Group #72252881 — subgroups

Subgroups — #72252881

Order 3456.

679 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 679 conjugacy classes of subgroups — too many to draw. Only its 207 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	#10588375⋊C₃	C₂	1	3	yes	maximal
2	#10588360⋊C₃	C₂	1	3	yes	maximal, complemented
3	C₈²×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#68057	—	3	1	no	maximal
5	#11782417	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×#3664⋊C₃	—	2	6	no	—
7	#11721394	C₂²	1	1	yes	characteristic
8	#63431⋊C₃	C₂²	1	3	yes	—
9	C₄×C₈×3₋¹⁺²	C₂²	1	3	yes	—
10	#63377⋊C₃	—	2	6	no	—
11	C₃×#4693	—	3	3	no	—
12	C₃×#4680	—	3	3	no	—
13	C₃²×C₈²	D₃	1	1	yes	characteristic
14	#10588375	C₂×C₃	1	3	yes	—
15	#10588360	C₂×C₃	1	3	yes	complemented
16	C₈²×C₉	—	2	1	no	—
17	C₈²×C₉	C₂×C₃	1	1	yes	characteristic, complemented
18	C₂×C₄⋊AGL(1,ℤ/9)	—	2	3	no	—
19	C₂×3₋¹⁺²⋊Q₈	—	2	3	no	—
20	C₂×C₈×3₋¹⁺²	D₄	1	6	yes	—
21	C₄²×3₋¹⁺²	C₂³	1	1	yes	characteristic
22	C₄⋊(3₋¹⁺²⋊C₄)	—	2	6	no	—
23	#3664⋊C₃	—	4	12	no	—
24	C₃×#1705	—	9	1	no	—
25	#68057	—	3	1	no	—
26	C₂×C₃×#204	—	6	6	no	—
27	C₃×#874	—	3	1	no	—
28	C₃×#869	—	3	3	no	—
29	C₃²×C₄×C₈	C₂×D₃	1	3	yes	—
30	C₃×#817	—	6	6	no	—
31	C₂×#3664	—	2	6	no	—
32	#63436	C₂²×C₃	1	1	yes	characteristic
33	#63431	C₂²×C₃	1	3	yes	—
34	C₄×C₈×C₉	—	2	3	no	—
35	C₄×C₈×C₉	C₂²×C₃	1	3	yes	—
36	#63377	—	2	6	no	—
37	C₂²×AGL(1,ℤ/9)	—	4	1	no	—
38	C₂×C₄×3₋¹⁺²	C₂×D₄	1	3	yes	—
39	C₂×3₋¹⁺²⋊C₄	—	4	3	no	—
40	C₄⋊AGL(1,ℤ/9)	—	4	6	no	—
41	3₋¹⁺²⋊Q₈	—	4	6	no	—
42	C₈×3₋¹⁺²	SD₁₆	1	12	yes	complemented
43	C₃×#498	—	9	3	no	—
44	C₃×#492	—	9	3	no	—
45	#4693	—	3	3	no	—
46	#4680	—	3	3	no	—
47	C₃×C₈²	—	3	1	no	—
48	C₃×C₈²	C₃×D₃	1	1	yes	characteristic
49	C₂×C₃×D₁₂	—	6	3	no	—
50	C₂×C₃×Dic₆	—	6	3	no	—
51	C₂×C₃²×C₈	D₁₂	1	6	yes	—
52	C₃²×C₄²	C₂²×D₃	1	1	yes	characteristic
53	C₃×C₄⋊Dic₃	—	6	6	no	—
54	C₃×#204	—	12	12	no	—
55	C₂×D₃₆	—	2	3	no	—
56	C₂×Dic₁₈	—	2	3	no	—
57	C₂×C₈×C₉	—	2	6	no	—
58	C₂×C₈×C₉	C₃×D₄	1	6	yes	—
59	C₄²×C₉	—	2	1	no	—
60	C₄²×C₉	C₂³×C₃	1	1	yes	characteristic
61	C₄⋊Dic₉	—	2	6	no	—
62	#3664	—	4	12	no	—
63	#1705	—	9	1	no	Sylow
64	C₂²×3₋¹⁺²	#127	1	1	yes	characteristic
65	C₂×AGL(1,ℤ/9)	—	8	3	no	—
66	C₄×3₋¹⁺²	C₂×SD₁₆	1	6	yes	—
67	3₋¹⁺²⋊C₄	—	8	3	no	—
68	C₂×C₃×SD₁₆	—	18	6	no	—
69	C₃×C₄⋊Q₈	—	9	3	no	—
70	C₃×#127	—	9	1	no	—
71	C₂×#204	—	6	6	no	—
72	#874	—	3	1	no	—
73	#869	—	3	3	no	—
74	C₃×#106	—	18	6	no	—
75	C₃×C₄×C₈	—	3	3	no	—
76	C₃×C₄×C₈	C₂×C₃×D₃	1	3	yes	—
77	#817	—	6	6	no	—
78	C₂²×C₃×D₃	—	12	1	no	—
79	C₂×C₃²×C₄	C₂×D₁₂	1	3	yes	—
80	C₂×C₃×Dic₃	—	12	3	no	—
81	C₃×D₁₂	—	12	6	no	—
82	C₃×Dic₆	—	12	6	no	—
83	C₂²×D₉	—	4	1	no	—
84	C₃²×C₈	#204	1	12	yes	—
85	C₂×C₄×C₉	—	2	3	no	—
86	C₂×C₄×C₉	C₂×C₃×D₄	1	3	yes	—
87	C₂×Dic₉	—	4	3	no	—
88	D₃₆	—	4	6	no	—
89	Dic₁₈	—	4	6	no	—
90	C₈×C₉	—	2	12	no	—
91	C₈×C₉	C₃×SD₁₆	1	12	yes	complemented
92	#498	—	9	3	no	—
93	#492	—	9	3	no	—
94	C₈²	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
95	C₂×3₋¹⁺²	#492	1	3	yes	—
96	AGL(1,ℤ/9)	—	16	1	no	—
97	C₂×C₃×Q₈	—	18	3	no	—
98	C₂×C₃×D₄	—	18	3	no	—
99	C₂×D₁₂	—	6	3	no	—
100	C₂×Dic₆	—	6	3	no	—
101	C₃×SD₁₆	—	36	12	no	—
102	C₂×C₃×C₈	—	3	6	no	—
103	C₂×C₃×C₈	C₃×D₁₂	1	6	yes	—
104	C₃×C₄⋊C₄	—	18	6	no	—
105	C₃×C₄²	—	3	1	no	—
106	C₃×C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
107	C₄⋊Dic₃	—	6	6	no	—
108	#204	—	12	12	no	—
109	C₂²×C₃²	#874	1	1	yes	characteristic
110	C₂×C₃×D₃	—	24	3	no	—
111	C₃²×C₄	C₂×#204	1	6	yes	—
112	C₃×Dic₃	—	24	3	no	—
113	C₂²×C₉	—	2	1	no	—
114	C₂²×C₉	C₃×#127	1	1	yes	characteristic
115	C₂×D₉	—	8	3	no	—
116	C₄×C₉	—	2	6	no	—
117	C₄×C₉	C₂×C₃×SD₁₆	1	6	yes	—
118	Dic₉	—	8	3	no	—
119	C₂×SD₁₆	—	18	6	no	—
120	C₄⋊Q₈	—	9	3	no	—
121	#127	—	9	1	no	—
122	#106	—	18	6	no	—
123	C₄×C₈	C₂×AGL(1,ℤ/9)	1	3	yes	—
124	3₋¹⁺²	#1705	1	1	yes	characteristic, complemented, Sylow
125	C₂³×C₃	—	36	1	no	—
126	C₂²×D₃	—	12	1	no	—
127	C₃×Q₈	—	36	6	no	—
128	C₃×D₄	—	36	6	no	—
129	C₂×C₃×C₄	—	3	3	no	—
130	C₂×C₃×C₄	—	36	3	no	—
131	C₂×C₃×C₄	C₂×C₃×D₁₂	1	3	yes	—
132	C₂×Dic₃	—	12	3	no	—
133	D₁₂	—	12	6	no	—
134	Dic₆	—	12	6	no	—
135	C₃×C₈	—	3	12	no	—
136	C₃×C₈	C₃×#204	1	12	yes	—
137	C₂×C₃²	#4693	1	3	yes	—
138	C₃×D₃	—	48	1	no	—
139	C₂×C₉	—	2	3	no	—
140	C₂×C₉	C₃×#492	1	3	yes	—
141	D₉	—	16	1	no	—
142	C₂×Q₈	—	18	3	no	—
143	C₂×D₄	—	18	3	no	—
144	SD₁₆	—	36	12	no	—
145	C₂×C₈	C₄⋊AGL(1,ℤ/9)	1	6	yes	—
146	C₄⋊C₄	—	18	6	no	—
147	C₄²	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
148	C₂²×C₃	—	3	1	no	—
149	C₂²×C₃	—	72	3	no	—
150	C₂²×C₃	C₃×#874	1	1	yes	characteristic
151	C₂×D₃	—	24	3	no	—
152	C₃×C₄	—	3	6	no	—
153	C₃×C₄	—	72	3	no	—
154	C₃×C₄	C₂×C₃×#204	1	6	yes	—
155	Dic₃	—	24	3	no	—
156	C₃²	#68057	1	1	yes	characteristic
157	C₉	—	2	1	no	—
158	C₉	C₃×#1705	1	1	yes	characteristic, complemented
159	C₂³	—	36	1	no	—
160	Q₈	—	36	6	no	—
161	D₄	—	36	6	no	—
162	C₂×C₄	—	36	3	no	—
163	C₂×C₄	C₂×C₄⋊AGL(1,ℤ/9)	1	3	yes	—
164	C₈	#3664⋊C₃	1	12	yes	complemented
165	C₂×C₃	—	3	3	no	—
166	C₂×C₃	—	144	1	no	—
167	C₂×C₃	C₃×#4693	1	3	yes	—
168	D₃	—	48	1	no	—
169	C₂²	—	72	3	no	—
170	C₂²	#11721394	1	1	yes	characteristic
171	C₄	—	72	3	no	—
172	C₄	C₂×#3664⋊C₃	1	6	yes	—
173	C₃	—	3	1	no	—
174	C₃	C₃×#68057	1	1	yes	characteristic
175	C₂	—	144	1	no	—
176	C₂	#10588375⋊C₃	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#10588375⋊C₃

C₂

yes

maximal

#10588360⋊C₃

C₂

yes

maximal, complemented

C₈²×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

C₃×#68057

—

maximal

#11782417

C₃

yes

maximal, characteristic, complemented

C₂×#3664⋊C₃

—

#11721394

C₂²

yes

characteristic

#63431⋊C₃

C₂²

yes

—

C₄×C₈×3₋¹⁺²

C₂²

yes

—

#63377⋊C₃

—

C₃×#4693

—

C₃×#4680

—

C₃²×C₈²

D₃

yes

characteristic

#10588375

C₂×C₃

yes

—

#10588360

C₂×C₃

yes

complemented

C₈²×C₉

—

C₈²×C₉

C₂×C₃

yes

characteristic, complemented

C₂×C₄⋊AGL(1,ℤ/9)

—

C₂×3₋¹⁺²⋊Q₈

—

C₂×C₈×3₋¹⁺²

D₄

yes

—

C₄²×3₋¹⁺²

C₂³

yes

characteristic

C₄⋊(3₋¹⁺²⋊C₄)

—

#3664⋊C₃

—

C₃×#1705

—

#68057

—

C₂×C₃×#204

—

C₃×#874

—

C₃×#869

—

C₃²×C₄×C₈

C₂×D₃

yes

—

C₃×#817

—

C₂×#3664

—

#63436

C₂²×C₃

yes

characteristic

#63431

C₂²×C₃

yes

—

C₄×C₈×C₉

—

C₄×C₈×C₉

C₂²×C₃

yes

—

#63377

—

C₂²×AGL(1,ℤ/9)

—

C₂×C₄×3₋¹⁺²

C₂×D₄

yes

—

C₂×3₋¹⁺²⋊C₄

—

C₄⋊AGL(1,ℤ/9)

—

3₋¹⁺²⋊Q₈

—

C₈×3₋¹⁺²

SD₁₆

yes

—

—

—

—

C₃×C₈²

—

C₃×C₈²

C₃×D₃

yes

characteristic

C₂×C₃×D₁₂

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

D₁₂

yes

—

C₃²×C₄²

C₂²×D₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₃×#204

—

C₂×D₃₆

—

C₂×Dic₁₈

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₃×D₄

yes

—

C₄²×C₉

—

C₄²×C₉

C₂³×C₃

yes

characteristic

C₄⋊Dic₉

—

—

—

C₂²×3₋¹⁺²

yes

—

C₄×3₋¹⁺²

C₂×SD₁₆

yes

—

3₋¹⁺²⋊C₄

—

C₂×C₃×SD₁₆

—

C₃×C₄⋊Q₈

—

—

—

—

—

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×C₃×D₃

yes

—

#817

—

C₂²×C₃×D₃

—

C₂×C₃²×C₄

C₂×D₁₂

yes

—

C₂×C₃×Dic₃

—

C₃×D₁₂

—

C₃×Dic₆

—

C₂²×D₉

—

C₃²×C₈

#204

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×D₄

yes

—

C₂×Dic₉

—

D₃₆

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

C₃×SD₁₆

yes

complemented

#498

—

#492

—

C₈²

AGL(1,ℤ/9)

yes

characteristic, complemented

C₂×3₋¹⁺²

#492

yes

—

AGL(1,ℤ/9)

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×D₁₂

—

100

C₂×Dic₆

—

101

C₃×SD₁₆

—

102

C₂×C₃×C₈

—

103

C₂×C₃×C₈

C₃×D₁₂

yes

—

104

C₃×C₄⋊C₄

—

105

C₃×C₄²

—

106

C₃×C₄²

C₂²×C₃×D₃

yes

characteristic

107

C₄⋊Dic₃

—

108

#204

—

109

C₂²×C₃²

#874

yes

characteristic

110

C₂×C₃×D₃

—

111

C₃²×C₄

C₂×#204

yes

—

112

C₃×Dic₃

—

113

C₂²×C₉

—

114

C₂²×C₉

C₃×#127

yes

characteristic

115

C₂×D₉

—

116

C₄×C₉

—

117

C₄×C₉

C₂×C₃×SD₁₆

yes

—

118

Dic₉

—

119

C₂×SD₁₆

—

120

C₄⋊Q₈

—

121

#127

—

122

#106

—

123

C₄×C₈

C₂×AGL(1,ℤ/9)

yes

—

124

3₋¹⁺²

#1705

yes

characteristic, complemented, Sylow

125

C₂³×C₃

—

126

C₂²×D₃

—

127

C₃×Q₈

—

128

C₃×D₄

—

129

C₂×C₃×C₄

—

130

C₂×C₃×C₄

—

131

C₂×C₃×C₄

C₂×C₃×D₁₂

yes

—

132

C₂×Dic₃

—

133

D₁₂

—

134

Dic₆

—

135

C₃×C₈

—

136

C₃×C₈

C₃×#204

yes

—

137

C₂×C₃²

#4693

yes

—

138

C₃×D₃

—

139

C₂×C₉

—

140

C₂×C₉

C₃×#492

yes

—

141

D₉

—

142

C₂×Q₈

—

143

C₂×D₄

—

144

SD₁₆

—

145

C₂×C₈

C₄⋊AGL(1,ℤ/9)

yes

—

146

C₄⋊C₄

—

147

C₄²

C₂²×AGL(1,ℤ/9)

yes

characteristic

148

C₂²×C₃

—

149

C₂²×C₃

—

150

C₂²×C₃

C₃×#874

yes

characteristic

151

C₂×D₃

—

152

C₃×C₄

—

153

C₃×C₄

—

154

C₃×C₄

C₂×C₃×#204

yes

—

155

Dic₃

—

156

C₃²

#68057

yes

characteristic

157

C₉

—

158

C₉

C₃×#1705

yes

characteristic, complemented

159

C₂³

—

160

Q₈

—

161

D₄

—

162

C₂×C₄

—

163

C₂×C₄

C₂×C₄⋊AGL(1,ℤ/9)

yes

—

164

C₈

#3664⋊C₃

yes

complemented

165

C₂×C₃

—

166

C₂×C₃

—

144

—

167

C₂×C₃

C₃×#4693

yes

—

168

D₃

—

169

C₂²

—

170

C₂²

#11721394

yes

characteristic

171

C₄

—

172

C₄

C₂×#3664⋊C₃

yes

—

173

C₃

—

174

C₃

C₃×#68057

yes

characteristic

175

C₂

—

144

—

176

C₂

#10588375⋊C₃

yes

—

	subgroup	G/H
center	C₂²	#11721394
commutator, cosocle	C₄²×C₉	C₂³×C₃
Frattini	C₃×C₄²	C₂²×C₃×D₃
Fitting	C₈²×3₋¹⁺²	C₂
socle	C₂²×C₃	C₃×#874
prime radical	#11721394	C₂²
hypercenter	C₈²	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#1705
LCS	G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#11721394

commutator, cosocle

C₄²×C₉

C₂³×C₃

Frattini

C₃×C₄²

C₂²×C₃×D₃

Fitting

C₈²×3₋¹⁺²

C₂

C₂²×C₃

C₂²

C₈²

C₉

G ▷ C₄²×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
2. #10588360⋊C₃	C₂	semidirect
3. C₈²×3₋¹⁺²	C₂	semidirect
5. #11782417	C₃	semidirect
15. #10588360	C₂×C₃	semidirect
17. C₈²×C₉	C₂×C₃	semidirect
42. C₈×3₋¹⁺²	SD₁₆	semidirect
91. C₈×C₉	C₃×SD₁₆	semidirect
94. C₈²	AGL(1,ℤ/9)	semidirect
124. 3₋¹⁺²	#1705	semidirect
158. C₉	C₃×#1705	semidirect
164. C₈	#3664⋊C₃	semidirect

H ≅ G/N

type

2. #10588360⋊C₃

C₂

semidirect

3. C₈²×3₋¹⁺²

C₂

semidirect

5. #11782417

C₃

semidirect

15. #10588360

C₂×C₃

semidirect

17. C₈²×C₉

C₂×C₃

semidirect

42. C₈×3₋¹⁺²

SD₁₆

semidirect

91. C₈×C₉

C₃×SD₁₆

semidirect

94. C₈²

AGL(1,ℤ/9)

semidirect

124. 3₋¹⁺²

#1705

semidirect

158. C₉

C₃×#1705

semidirect

164. C₈

#3664⋊C₃

semidirect

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

subgroup	index
#10588375⋊C₃	2
#10588360⋊C₃	2
C₈²×3₋¹⁺²	2
C₃×#68057	3
#11782417	3

subgroup

index

C₈²×3₋¹⁺²

Subgroups — #72252881

Conjugacy classes of subgroups

Complements

Maximal subgroups