Group #14517251 — subgroups

Subgroups — #14517251

Order 1944.

218 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 67 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
2	C₃×#6204	C₃	1	1	yes	maximal, characteristic
3	#10619584	C₃	1	2	yes	maximal
4	C₉⋊(Q₈⋊C₉)	C₃	1	2	yes	maximal, complemented
5	(Q₈⋊C₉)⋊C₉	C₃	1	4	yes	maximal, complemented
6	(Q₈⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
7	C₉×Q₈⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	C₂×#6896	—	4	1	no	maximal
9	C₃×C₄×3₋¹⁺²	—	3	1	no	—
10	#6896	—	4	1	no	Sylow
11	C₃³×Q₈	C₃²	1	1	yes	characteristic
12	Q₈×3₋¹⁺²	—	3	3	no	—
13	C₃×Q₈×C₉	C₃²	1	2	yes	—
14	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
15	#6204	—	3	3	no	—
16	C₃×Q₈⋊C₉	C₃²	1	2	yes	—
17	C₃×Q₈⋊C₉	C₃²	1	2	yes	—
18	C₃×Q₈⋊C₉	C₃²	1	4	yes	—
19	C₃×Q₈⋊C₉	C₃²	1	1	yes	characteristic
20	C₂×C₃×3₋¹⁺²	—	4	1	no	—
21	C₂×C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
22	C₂×C₉⋊C₉	—	4	2	no	—
23	C₂×C₉⋊C₉	—	4	4	no	—
24	C₂×C₉⋊C₉	—	4	2	no	—
25	C₂×C₃²⋊C₉	—	4	2	no	—
26	C₂×C₉²	—	4	1	no	—
27	C₃³×C₄	—	3	1	no	—
28	C₄×3₋¹⁺²	—	3	9	no	—
29	C₃×C₄×C₉	—	3	2	no	—
30	C₃×C₄×C₉	—	3	1	no	—
31	C₃×3₋¹⁺²	—	4	1	no	—
32	C₃×3₋¹⁺²	SL(2,𝔽₃)	1	1	yes	characteristic
33	C₉⋊C₉	—	4	2	no	—
34	C₉⋊C₉	—	4	4	no	—
35	C₉⋊C₉	—	4	2	no	—
36	C₃²⋊C₉	—	4	2	no	—
37	C₉²	—	4	1	no	—
38	C₃²×Q₈	—	3	1	no	—
39	C₃²×Q₈	—	3	1	no	—
40	C₃²×Q₈	—	3	2	no	—
41	C₃²×Q₈	C₃³	1	1	yes	characteristic
42	Q₈×C₉	—	3	2	no	—
43	Q₈×C₉	3₋¹⁺²	1	3	yes	complemented
44	Q₈⋊C₉	—	3	2	no	—
45	Q₈⋊C₉	—	3	2	no	—
46	Q₈⋊C₉	—	3	4	no	—
47	Q₈⋊C₉	3₋¹⁺²	1	3	yes	complemented
48	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
49	C₂×3₋¹⁺²	—	3	3	no	—
50	C₂×3₋¹⁺²	—	12	3	no	—
51	C₂×C₃×C₉	—	4	1	no	—
52	C₂×C₃×C₉	—	4	2	no	—
53	C₂×C₃×C₉	—	4	4	no	—
54	C₂×C₃×C₉	—	4	2	no	—
55	C₂×C₃×C₉	C₃×A₄	1	2	yes	—
56	C₂×C₃×C₉	C₃×A₄	1	1	yes	characteristic
57	C₃²×C₄	—	3	1	no	—
58	C₃²×C₄	—	3	3	no	—
59	C₃²×C₄	—	9	1	no	—
60	C₃²×C₄	—	9	2	no	—
61	C₄×C₉	—	3	6	no	—
62	C₄×C₉	—	3	3	no	—
63	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
64	3₋¹⁺²	—	3	3	no	—
65	3₋¹⁺²	—	12	3	no	—
66	C₃×C₉	—	4	1	no	—
67	C₃×C₉	—	4	2	no	—
68	C₃×C₉	—	4	4	no	—
69	C₃×C₉	—	4	2	no	—
70	C₃×C₉	C₃×SL(2,𝔽₃)	1	2	yes	—
71	C₃×C₉	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
72	C₃×Q₈	—	9	1	no	—
73	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
74	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
75	C₃×Q₈	#737	1	2	yes	—
76	C₂×C₃²	—	3	1	no	—
77	C₂×C₃²	—	3	2	no	—
78	C₂×C₃²	—	3	1	no	—
79	C₂×C₃²	C₃²×A₄	1	1	yes	characteristic
80	C₂×C₉	—	3	2	no	—
81	C₂×C₉	—	4	3	no	—
82	C₂×C₉	—	12	2	no	—
83	C₂×C₉	—	12	2	no	—
84	C₂×C₉	—	12	4	no	—
85	C₂×C₉	#1094	1	3	yes	—
86	C₃×C₄	—	3	1	no	—
87	C₃×C₄	—	3	2	no	—
88	C₃×C₄	—	3	1	no	—
89	C₃×C₄	—	9	3	no	—
90	C₃²	—	3	1	no	—
91	C₃²	—	3	2	no	—
92	C₃²	—	3	1	no	—
93	C₃²	C₃²×SL(2,𝔽₃)	1	1	yes	characteristic
94	C₉	—	3	2	no	—
95	C₉	—	4	3	no	—
96	C₉	—	12	2	no	—
97	C₉	—	12	2	no	—
98	C₉	—	12	4	no	—
99	C₉	#6204	1	3	yes	complemented
100	Q₈	#6896	1	1	yes	characteristic, complemented, Sylow
101	C₂×C₃	—	9	1	no	—
102	C₂×C₃	A₄×3₋¹⁺²	1	1	yes	characteristic
103	C₂×C₃	#66440	1	2	yes	—
104	C₂×C₃	C₃×#1094	1	1	yes	characteristic
105	C₄	—	3	1	no	—
106	C₃	—	9	1	no	—
107	C₃	SL(2,𝔽₃)×3₋¹⁺²	1	1	yes	characteristic
108	C₃	#10619916	1	2	yes	—
109	C₃	C₃×#6204	1	1	yes	characteristic
110	C₂	#11757844	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×Q₈×3₋¹⁺²

C₃

yes

maximal, characteristic

C₃×#6204

C₃

yes

maximal, characteristic

#10619584

C₃

yes

maximal

C₉⋊(Q₈⋊C₉)

C₃

yes

maximal, complemented

(Q₈⋊C₉)⋊C₉

C₃

yes

maximal, complemented

(Q₈⋊C₉)⋊C₉

C₃

yes

maximal, complemented

C₉×Q₈⋊C₉

C₃

yes

maximal, characteristic, complemented

C₂×#6896

—

maximal

C₃×C₄×3₋¹⁺²

—

#6896

—

Sylow

C₃³×Q₈

C₃²

yes

characteristic

Q₈×3₋¹⁺²

—

C₃×Q₈×C₉

C₃²

yes

—

C₃×Q₈×C₉

C₃²

yes

characteristic

#6204

—

C₃×Q₈⋊C₉

C₃²

yes

—

C₃×Q₈⋊C₉

C₃²

yes

—

C₃×Q₈⋊C₉

C₃²

yes

—

C₃×Q₈⋊C₉

C₃²

yes

characteristic

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

A₄

yes

characteristic

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₉⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₉²

—

C₃³×C₄

—

C₄×3₋¹⁺²

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

SL(2,𝔽₃)

yes

characteristic

C₉⋊C₉

—

C₉⋊C₉

—

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

C₃³

yes

characteristic

Q₈×C₉

—

Q₈×C₉

3₋¹⁺²

yes

complemented

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

—

Q₈⋊C₉

3₋¹⁺²

yes

complemented

C₂×C₃³

C₃×A₄

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×A₄

yes

—

C₂×C₃×C₉

C₃×A₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₄×C₉

—

C₄×C₉

—

C₃³

C₃×SL(2,𝔽₃)

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×SL(2,𝔽₃)

yes

—

C₃×C₉

C₃×SL(2,𝔽₃)

yes

characteristic

C₃×Q₈

—

C₃×Q₈

C₃×3₋¹⁺²

yes

characteristic

C₃×Q₈

C₃×3₋¹⁺²

yes

characteristic

C₃×Q₈

#737

yes

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃²×A₄

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

#1094

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃²×SL(2,𝔽₃)

yes

characteristic

C₉

—

C₉

—

C₉

—

C₉

—

C₉

—

C₉

#6204

yes

complemented

100

Q₈

#6896

yes

characteristic, complemented, Sylow

101

C₂×C₃

—

102

C₂×C₃

A₄×3₋¹⁺²

yes

characteristic

103

C₂×C₃

#66440

yes

—

104

C₂×C₃

C₃×#1094

yes

characteristic

105

C₄

—

106

C₃

—

107

C₃

SL(2,𝔽₃)×3₋¹⁺²

yes

characteristic

108

C₃

#10619916

yes

—

109

C₃

C₃×#6204

yes

characteristic

110

C₂

#11757844

yes

characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃²×A₄
commutator, cosocle	C₃²×Q₈	C₃³
Fitting	C₃×Q₈×3₋¹⁺²	C₃
prime radical	C₂×C₃³	C₃×A₄
hypercenter	C₂×C₃×3₋¹⁺²	A₄
hypocenter	Q₈	#6896
LCS	G ▷ C₃²×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×3₋¹⁺²

subgroup

G/H

center, Frattini, socle

C₂×C₃²

C₃²×A₄

commutator, cosocle

C₃²×Q₈

C₃³

Fitting

C₃×Q₈×3₋¹⁺²

C₃

prime radical

C₂×C₃³

C₃×A₄

hypercenter

C₂×C₃×3₋¹⁺²

A₄

Q₈

G ▷ C₃²×Q₈ ▷ Q₈

C₁ ◁ C₂×C₃² ◁ C₂×C₃×3₋¹⁺²

N	H ≅ G/N	type
4. C₉⋊(Q₈⋊C₉)	C₃	semidirect
5. (Q₈⋊C₉)⋊C₉	C₃	semidirect
6. (Q₈⋊C₉)⋊C₉	C₃	semidirect
7. C₉×Q₈⋊C₉	C₃	semidirect
43. Q₈×C₉	3₋¹⁺²	semidirect
47. Q₈⋊C₉	3₋¹⁺²	semidirect
99. C₉	#6204	semidirect
100. Q₈	#6896	semidirect

H ≅ G/N

type

4. C₉⋊(Q₈⋊C₉)

C₃

semidirect

5. (Q₈⋊C₉)⋊C₉

C₃

semidirect

6. (Q₈⋊C₉)⋊C₉

C₃

semidirect

7. C₉×Q₈⋊C₉

C₃

semidirect

43. Q₈×C₉

3₋¹⁺²

semidirect

47. Q₈⋊C₉

3₋¹⁺²

semidirect

99. C₉

#6204

semidirect

100. Q₈

#6896

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×#6204	3
#10619584	3
C₉⋊(Q₈⋊C₉)	3
(Q₈⋊C₉)⋊C₉	3
C₉×Q₈⋊C₉	3
C₂×#6896	4

subgroup

index

C₃×Q₈×3₋¹⁺²

C₃×#6204

#10619584

C₉⋊(Q₈⋊C₉)

(Q₈⋊C₉)⋊C₉

C₉×Q₈⋊C₉

C₂×#6896

Subgroups — #14517251

Conjugacy classes of subgroups

Complements

Maximal subgroups