Group #14518866 — subgroups

Subgroups — #14518866

Order 1944.

728 conjugacy classes of subgroups; only the 271 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 271 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	C₃×Q₈×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×Q₈⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×#6204	C₃	1	2	yes	maximal, complemented
4	#10619916	C₃	1	18	yes	maximal, complemented
5	SL(2,𝔽₃)×3₋¹⁺²	C₃	1	9	yes	maximal, complemented
6	#10619912	C₃	1	9	yes	maximal, complemented
7	C₂×3₋¹⁺⁴	—	4	1	no	maximal
8	C₃×C₄×3₋¹⁺²	—	3	1	no	—
9	3₋¹⁺⁴	—	4	1	no	Sylow
10	C₃³×Q₈	C₃²	1	1	yes	characteristic
11	C₃²×SL(2,𝔽₃)	C₃²	1	3	yes	—
12	Q₈×3₋¹⁺²	C₃²	1	3	yes	—
13	Q₈×3₋¹⁺²	C₃²	1	6	yes	complemented
14	C₃×Q₈×C₉	C₃²	1	3	yes	complemented
15	Q₈⋊H(3,𝔽₃)	C₃²	1	3	yes	—
16	Q₈⋊H(3,𝔽₃)	C₃²	1	6	yes	—
17	#6204	C₃²	1	6	yes	—
18	#6204	C₃²	1	6	yes	complemented
19	#6204	C₃²	1	6	yes	complemented
20	C₃×Q₈⋊C₉	C₃²	1	6	yes	complemented
21	C₉⋊SL(2,𝔽₃)	C₃²	1	54	yes	complemented
22	C₉×SL(2,𝔽₃)	C₃²	1	27	yes	complemented
23	C₂×#737	—	4	18	no	—
24	C₂×#737	—	4	9	no	—
25	C₂×C₃×3₋¹⁺²	—	4	2	no	—
26	C₂×C₃×3₋¹⁺²	—	4	9	no	—
27	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
28	C₂×C₃×H(3,𝔽₃)	—	4	1	no	—
29	C₃³×C₄	—	3	1	no	—
30	C₄×3₋¹⁺²	—	3	3	no	—
31	C₄×3₋¹⁺²	—	3	6	no	—
32	C₃×C₄×C₉	—	3	3	no	—
33	#737	—	4	18	no	—
34	#737	—	4	9	no	—
35	C₃×3₋¹⁺²	—	4	2	no	—
36	C₃×3₋¹⁺²	—	4	9	no	—
37	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
38	C₃×H(3,𝔽₃)	—	4	1	no	—
39	C₃²×Q₈	—	9	1	no	—
40	C₃²×Q₈	C₃³	1	2	yes	—
41	C₃²×Q₈	C₃³	1	1	yes	characteristic
42	C₃²×Q₈	C₃³	1	1	yes	characteristic
43	C₃×SL(2,𝔽₃)	—	9	3	no	—
44	C₃×SL(2,𝔽₃)	C₃³	1	9	yes	—
45	Q₈×C₉	C₃³	1	9	yes	—
46	Q₈⋊C₉	C₃³	1	18	yes	—
47	C₂×C₃³	—	4	3	no	—
48	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
49	C₂×3₋¹⁺²	—	4	6	no	—
50	C₂×3₋¹⁺²	—	4	6	no	—
51	C₂×3₋¹⁺²	—	4	6	no	—
52	C₂×3₋¹⁺²	—	4	54	no	—
53	C₂×3₋¹⁺²	C₃×A₄	1	3	yes	—
54	C₂×3₋¹⁺²	C₃×A₄	1	6	yes	—
55	C₂×H(3,𝔽₃)	—	4	6	no	—
56	C₂×H(3,𝔽₃)	—	4	3	no	—
57	C₂×C₃×C₉	—	4	6	no	—
58	C₂×C₃×C₉	—	4	27	no	—
59	C₂×C₃×C₉	C₃×A₄	1	3	yes	—
60	C₃²×C₄	—	3	1	no	—
61	C₃²×C₄	—	3	2	no	—
62	C₃²×C₄	—	3	1	no	—
63	C₃²×C₄	—	9	3	no	—
64	C₄×C₉	—	3	9	no	—
65	C₃³	—	4	3	no	—
66	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
67	3₋¹⁺²	—	4	6	no	—
68	3₋¹⁺²	—	4	6	no	—
69	3₋¹⁺²	—	4	6	no	—
70	3₋¹⁺²	—	4	54	no	—
71	3₋¹⁺²	C₃×SL(2,𝔽₃)	1	3	yes	—
72	3₋¹⁺²	C₃×SL(2,𝔽₃)	1	6	yes	complemented
73	H(3,𝔽₃)	—	4	6	no	—
74	H(3,𝔽₃)	—	4	3	no	—
75	C₃×C₉	—	4	6	no	—
76	C₃×C₉	—	4	27	no	—
77	C₃×C₉	C₃×SL(2,𝔽₃)	1	3	yes	complemented
78	C₃×Q₈	—	3	1	no	—
79	C₃×Q₈	—	3	1	no	—
80	C₃×Q₈	—	3	2	no	—
81	C₃×Q₈	C₃⁴	1	1	yes	characteristic
82	SL(2,𝔽₃)	—	3	9	no	—
83	C₂×C₃²	—	4	9	no	—
84	C₂×C₃²	—	9	1	no	—
85	C₂×C₃²	—	36	3	no	—
86	C₂×C₃²	C₃²×A₄	1	2	yes	—
87	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
88	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
89	C₂×C₉	—	4	18	no	—
90	C₂×C₉	C₃²×A₄	1	9	yes	—
91	C₃×C₄	—	3	3	no	—
92	C₃×C₄	—	3	1	no	—
93	C₃×C₄	—	9	1	no	—
94	C₃×C₄	—	9	2	no	—
95	C₃²	—	4	9	no	—
96	C₃²	—	9	1	no	—
97	C₃²	—	36	3	no	—
98	C₃²	C₃²×SL(2,𝔽₃)	1	2	yes	—
99	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
100	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
101	C₉	—	4	18	no	—
102	C₉	C₃²×SL(2,𝔽₃)	1	9	yes	—
103	Q₈	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
104	C₂×C₃	—	3	1	no	—
105	C₂×C₃	—	3	1	no	—
106	C₂×C₃	—	3	2	no	—
107	C₂×C₃	—	12	9	no	—
108	C₂×C₃	C₃³×A₄	1	1	yes	characteristic
109	C₄	—	3	1	no	—
110	C₃	—	3	1	no	—
111	C₃	—	3	1	no	—
112	C₃	—	3	2	no	—
113	C₃	—	12	9	no	—
114	C₃	C₃³×SL(2,𝔽₃)	1	1	yes	characteristic
115	C₂	#11758104	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×Q₈×3₋¹⁺²

C₃

yes

maximal, characteristic, complemented

C₃×Q₈⋊H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃×#6204

C₃

yes

maximal, complemented

#10619916

C₃

yes

maximal, complemented

SL(2,𝔽₃)×3₋¹⁺²

C₃

yes

maximal, complemented

#10619912

C₃

yes

maximal, complemented

C₂×3₋¹⁺⁴

—

maximal

C₃×C₄×3₋¹⁺²

—

3₋¹⁺⁴

—

Sylow

C₃³×Q₈

C₃²

yes

characteristic

C₃²×SL(2,𝔽₃)

C₃²

yes

—

Q₈×3₋¹⁺²

C₃²

yes

—

Q₈×3₋¹⁺²

C₃²

yes

complemented

C₃×Q₈×C₉

C₃²

yes

complemented

Q₈⋊H(3,𝔽₃)

C₃²

yes

—

Q₈⋊H(3,𝔽₃)

C₃²

yes

—

#6204

C₃²

yes

—

#6204

C₃²

yes

complemented

#6204

C₃²

yes

complemented

C₃×Q₈⋊C₉

C₃²

yes

complemented

C₉⋊SL(2,𝔽₃)

C₃²

yes

complemented

C₉×SL(2,𝔽₃)

C₃²

yes

complemented

C₂×#737

—

C₂×#737

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

A₄

yes

characteristic

C₂×C₃×H(3,𝔽₃)

—

C₃³×C₄

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₃×C₄×C₉

—

#737

—

#737

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

SL(2,𝔽₃)

yes

characteristic, complemented

C₃×H(3,𝔽₃)

—

C₃²×Q₈

—

C₃²×Q₈

C₃³

yes

—

C₃²×Q₈

C₃³

yes

characteristic

C₃²×Q₈

C₃³

yes

characteristic

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

C₃³

yes

—

Q₈×C₉

C₃³

yes

—

Q₈⋊C₉

C₃³

yes

—

C₂×C₃³

—

C₂×C₃³

C₃×A₄

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₃×A₄

yes

—

C₂×3₋¹⁺²

C₃×A₄

yes

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×A₄

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₄×C₉

—

C₃³

—

C₃³

C₃×SL(2,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×SL(2,𝔽₃)

yes

—

3₋¹⁺²

C₃×SL(2,𝔽₃)

yes

complemented

H(3,𝔽₃)

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×SL(2,𝔽₃)

yes

complemented

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₃⁴

yes

characteristic

SL(2,𝔽₃)

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃²×A₄

yes

—

C₂×C₃²

C₃²×A₄

yes

characteristic

C₂×C₃²

C₃²×A₄

yes

characteristic

C₂×C₉

—

C₂×C₉

C₃²×A₄

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×SL(2,𝔽₃)

yes

—

C₃²

C₃²×SL(2,𝔽₃)

yes

characteristic

100

C₃²

C₃²×SL(2,𝔽₃)

yes

characteristic

101

C₉

—

102

C₉

C₃²×SL(2,𝔽₃)

yes

—

103

Q₈

3₋¹⁺⁴

yes

characteristic, complemented, Sylow

104

C₂×C₃

—

105

C₂×C₃

—

106

C₂×C₃

—

107

C₂×C₃

—

108

C₂×C₃

C₃³×A₄

yes

characteristic

109

C₄

—

110

C₃

—

111

C₃

—

112

C₃

—

113

C₃

—

114

C₃

C₃³×SL(2,𝔽₃)

yes

characteristic

115

C₂

#11758104

yes

characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₃³×A₄
commutator, cosocle	C₃×Q₈	C₃⁴
Fitting	C₃×Q₈×3₋¹⁺²	C₃
prime radical	C₃×Q₈⋊H(3,𝔽₃)	C₃
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	3₋¹⁺⁴
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×3₋¹⁺²

subgroup

G/H

center, Frattini, socle

C₂×C₃

C₃³×A₄

commutator, cosocle

C₃×Q₈

C₃⁴

Fitting

C₃×Q₈×3₋¹⁺²

C₃

prime radical

C₃×Q₈⋊H(3,𝔽₃)

C₃

hypercenter

C₂×C₃×3₋¹⁺²

A₄

hypocenter

Q₈

3₋¹⁺⁴

LCS

G ▷ C₃×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂×C₃ ◁ C₂×C₃×3₋¹⁺²

N	H ≅ G/N	type
1. C₃×Q₈×3₋¹⁺²	C₃	semidirect
3. C₃×#6204	C₃	semidirect
4. #10619916	C₃	semidirect
5. SL(2,𝔽₃)×3₋¹⁺²	C₃	semidirect
6. #10619912	C₃	semidirect
13. Q₈×3₋¹⁺²	C₃²	semidirect
14. C₃×Q₈×C₉	C₃²	semidirect
18. #6204	C₃²	semidirect
19. #6204	C₃²	semidirect
20. C₃×Q₈⋊C₉	C₃²	semidirect
21. C₉⋊SL(2,𝔽₃)	C₃²	semidirect
22. C₉×SL(2,𝔽₃)	C₃²	semidirect
37. C₃×3₋¹⁺²	SL(2,𝔽₃)	semidirect
72. 3₋¹⁺²	C₃×SL(2,𝔽₃)	semidirect
77. C₃×C₉	C₃×SL(2,𝔽₃)	semidirect
103. Q₈	3₋¹⁺⁴	semidirect

H ≅ G/N

type

1. C₃×Q₈×3₋¹⁺²

C₃

semidirect

3. C₃×#6204

C₃

semidirect

4. #10619916

C₃

semidirect

5. SL(2,𝔽₃)×3₋¹⁺²

C₃

semidirect

6. #10619912

C₃

semidirect

13. Q₈×3₋¹⁺²

C₃²

semidirect

14. C₃×Q₈×C₉

C₃²

semidirect

18. #6204

C₃²

semidirect

19. #6204

C₃²

semidirect

20. C₃×Q₈⋊C₉

C₃²

semidirect

21. C₉⋊SL(2,𝔽₃)

C₃²

semidirect

22. C₉×SL(2,𝔽₃)

C₃²

semidirect

37. C₃×3₋¹⁺²

SL(2,𝔽₃)

semidirect

72. 3₋¹⁺²

C₃×SL(2,𝔽₃)

semidirect

77. C₃×C₉

C₃×SL(2,𝔽₃)

semidirect

103. Q₈

3₋¹⁺⁴

semidirect

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

subgroup	index
C₃×Q₈×3₋¹⁺²	3
C₃×Q₈⋊H(3,𝔽₃)	3
C₃×#6204	3
#10619916	3
SL(2,𝔽₃)×3₋¹⁺²	3
#10619912	3
C₂×3₋¹⁺⁴	4

subgroup

index

C₃×Q₈×3₋¹⁺²

C₃×Q₈⋊H(3,𝔽₃)

C₃×#6204

#10619916

SL(2,𝔽₃)×3₋¹⁺²

#10619912

C₂×3₋¹⁺⁴

Subgroups — #14518866

Conjugacy classes of subgroups

Complements

Maximal subgroups