C₃×C₈×C₉

semidirect

105. C₉

C₃×C₃⋊(C₃⋊C₈)

direct (N × H)

115. C₃

C₃×C₉×C₃⋊C₈

semidirect

116. C₃

C₉×C₃⋊(C₃⋊C₈)

direct (N × H)

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

subgroup	index
C₃³×C₄×C₉	2
C₃²×C₃⋊(C₃⋊C₈)	3
C₉×C₃⋊(C₃⋊C₈)	3
C₃×C₉×C₃⋊C₈	3

subgroup

index

C₃³×C₄×C₉

C₃²×C₃⋊(C₃⋊C₈)

C₉×C₃⋊(C₃⋊C₈)

C₃×C₉×C₃⋊C₈

Subgroups — C₃×C₉×C₃⋊(C₃⋊C₈) #14517867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C9×C3⋊(C3⋊C8) #14517867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×C₃⋊(C₃⋊C₈) #14517867