Group #14518868 — subgroups

Subgroups — #14518868

Order 1944.

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

normal subgroups only

This group has 723 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	Q₈×#737	C₃	1	1	yes	maximal, characteristic, complemented
2	#10619916	C₃	1	16	yes	maximal, complemented
3	SL(2,𝔽₃)×3₋¹⁺²	C₃	1	8	yes	maximal, complemented
4	SL(2,𝔽₃)×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#10619913	C₃	1	2	yes	maximal, complemented
6	#10619912	C₃	1	8	yes	maximal, complemented
7	C₃×C₉⋊SL(2,𝔽₃)	C₃	1	4	yes	maximal, complemented
8	C₂×3₋¹⁺⁴	—	4	1	no	maximal
9	C₄×#737	—	3	1	no	—
10	3₋¹⁺⁴	—	4	1	no	Sylow
11	C₃²×SL(2,𝔽₃)	C₃²	1	4	yes	—
12	Q₈×3₋¹⁺²	C₃²	1	8	yes	complemented
13	Q₈×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃×Q₈×C₉	C₃²	1	4	yes	complemented
15	Q₈⋊H(3,𝔽₃)	C₃²	1	8	yes	—
16	#6204	C₃²	1	16	yes	complemented
17	C₃×Q₈⋊C₉	C₃²	1	8	yes	complemented
18	C₉⋊SL(2,𝔽₃)	C₃²	1	1	yes	characteristic
19	C₉⋊SL(2,𝔽₃)	C₃²	1	8	yes	—
20	C₉⋊SL(2,𝔽₃)	C₃²	1	8	yes	complemented
21	C₉⋊SL(2,𝔽₃)	C₃²	1	16	yes	complemented
22	C₉⋊SL(2,𝔽₃)	C₃²	1	24	yes	complemented
23	C₉×SL(2,𝔽₃)	C₃²	1	24	yes	complemented
24	C₂×#737	—	4	16	no	—
25	C₂×#737	—	4	2	no	—
26	C₂×#737	—	4	8	no	—
27	C₂×#737	A₄	1	1	yes	characteristic
28	C₂×C₃×3₋¹⁺²	—	4	8	no	—
29	C₂×C₃×3₋¹⁺²	—	4	4	no	—
30	C₂×C₃×H(3,𝔽₃)	—	4	1	no	—
31	C₄×3₋¹⁺²	—	3	8	no	—
32	C₄×H(3,𝔽₃)	—	3	1	no	—
33	C₃×C₄×C₉	—	3	4	no	—
34	#737	—	4	16	no	—
35	#737	—	4	2	no	—
36	#737	—	4	8	no	—
37	#737	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
38	C₃×3₋¹⁺²	—	4	8	no	—
39	C₃×3₋¹⁺²	—	4	4	no	—
40	C₃×H(3,𝔽₃)	—	4	1	no	—
41	C₃²×Q₈	C₃³	1	4	yes	—
42	C₃×SL(2,𝔽₃)	—	9	4	no	—
43	C₃×SL(2,𝔽₃)	C₃³	1	8	yes	—
44	C₃×SL(2,𝔽₃)	C₃³	1	1	yes	characteristic
45	Q₈×C₉	C₃³	1	8	yes	—
46	Q₈×C₉	C₃³	1	1	yes	characteristic
47	Q₈⋊C₉	C₃³	1	16	yes	—
48	Q₈⋊C₉	C₃³	1	2	yes	—
49	C₂×C₃³	—	4	4	no	—
50	C₂×3₋¹⁺²	—	4	16	no	—
51	C₂×3₋¹⁺²	—	4	8	no	—
52	C₂×3₋¹⁺²	—	4	1	no	—
53	C₂×3₋¹⁺²	—	4	16	no	—
54	C₂×3₋¹⁺²	—	4	24	no	—
55	C₂×3₋¹⁺²	—	4	8	no	—
56	C₂×3₋¹⁺²	C₃×A₄	1	8	yes	—
57	C₂×H(3,𝔽₃)	—	4	8	no	—
58	C₂×H(3,𝔽₃)	C₃×A₄	1	1	yes	characteristic
59	C₂×C₃×C₉	—	4	8	no	—
60	C₂×C₃×C₉	—	4	24	no	—
61	C₂×C₃×C₉	C₃×A₄	1	4	yes	—
62	C₃²×C₄	—	3	4	no	—
63	C₄×C₉	—	3	1	no	—
64	C₄×C₉	—	3	8	no	—
65	C₃³	—	4	4	no	—
66	3₋¹⁺²	—	4	16	no	—
67	3₋¹⁺²	—	4	8	no	—
68	3₋¹⁺²	—	4	1	no	—
69	3₋¹⁺²	—	4	24	no	—
70	3₋¹⁺²	—	4	16	no	—
71	3₋¹⁺²	—	4	8	no	—
72	3₋¹⁺²	C₃×SL(2,𝔽₃)	1	8	yes	complemented
73	H(3,𝔽₃)	—	4	8	no	—
74	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
75	C₃×C₉	—	4	8	no	—
76	C₃×C₉	—	4	24	no	—
77	C₃×C₉	C₃×SL(2,𝔽₃)	1	4	yes	complemented
78	C₃×Q₈	—	3	4	no	—
79	C₃×Q₈	C₃⁴	1	1	yes	characteristic
80	SL(2,𝔽₃)	—	3	1	no	—
81	SL(2,𝔽₃)	—	3	8	no	—
82	C₂×C₃²	—	4	8	no	—
83	C₂×C₃²	—	4	1	no	—
84	C₂×C₃²	—	36	4	no	—
85	C₂×C₃²	C₃²×A₄	1	4	yes	—
86	C₂×C₉	—	4	2	no	—
87	C₂×C₉	—	4	16	no	—
88	C₂×C₉	C₃²×A₄	1	8	yes	—
89	C₂×C₉	C₃²×A₄	1	1	yes	characteristic
90	C₃×C₄	—	3	1	no	—
91	C₃×C₄	—	9	4	no	—
92	C₃²	—	4	8	no	—
93	C₃²	—	4	1	no	—
94	C₃²	—	36	4	no	—
95	C₃²	C₃²×SL(2,𝔽₃)	1	4	yes	—
96	C₉	—	4	2	no	—
97	C₉	—	4	16	no	—
98	C₉	C₃²×SL(2,𝔽₃)	1	8	yes	—
99	C₉	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
100	Q₈	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₃	—	3	4	no	—
102	C₂×C₃	—	12	1	no	—
103	C₂×C₃	—	12	8	no	—
104	C₂×C₃	C₃³×A₄	1	1	yes	characteristic
105	C₄	—	3	1	no	—
106	C₃	—	3	4	no	—
107	C₃	—	12	8	no	—
108	C₃	—	12	1	no	—
109	C₃	C₃³×SL(2,𝔽₃)	1	1	yes	characteristic
110	C₂	#11758106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#737

C₃

yes

maximal, characteristic, complemented

#10619916

C₃

yes

maximal, complemented

SL(2,𝔽₃)×3₋¹⁺²

C₃

yes

maximal, complemented

SL(2,𝔽₃)×H(3,𝔽₃)

C₃

yes

maximal, characteristic

#10619913

C₃

yes

maximal, complemented

#10619912

C₃

yes

maximal, complemented

C₃×C₉⋊SL(2,𝔽₃)

C₃

yes

maximal, complemented

C₂×3₋¹⁺⁴

—

maximal

C₄×#737

—

3₋¹⁺⁴

—

Sylow

C₃²×SL(2,𝔽₃)

C₃²

yes

—

Q₈×3₋¹⁺²

C₃²

yes

complemented

Q₈×H(3,𝔽₃)

C₃²

yes

characteristic

C₃×Q₈×C₉

C₃²

yes

complemented

Q₈⋊H(3,𝔽₃)

C₃²

yes

—

#6204

C₃²

yes

complemented

C₃×Q₈⋊C₉

C₃²

yes

complemented

C₉⋊SL(2,𝔽₃)

C₃²

yes

characteristic

C₉⋊SL(2,𝔽₃)

C₃²

yes

—

C₉⋊SL(2,𝔽₃)

C₃²

yes

complemented

C₉⋊SL(2,𝔽₃)

C₃²

yes

complemented

C₉⋊SL(2,𝔽₃)

C₃²

yes

complemented

C₉×SL(2,𝔽₃)

C₃²

yes

—

—

—

A₄

yes

characteristic

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×H(3,𝔽₃)

—

C₄×3₋¹⁺²

—

C₄×H(3,𝔽₃)

—

C₃×C₄×C₉

—

—

—

—

yes

characteristic, complemented

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×H(3,𝔽₃)

—

C₃²×Q₈

C₃³

yes

—

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

C₃³

yes

—

C₃×SL(2,𝔽₃)

C₃³

yes

characteristic

Q₈×C₉

C₃³

yes

—

Q₈×C₉

C₃³

yes

characteristic

Q₈⋊C₉

C₃³

yes

—

Q₈⋊C₉

C₃³

yes

—

C₂×C₃³

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₃×A₄

yes

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₃×A₄

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×A₄

yes

—

C₃²×C₄

—

C₄×C₉

—

C₄×C₉

—

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×SL(2,𝔽₃)

yes

complemented

H(3,𝔽₃)

—

H(3,𝔽₃)

C₃×SL(2,𝔽₃)

yes

characteristic

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×SL(2,𝔽₃)

yes

complemented

C₃×Q₈

—

C₃×Q₈

C₃⁴

yes

characteristic

SL(2,𝔽₃)

—

SL(2,𝔽₃)

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃²×A₄

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₃²×A₄

yes

—

C₂×C₉

C₃²×A₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×SL(2,𝔽₃)

yes

—

C₉

—

C₉

—

C₉

C₃²×SL(2,𝔽₃)

yes

—

C₉

C₃²×SL(2,𝔽₃)

yes

characteristic

100

Q₈

3₋¹⁺⁴

yes

characteristic, complemented, Sylow

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂×C₃

—

104

C₂×C₃

C₃³×A₄

yes

characteristic

105

C₄

—

106

C₃

—

107

C₃

—

108

C₃

—

109

C₃

C₃³×SL(2,𝔽₃)

yes

characteristic

110

C₂

#11758106

yes

characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₃³×A₄
commutator, cosocle	C₃×Q₈	C₃⁴
Fitting	Q₈×#737	C₃
prime radical	SL(2,𝔽₃)×H(3,𝔽₃)	C₃
hypercenter	C₂×#737	A₄
hypocenter	Q₈	3₋¹⁺⁴
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃ ◁ C₂×#737

subgroup

G/H

center, Frattini, socle

C₂×C₃

C₃³×A₄

commutator, cosocle

C₃×Q₈

C₃⁴

C₃

C₃

A₄

Q₈

3₋¹⁺⁴

LCS

G ▷ C₃×Q₈ ▷ Q₈

UCS

C₁ ◁ C₂×C₃ ◁ C₂×#737

N	H ≅ G/N	type
1. Q₈×#737	C₃	semidirect
2. #10619916	C₃	semidirect
3. SL(2,𝔽₃)×3₋¹⁺²	C₃	semidirect
5. #10619913	C₃	semidirect
6. #10619912	C₃	semidirect
7. C₃×C₉⋊SL(2,𝔽₃)	C₃	semidirect
12. Q₈×3₋¹⁺²	C₃²	semidirect
14. C₃×Q₈×C₉	C₃²	semidirect
16. #6204	C₃²	semidirect
17. C₃×Q₈⋊C₉	C₃²	semidirect
20. C₉⋊SL(2,𝔽₃)	C₃²	semidirect
21. C₉⋊SL(2,𝔽₃)	C₃²	semidirect
22. C₉⋊SL(2,𝔽₃)	C₃²	semidirect
23. C₉×SL(2,𝔽₃)	C₃²	semidirect
37. #737	SL(2,𝔽₃)	semidirect
72. 3₋¹⁺²	C₃×SL(2,𝔽₃)	semidirect
77. C₃×C₉	C₃×SL(2,𝔽₃)	semidirect
100. Q₈	3₋¹⁺⁴	semidirect

H ≅ G/N

type

1. Q₈×#737

C₃

semidirect

2. #10619916

C₃

semidirect

3. SL(2,𝔽₃)×3₋¹⁺²

C₃

semidirect

5. #10619913

C₃

semidirect

6. #10619912

C₃

semidirect

7. C₃×C₉⋊SL(2,𝔽₃)

C₃

semidirect

12. Q₈×3₋¹⁺²

C₃²

semidirect

14. C₃×Q₈×C₉

C₃²

semidirect

16. #6204

C₃²

semidirect

17. C₃×Q₈⋊C₉

C₃²

semidirect

20. C₉⋊SL(2,𝔽₃)

C₃²

semidirect

21. C₉⋊SL(2,𝔽₃)

C₃²

semidirect

22. C₉⋊SL(2,𝔽₃)

C₃²

semidirect

23. C₉×SL(2,𝔽₃)

C₃²

semidirect

37. #737

SL(2,𝔽₃)

semidirect

72. 3₋¹⁺²

C₃×SL(2,𝔽₃)

semidirect

77. C₃×C₉

C₃×SL(2,𝔽₃)

semidirect

100. Q₈

3₋¹⁺⁴

semidirect

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

subgroup	index
Q₈×#737	3
#10619916	3
SL(2,𝔽₃)×3₋¹⁺²	3
SL(2,𝔽₃)×H(3,𝔽₃)	3
#10619913	3
#10619912	3
C₃×C₉⋊SL(2,𝔽₃)	3
C₂×3₋¹⁺⁴	4

subgroup

index

SL(2,𝔽₃)×3₋¹⁺²

C₃×C₉⋊SL(2,𝔽₃)

C₂×3₋¹⁺⁴

Subgroups — #14518868

Conjugacy classes of subgroups

Complements

Maximal subgroups