Group #11722642 — subgroups

Subgroups — #11722642

Order 864.

458 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 187 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	#41×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×AGL(1,ℤ/9)	C₂	1	2	yes	maximal, complemented
3	#3699⋊C₃	C₂	1	6	yes	maximal, complemented
4	#3697⋊C₃	C₂	1	3	yes	maximal, complemented
5	C₂×3₋¹⁺²⋊Q₈	C₂	1	3	yes	maximal, complemented
6	C₃×#1010	—	3	1	no	maximal
7	#63716	C₃	1	1	yes	maximal, characteristic, complemented
8	Q₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	D₄×3₋¹⁺²	C₂²	1	3	yes	—
10	C₂×C₄×3₋¹⁺²	C₂²	1	3	yes	—
11	#611⋊C₃	C₂²	1	6	yes	—
12	C₂×3₋¹⁺²⋊C₄	C₂²	1	6	yes	—
13	C₄⋊AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
14	C₄×AGL(1,ℤ/9)	C₂²	1	6	yes	—
15	3₋¹⁺²⋊Q₈	C₂²	1	3	yes	—
16	3₋¹⁺²⋊Q₈	C₂²	1	6	yes	—
17	C₃²×#41	D₃	1	1	yes	characteristic
18	C₃×D₃×Q₈	—	3	2	no	—
19	C₃×#237	—	3	6	no	—
20	C₃×#235	—	3	3	no	—
21	C₂×C₃×Dic₆	—	3	3	no	—
22	C₉×#41	—	2	1	no	—
23	C₉×#41	C₂×C₃	1	1	yes	characteristic, complemented
24	Q₈×D₉	C₂×C₃	1	2	yes	complemented
25	#3699	C₂×C₃	1	6	yes	complemented
26	#3697	C₂×C₃	1	3	yes	complemented
27	C₂×Dic₁₈	C₂×C₃	1	3	yes	complemented
28	C₂²×3₋¹⁺²	C₂³	1	3	yes	—
29	C₂×AGL(1,ℤ/9)	C₂³	1	2	yes	—
30	C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
31	C₄×3₋¹⁺²	C₂³	1	3	yes	—
32	3₋¹⁺²⋊C₄	C₂³	1	6	yes	—
33	C₃×2₋¹⁺⁴	—	9	1	no	—
34	#1010	—	3	1	no	—
35	C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
36	C₃²×D₄	C₂×D₃	1	3	yes	—
37	C₂×C₃²×C₄	C₂×D₃	1	3	yes	—
38	C₃×#67	—	3	6	no	—
39	C₂×C₃×Dic₃	—	3	6	no	—
40	C₃×D₁₂	—	3	1	no	—
41	C₃×C₄×D₃	—	3	6	no	—
42	C₃×Dic₆	—	3	3	no	—
43	C₃×Dic₆	—	3	6	no	—
44	Q₈×C₉	—	2	1	no	—
45	Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
46	D₄×C₉	—	2	3	no	—
47	D₄×C₉	C₂²×C₃	1	3	yes	—
48	C₂×C₄×C₉	—	2	3	no	—
49	C₂×C₄×C₉	C₂²×C₃	1	3	yes	—
50	#611	C₂²×C₃	1	6	yes	—
51	C₂×Dic₉	C₂²×C₃	1	6	yes	—
52	D₃₆	C₂²×C₃	1	1	yes	characteristic
53	C₄×D₉	C₂²×C₃	1	6	yes	—
54	Dic₁₈	C₂²×C₃	1	3	yes	—
55	Dic₁₈	C₂²×C₃	1	6	yes	—
56	C₂×3₋¹⁺²	—	2	3	no	—
57	C₂×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
58	AGL(1,ℤ/9)	—	2	2	no	—
59	C₃×#41	—	6	1	no	—
60	C₃×#41	—	9	3	no	—
61	C₃×#41	—	9	6	no	—
62	C₃×#41	C₃×D₃	1	1	yes	characteristic
63	C₂×C₃×Q₈	—	9	3	no	—
64	C₂×C₃×Q₈	—	9	2	no	—
65	D₃×Q₈	—	3	2	no	—
66	#237	—	3	6	no	—
67	#235	—	3	3	no	—
68	C₂×Dic₆	—	3	3	no	—
69	C₂²×C₃²	C₂²×D₃	1	3	yes	—
70	C₂×C₃×D₃	—	3	2	no	—
71	C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
72	C₃²×C₄	C₂²×D₃	1	3	yes	—
73	C₃×Dic₃	—	3	6	no	—
74	C₂²×C₉	—	2	3	no	—
75	C₂²×C₉	C₂³×C₃	1	3	yes	—
76	C₂×D₉	C₂³×C₃	1	2	yes	—
77	C₄×C₉	—	2	1	no	—
78	C₄×C₉	—	2	3	no	—
79	C₄×C₉	C₂³×C₃	1	1	yes	characteristic
80	C₄×C₉	C₂³×C₃	1	3	yes	—
81	Dic₉	C₂³×C₃	1	6	yes	—
82	2₋¹⁺⁴	—	9	1	no	Sylow
83	3₋¹⁺²	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
84	C₃×Q₈	—	6	1	no	—
85	C₃×Q₈	—	9	3	no	—
86	C₃×Q₈	—	9	6	no	—
87	C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
88	C₃×D₄	—	6	3	no	—
89	C₃×D₄	—	9	1	no	—
90	C₃×D₄	—	9	6	no	—
91	C₃×D₄	C₂×C₃×D₃	1	3	yes	—
92	C₂×C₃×C₄	—	6	3	no	—
93	C₂×C₃×C₄	—	9	6	no	—
94	C₂×C₃×C₄	—	9	6	no	—
95	C₂×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
96	#67	—	3	6	no	—
97	C₂×Dic₃	—	3	6	no	—
98	D₁₂	—	3	1	no	—
99	C₄×D₃	—	3	6	no	—
100	Dic₆	—	3	3	no	—
101	Dic₆	—	3	6	no	—
102	C₂×C₃²	—	2	3	no	—
103	C₂×C₃²	C₂³×D₃	1	1	yes	characteristic
104	C₃×D₃	—	6	2	no	—
105	C₂×C₉	—	2	3	no	—
106	C₂×C₉	—	2	1	no	—
107	C₂×C₉	—	4	3	no	—
108	C₂×C₉	C₂⁴×C₃	1	1	yes	characteristic
109	D₉	—	2	2	no	—
110	#41	—	9	6	no	—
111	#41	—	9	3	no	—
112	#41	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
113	C₂×Q₈	—	9	2	no	—
114	C₂×Q₈	—	9	3	no	—
115	C₂²×C₃	—	6	3	no	—
116	C₂²×C₃	—	9	2	no	—
117	C₂²×C₃	C₂²×C₃×D₃	1	3	yes	—
118	C₂×D₃	—	3	2	no	—
119	C₃×C₄	—	6	1	no	—
120	C₃×C₄	—	6	3	no	—
121	C₃×C₄	—	9	6	no	—
122	C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
123	C₃×C₄	C₂²×C₃×D₃	1	3	yes	—
124	Dic₃	—	3	6	no	—
125	C₃²	#1010	1	1	yes	characteristic
126	C₉	—	2	1	no	—
127	C₉	C₃×2₋¹⁺⁴	1	1	yes	characteristic, complemented
128	Q₈	—	9	6	no	—
129	Q₈	—	9	3	no	—
130	Q₈	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
131	D₄	—	9	6	no	—
132	D₄	—	9	1	no	—
133	D₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
134	C₂×C₄	—	9	6	no	—
135	C₂×C₄	—	9	6	no	—
136	C₂×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
137	C₂×C₃	—	2	3	no	—
138	C₂×C₃	—	6	1	no	—
139	C₂×C₃	—	12	3	no	—
140	C₂×C₃	—	18	2	no	—
141	C₂×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
142	D₃	—	6	2	no	—
143	C₂²	—	9	2	no	—
144	C₂²	C₂²×AGL(1,ℤ/9)	1	3	yes	—
145	C₄	—	9	6	no	—
146	C₄	C₂²×AGL(1,ℤ/9)	1	3	yes	—
147	C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
148	C₃	—	6	1	no	—
149	C₃	C₃×#1010	1	1	yes	characteristic
150	C₂	—	2	3	no	—
151	C₂	—	18	2	no	—
152	C₂	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

Q₈×AGL(1,ℤ/9)

C₂

yes

maximal, complemented

#3699⋊C₃

C₂

yes

maximal, complemented

#3697⋊C₃

C₂

yes

maximal, complemented

C₂×3₋¹⁺²⋊Q₈

C₂

yes

maximal, complemented

C₃×#1010

—

maximal

#63716

C₃

yes

maximal, characteristic, complemented

Q₈×3₋¹⁺²

C₂²

yes

characteristic

D₄×3₋¹⁺²

C₂²

yes

—

C₂×C₄×3₋¹⁺²

C₂²

yes

—

#611⋊C₃

C₂²

yes

—

C₂×3₋¹⁺²⋊C₄

C₂²

yes

—

C₄⋊AGL(1,ℤ/9)

C₂²

yes

characteristic

C₄×AGL(1,ℤ/9)

C₂²

yes

—

3₋¹⁺²⋊Q₈

C₂²

yes

—

3₋¹⁺²⋊Q₈

C₂²

yes

—

C₃²×#41

D₃

yes

characteristic

C₃×D₃×Q₈

—

C₃×#237

—

C₃×#235

—

C₂×C₃×Dic₆

—

C₉×#41

—

C₉×#41

C₂×C₃

yes

characteristic, complemented

Q₈×D₉

C₂×C₃

yes

complemented

#3699

C₂×C₃

yes

complemented

#3697

C₂×C₃

yes

complemented

C₂×Dic₁₈

C₂×C₃

yes

complemented

C₂²×3₋¹⁺²

C₂³

yes

—

C₂×AGL(1,ℤ/9)

C₂³

yes

—

C₄×3₋¹⁺²

C₂³

yes

characteristic

C₄×3₋¹⁺²

C₂³

yes

—

3₋¹⁺²⋊C₄

C₂³

yes

—

C₃×2₋¹⁺⁴

—

#1010

—

C₃²×Q₈

C₂×D₃

yes

characteristic

C₃²×D₄

C₂×D₃

yes

—

C₂×C₃²×C₄

C₂×D₃

yes

—

C₃×#67

—

C₂×C₃×Dic₃

—

C₃×D₁₂

—

C₃×C₄×D₃

—

C₃×Dic₆

—

C₃×Dic₆

—

Q₈×C₉

—

Q₈×C₉

C₂²×C₃

yes

characteristic

D₄×C₉

—

D₄×C₉

C₂²×C₃

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²×C₃

yes

—

#611

C₂²×C₃

yes

—

C₂×Dic₉

C₂²×C₃

yes

—

D₃₆

C₂²×C₃

yes

characteristic

C₄×D₉

C₂²×C₃

yes

—

Dic₁₈

C₂²×C₃

yes

—

Dic₁₈

C₂²×C₃

yes

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂⁴

yes

characteristic

AGL(1,ℤ/9)

—

C₃×#41

—

C₃×#41

—

C₃×#41

—

C₃×#41

C₃×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

D₃×Q₈

—

#237

—

#235

—

C₂×Dic₆

—

C₂²×C₃²

C₂²×D₃

yes

—

C₂×C₃×D₃

—

C₃²×C₄

C₂²×D₃

yes

characteristic

C₃²×C₄

C₂²×D₃

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂³×C₃

yes

—

C₂×D₉

C₂³×C₃

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂³×C₃

yes

characteristic

C₄×C₉

C₂³×C₃

yes

—

Dic₉

C₂³×C₃

yes

—

2₋¹⁺⁴

—

Sylow

3₋¹⁺²

2₋¹⁺⁴

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₂×C₃×D₃

yes

characteristic

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

C₂×C₃×D₃

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×D₃

yes

—

#67

—

C₂×Dic₃

—

D₁₂

—

C₄×D₃

—

100

Dic₆

—

101

Dic₆

—

102

C₂×C₃²

—

103

C₂×C₃²

C₂³×D₃

yes

characteristic

104

C₃×D₃

—

105

C₂×C₉

—

106

C₂×C₉

—

107

C₂×C₉

—

108

C₂×C₉

C₂⁴×C₃

yes

characteristic

109

D₉

—

110

#41

—

111

#41

—

112

#41

AGL(1,ℤ/9)

yes

characteristic, complemented

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₃

—

116

C₂²×C₃

—

117

C₂²×C₃

C₂²×C₃×D₃

yes

—

118

C₂×D₃

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

—

122

C₃×C₄

C₂²×C₃×D₃

yes

characteristic

123

C₃×C₄

C₂²×C₃×D₃

yes

—

124

Dic₃

—

125

C₃²

#1010

yes

characteristic

126

C₉

—

127

C₉

C₃×2₋¹⁺⁴

yes

characteristic, complemented

128

Q₈

—

129

Q₈

—

130

Q₈

C₂×AGL(1,ℤ/9)

yes

characteristic

131

D₄

—

132

D₄

—

133

D₄

C₂×AGL(1,ℤ/9)

yes

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₂×AGL(1,ℤ/9)

yes

—

137

C₂×C₃

—

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

C₂³×C₃×D₃

yes

characteristic

142

D₃

—

143

C₂²

—

144

C₂²

C₂²×AGL(1,ℤ/9)

yes

—

145

C₄

—

146

C₄

C₂²×AGL(1,ℤ/9)

yes

—

147

C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

148

C₃

—

149

C₃

C₃×#1010

yes

characteristic

150

C₂

—

151

C₂

—

152

C₂

C₂³×AGL(1,ℤ/9)

yes

characteristic

	subgroup	G/H
center	C₂	C₂³×AGL(1,ℤ/9)
commutator, cosocle	C₂×C₉	C₂⁴×C₃
Frattini, socle	C₂×C₃	C₂³×C₃×D₃
Fitting	#41×3₋¹⁺²	C₂
hypercenter	#41	AGL(1,ℤ/9)
hypocenter	C₉	C₃×2₋¹⁺⁴
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ #41

subgroup

G/H

center

C₂

C₂³×AGL(1,ℤ/9)

commutator, cosocle

C₂×C₉

C₂⁴×C₃

Frattini, socle

C₂×C₃

C₂³×C₃×D₃

Fitting

#41×3₋¹⁺²

C₂

C₉

C₃×2₋¹⁺⁴

LCS

G ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂ ◁ #41

N	H ≅ G/N	type
1. #41×3₋¹⁺²	C₂	semidirect
2. Q₈×AGL(1,ℤ/9)	C₂	semidirect
3. #3699⋊C₃	C₂	semidirect
4. #3697⋊C₃	C₂	semidirect
5. C₂×3₋¹⁺²⋊Q₈	C₂	semidirect
7. #63716	C₃	semidirect
23. C₉×#41	C₂×C₃	semidirect
24. Q₈×D₉	C₂×C₃	semidirect
25. #3699	C₂×C₃	semidirect
26. #3697	C₂×C₃	semidirect
27. C₂×Dic₁₈	C₂×C₃	semidirect
83. 3₋¹⁺²	2₋¹⁺⁴	semidirect
112. #41	AGL(1,ℤ/9)	semidirect
127. C₉	C₃×2₋¹⁺⁴	semidirect

H ≅ G/N

type

1. #41×3₋¹⁺²

C₂

semidirect

2. Q₈×AGL(1,ℤ/9)

C₂

semidirect

3. #3699⋊C₃

C₂

semidirect

4. #3697⋊C₃

C₂

semidirect

5. C₂×3₋¹⁺²⋊Q₈

C₂

semidirect

7. #63716

C₃

semidirect

23. C₉×#41

C₂×C₃

semidirect

24. Q₈×D₉

C₂×C₃

semidirect

25. #3699

C₂×C₃

semidirect

26. #3697

C₂×C₃

semidirect

27. C₂×Dic₁₈

C₂×C₃

semidirect

83. 3₋¹⁺²

2₋¹⁺⁴

semidirect

112. #41

AGL(1,ℤ/9)

semidirect

127. C₉

C₃×2₋¹⁺⁴

semidirect

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

subgroup	index
#41×3₋¹⁺²	2
Q₈×AGL(1,ℤ/9)	2
#3699⋊C₃	2
#3697⋊C₃	2
C₂×3₋¹⁺²⋊Q₈	2
C₃×#1010	3
#63716	3

subgroup

index

#41×3₋¹⁺²

C₂×3₋¹⁺²⋊Q₈

Subgroups — #11722642

Conjugacy classes of subgroups

Complements

Maximal subgroups