Group #72269806 — subgroups

Subgroups — #72269806

Order 2304.

468 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 153 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	C₈×#3745	C₂	1	1	yes	maximal, characteristic
2	#11772164	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#1312	C₂	1	1	yes	maximal, characteristic, complemented
4	#10681843	—	3	4	no	maximal
5	C₄×#3745	C₄	1	1	yes	characteristic
6	C₈×C₄⋊(C₃⋊D₃)	—	2	1	no	—
7	C₃²×C₄×C₁₆	—	2	1	no	—
8	C₃²×C₈²	C₂²	1	1	yes	characteristic
9	C₄⋊(C₃⋊(C₃⋊C₁₆))	—	2	1	no	—
10	#10589325	—	2	1	no	—
11	#10589324	C₄	1	1	yes	characteristic
12	C₈×D₂₄	—	3	4	no	—
13	C₃×#1312	D₃	1	4	yes	complemented
14	#67439	—	3	4	no	—
15	C₂×#3745	C₈	1	1	yes	characteristic
16	C₂×C₈×C₃⋊D₃	—	4	1	no	—
17	C₄×C₄⋊(C₃⋊D₃)	—	2	1	no	—
18	C₂×C₃²×C₁₆	—	2	2	no	—
19	C₃²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₃²×C₄×C₈	D₄	1	1	yes	characteristic
21	C₃²×C₄×C₈	D₄	1	1	yes	characteristic
22	#63648	—	2	1	no	—
23	#63647	—	4	1	no	—
24	#63645	C₈	1	1	yes	characteristic
25	C₂×C₃⋊(C₃⋊C₁₆)	—	4	1	no	—
26	C₄⋊(C₃⋊(C₃⋊C₈))	—	2	1	no	—
27	#7070	—	9	1	no	Sylow
28	C₄×D₂₄	—	3	4	no	—
29	C₈×D₁₂	—	6	4	no	—
30	C₃×C₄×C₁₆	—	2	4	no	—
31	C₃×C₈²	C₂×D₃	1	4	yes	—
32	C₄⋊(C₃⋊C₁₆)	—	6	4	no	—
33	#4459	—	6	4	no	—
34	#4458	—	3	4	no	—
35	C₂×C₄⋊(C₃⋊D₃)	—	2	1	no	—
36	C₂×C₄×C₃⋊D₃	—	4	1	no	—
37	C₂×C₃²×C₈	—	2	1	no	—
38	C₂×C₃²×C₈	C₂×C₈	1	1	yes	characteristic
39	C₂×C₃²×C₈	M₁₆	1	1	yes	characteristic
40	C₂×C₃²×C₈	D₈	1	1	yes	characteristic
41	C₂×C₃²×C₈	SD₁₆	1	1	yes	characteristic
42	C₃²×C₄²	C₂²⋊C₄	1	1	yes	characteristic
43	#3752	—	4	1	no	—
44	C₄⋊(C₃⋊Dic₃)	—	2	1	no	—
45	C₂×C₃⋊(C₃⋊C₈)	—	4	1	no	—
46	#3745	—	2	1	no	—
47	#3745	C₁₆	1	2	yes	complemented
48	C₈×C₃⋊D₃	—	8	2	no	—
49	C₃²×C₁₆	—	4	2	no	—
50	C₃⋊(C₃⋊C₁₆)	—	8	1	no	—
51	C₈×D₈	—	9	1	no	—
52	#1367	—	9	1	no	—
53	#1312	C₃⋊D₃	1	1	yes	characteristic, complemented
54	C₂×D₂₄	—	3	4	no	—
55	C₂×D₃×C₈	—	12	4	no	—
56	C₄×D₁₂	—	6	4	no	—
57	C₂×C₃×C₁₆	—	2	8	no	—
58	C₃×C₄×C₈	C₄×D₃	1	4	yes	—
59	C₃×C₄×C₈	D₁₂	1	4	yes	—
60	C₃×C₄×C₈	#67	1	4	yes	—
61	#821	—	6	4	no	—
62	(C₂×D₃)⋊C₈	—	12	4	no	—
63	#818	—	3	4	no	—
64	C₂×C₃⋊C₁₆	—	12	4	no	—
65	C₄⋊(C₃⋊C₈)	—	6	4	no	—
66	C₂²×C₃⋊D₃	—	4	1	no	—
67	C₂×C₃²×C₄	C₂²⋊C₈	1	1	yes	characteristic
68	C₂×C₃²×C₄	#102	1	1	yes	characteristic
69	C₂×C₃²×C₄	#104	1	1	yes	characteristic
70	C₂×C₃⋊Dic₃	—	4	1	no	—
71	C₄⋊(C₃⋊D₃)	—	2	2	no	—
72	C₄⋊(C₃⋊D₃)	—	4	1	no	—
73	C₄×C₃⋊D₃	—	8	1	no	—
74	C₄×C₃⋊D₃	—	8	1	no	—
75	C₃²×C₈	—	2	2	no	—
76	C₃²×C₈	—	2	1	no	—
77	C₃²×C₈	—	2	1	no	—
78	C₃²×C₈	—	2	1	no	—
79	C₃²×C₈	C₂×C₁₆	1	1	yes	characteristic
80	C₃²×C₈	M₃₂	1	1	yes	characteristic
81	C₃⋊(C₃⋊C₈)	—	8	1	no	—
82	C₄×D₈	—	9	1	no	—
83	C₈×D₄	—	18	1	no	—
84	C₄⋊C₁₆	—	18	1	no	—
85	C₄×C₁₆	—	2	1	no	—
86	#335	—	9	1	no	—
87	D₄⋊C₈	—	18	1	no	—
88	C₈²	C₂×C₃⋊D₃	1	1	yes	characteristic
89	C₂×D₁₂	—	6	4	no	—
90	C₂×C₄×D₃	—	12	4	no	—
91	C₂×C₃×C₈	—	2	4	no	—
92	C₂×C₃×C₈	D₃×C₈	1	4	yes	—
93	C₂×C₃×C₈	ΓL(1,𝔽₂₅)	1	4	yes	—
94	C₂×C₃×C₈	#204	1	4	yes	—
95	C₂×C₃×C₈	D₂₄	1	4	yes	—
96	C₃×C₄²	(C₂×D₃)⋊C₄	1	4	yes	—
97	(C₂×D₃)⋊C₄	—	12	4	no	—
98	C₄⋊Dic₃	—	6	4	no	—
99	C₂×C₃⋊C₈	—	12	4	no	—
100	D₂₄	—	3	8	no	—
101	D₂₄	—	6	4	no	—
102	D₃×C₈	—	24	8	no	—
103	C₃×C₁₆	—	4	8	no	—
104	C₃⋊C₁₆	—	24	4	no	—
105	C₂²×C₃²	D₄⋊C₈	1	1	yes	characteristic
106	C₂×C₃⋊D₃	—	4	2	no	—
107	C₂×C₃⋊D₃	—	8	1	no	—
108	C₂×C₃⋊D₃	—	8	1	no	—
109	C₃²×C₄	—	2	1	no	—
110	C₃²×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
111	C₃²×C₄	#350	1	1	yes	characteristic
112	C₃²×C₄	#360	1	1	yes	characteristic
113	C₃²×C₄	#361	1	1	yes	characteristic
114	C₃⋊Dic₃	—	8	1	no	—
115	C₂×D₈	—	9	1	no	—
116	C₂²×C₈	—	36	1	no	—
117	C₄×D₄	—	18	1	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	36	1	no	—
120	#107	—	9	1	no	—
121	C₄⋊C₈	—	18	1	no	—
122	#102	—	18	1	no	—
123	C₂²⋊C₈	—	36	1	no	—
124	C₄×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
125	C₄×C₈	C₄⋊(C₃⋊D₃)	1	1	yes	characteristic
126	C₄×C₈	#638	1	1	yes	characteristic
127	C₂²×D₃	—	12	4	no	—
128	C₂×C₃×C₄	#805	1	4	yes	—
129	C₂×C₃×C₄	(C₂×D₃)⋊C₈	1	4	yes	—
130	C₂×C₃×C₄	#821	1	4	yes	—
131	C₂×Dic₃	—	12	4	no	—
132	D₁₂	—	6	8	no	—
133	D₁₂	—	12	4	no	—
134	C₄×D₃	—	24	4	no	—
135	C₄×D₃	—	24	4	no	—
136	C₃×C₈	—	2	8	no	—
137	C₃×C₈	—	2	4	no	—
138	C₃×C₈	—	2	4	no	—
139	C₃×C₈	—	2	4	no	—
140	C₃×C₈	D₃×C₁₆	1	4	yes	—
141	C₃×C₈	#798	1	4	yes	—
142	C₃⋊C₈	—	24	4	no	—
143	C₂×C₃²	D₄⋊C₁₆	1	1	yes	characteristic
144	C₂×C₃²	#1333	1	1	yes	characteristic
145	C₂×C₃²	#1336	1	1	yes	characteristic
146	C₃⋊D₃	—	8	2	no	—
147	C₂×D₄	—	18	1	no	—
148	C₂²×C₄	—	36	1	no	—
149	D₈	—	9	2	no	—
150	D₈	—	18	1	no	—
151	C₂×C₈	—	2	1	no	—
152	C₂×C₈	—	36	1	no	—
153	C₂×C₈	—	72	2	no	—
154	C₂×C₈	C₈×C₃⋊D₃	1	1	yes	characteristic
155	C₂×C₈	#3743	1	1	yes	characteristic
156	C₂×C₈	#3744	1	1	yes	characteristic
157	C₂×C₈	#3745	1	1	yes	characteristic
158	C₄⋊C₄	—	18	1	no	—
159	C₂²⋊C₄	—	36	1	no	—
160	C₄²	#3752	1	1	yes	characteristic
161	C₁₆	—	4	2	no	—
162	C₁₆	—	72	1	no	—
163	C₂²×C₃	#4459	1	4	yes	—
164	C₂×D₃	—	12	8	no	—
165	C₂×D₃	—	24	4	no	—
166	C₂×D₃	—	24	4	no	—
167	C₃×C₄	—	2	4	no	—
168	C₃×C₄	(C₂×D₃)⋊C₁₆	1	4	yes	—
169	C₃×C₄	#4508	1	4	yes	—
170	C₃×C₄	#4516	1	4	yes	—
171	C₃×C₄	#4518	1	4	yes	—
172	Dic₃	—	24	4	no	—
173	C₃²	#7070	1	1	yes	characteristic, complemented, Sylow
174	C₂³	—	36	1	no	—
175	D₄	—	18	2	no	—
176	D₄	—	36	1	no	—
177	C₂×C₄	—	36	1	no	—
178	C₂×C₄	—	72	1	no	—
179	C₂×C₄	—	72	1	no	—
180	C₂×C₄	#63632	1	1	yes	characteristic
181	C₂×C₄	#63647	1	1	yes	characteristic
182	C₂×C₄	#63648	1	1	yes	characteristic
183	C₈	—	2	2	no	—
184	C₈	—	2	1	no	—
185	C₈	—	2	1	no	—
186	C₈	—	2	1	no	—
187	C₈	—	72	1	no	—
188	C₈	C₁₆×C₃⋊D₃	1	1	yes	characteristic
189	C₈	#63625	1	1	yes	characteristic
190	C₂×C₃	#67443	1	4	yes	—
191	C₂×C₃	#67503	1	4	yes	—
192	C₂×C₃	#67516	1	4	yes	—
193	D₃	—	24	8	no	—
194	C₂²	—	36	2	no	—
195	C₂²	—	72	1	no	—
196	C₂²	—	72	1	no	—
197	C₂²	#10589325	1	1	yes	characteristic
198	C₄	—	2	1	no	—
199	C₄	—	72	1	no	—
200	C₄	#10589373	1	1	yes	characteristic
201	C₄	#10589374	1	1	yes	characteristic
202	C₄	#10589382	1	1	yes	characteristic
203	C₄	#10589384	1	1	yes	characteristic
204	C₃	#10681843	1	4	yes	complemented
205	C₂	—	72	2	no	—
206	C₂	#11771532	1	1	yes	characteristic
207	C₂	#11771606	1	1	yes	characteristic
208	C₂	#11771661	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×#3745

C₂

yes

maximal, characteristic

#11772164

C₂

yes

maximal, characteristic, complemented

C₃²×#1312

C₂

yes

maximal, characteristic, complemented

#10681843

—

maximal

C₄×#3745

C₄

yes

characteristic

C₈×C₄⋊(C₃⋊D₃)

—

C₃²×C₄×C₁₆

—

C₃²×C₈²

C₂²

yes

characteristic

C₄⋊(C₃⋊(C₃⋊C₁₆))

—

#10589325

—

#10589324

C₄

yes

characteristic

C₈×D₂₄

—

C₃×#1312

D₃

yes

complemented

#67439

—

C₂×#3745

C₈

yes

characteristic

C₂×C₈×C₃⋊D₃

—

C₄×C₄⋊(C₃⋊D₃)

—

C₂×C₃²×C₁₆

—

C₃²×C₄×C₈

C₂×C₄

yes

characteristic

C₃²×C₄×C₈

D₄

yes

characteristic

C₃²×C₄×C₈

D₄

yes

—

—

C₈

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

—

C₄⋊(C₃⋊(C₃⋊C₈))

—

#7070

—

Sylow

C₄×D₂₄

—

C₈×D₁₂

—

C₃×C₄×C₁₆

—

C₃×C₈²

C₂×D₃

yes

—

C₄⋊(C₃⋊C₁₆)

—

#4459

—

#4458

—

C₂×C₄⋊(C₃⋊D₃)

—

C₂×C₄×C₃⋊D₃

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₂×C₈

yes

characteristic

C₂×C₃²×C₈

M₁₆

yes

characteristic

C₂×C₃²×C₈

D₈

yes

characteristic

C₂×C₃²×C₈

SD₁₆

yes

characteristic

C₃²×C₄²

C₂²⋊C₄

yes

characteristic

#3752

—

C₄⋊(C₃⋊Dic₃)

—

C₂×C₃⋊(C₃⋊C₈)

—

#3745

—

#3745

C₁₆

yes

complemented

C₈×C₃⋊D₃

—

C₃²×C₁₆

—

C₃⋊(C₃⋊C₁₆)

—

C₈×D₈

—

#1367

—

#1312

C₃⋊D₃

yes

characteristic, complemented

C₂×D₂₄

—

C₂×D₃×C₈

—

C₄×D₁₂

—

C₂×C₃×C₁₆

—

C₃×C₄×C₈

C₄×D₃

yes

—

C₃×C₄×C₈

D₁₂

yes

—

C₃×C₄×C₈

#67

yes

—

#821

—

(C₂×D₃)⋊C₈

—

#818

—

C₂×C₃⋊C₁₆

—

C₄⋊(C₃⋊C₈)

—

C₂²×C₃⋊D₃

—

C₂×C₃²×C₄

C₂²⋊C₈

yes

C₂×C₃²×C₄

yes

C₂×C₃²×C₄

yes

C₂×C₃⋊Dic₃

—

C₄⋊(C₃⋊D₃)

—

C₄⋊(C₃⋊D₃)

—

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

—

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

C₂×C₁₆

yes

characteristic

C₃²×C₈

M₃₂

yes

characteristic

C₃⋊(C₃⋊C₈)

—

C₄×D₈

—

C₈×D₄

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

—

D₄⋊C₈

—

C₈²

C₂×C₃⋊D₃

yes

characteristic

C₂×D₁₂

—

C₂×C₄×D₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₃×C₈

yes

—

C₂×C₃×C₈

ΓL(1,𝔽₂₅)

yes

—

C₂×C₃×C₈

#204

yes

—

C₂×C₃×C₈

D₂₄

yes

—

C₃×C₄²

(C₂×D₃)⋊C₄

yes

—

(C₂×D₃)⋊C₄

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

100

D₂₄

—

101

D₂₄

—

102

D₃×C₈

—

103

C₃×C₁₆

—

104

C₃⋊C₁₆

—

105

C₂²×C₃²

D₄⋊C₈

yes

characteristic

106

C₂×C₃⋊D₃

—

107

C₂×C₃⋊D₃

—

108

C₂×C₃⋊D₃

—

109

C₃²×C₄

—

110

C₃²×C₄

C₂²⋊C₁₆

yes

characteristic

111

C₃²×C₄

#350

yes

characteristic

112

C₃²×C₄

#360

yes

characteristic

113

C₃²×C₄

#361

yes

characteristic

114

C₃⋊Dic₃

—

115

C₂×D₈

—

116

C₂²×C₈

—

117

C₄×D₄

—

118

C₂×C₁₆

—

119

C₂×C₁₆

—

120

#107

—

121

C₄⋊C₈

—

122

#102

—

123

C₂²⋊C₈

—

124

C₄×C₈

C₄×C₃⋊D₃

yes

characteristic

125

C₄×C₈

C₄⋊(C₃⋊D₃)

yes

characteristic

126

C₄×C₈

#638

yes

characteristic

127

C₂²×D₃

—

128

C₂×C₃×C₄

#805

yes

—

129

C₂×C₃×C₄

(C₂×D₃)⋊C₈

yes

—

130

C₂×C₃×C₄

#821

yes

—

131

C₂×Dic₃

—

132

D₁₂

—

133

D₁₂

—

134

C₄×D₃

—

135

C₄×D₃

—

136

C₃×C₈

—

137

C₃×C₈

—

138

C₃×C₈

—

139

C₃×C₈

—

140

C₃×C₈

D₃×C₁₆

yes

—

141

C₃×C₈

#798

yes

—

142

C₃⋊C₈

—

143

C₂×C₃²

D₄⋊C₁₆

yes

characteristic

144

C₂×C₃²

#1333

yes

characteristic

145

C₂×C₃²

#1336

yes

characteristic

146

C₃⋊D₃

—

147

C₂×D₄

—

148

C₂²×C₄

—

149

D₈

—

150

D₈

—

151

C₂×C₈

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

C₈×C₃⋊D₃

yes

characteristic

155

C₂×C₈

#3743

yes

characteristic

156

C₂×C₈

#3744

yes

characteristic

157

C₂×C₈

#3745

yes

characteristic

158

C₄⋊C₄

—

159

C₂²⋊C₄

—

160

C₄²

#3752

yes

characteristic

161

C₁₆

—

162

C₁₆

—

163

C₂²×C₃

#4459

yes

—

164

C₂×D₃

—

165

C₂×D₃

—

166

C₂×D₃

—

167

C₃×C₄

—

168

C₃×C₄

(C₂×D₃)⋊C₁₆

yes

—

169

C₃×C₄

#4508

yes

—

170

C₃×C₄

#4516

yes

—

171

C₃×C₄

#4518

yes

—

172

Dic₃

—

173

C₃²

#7070

yes

characteristic, complemented, Sylow

174

C₂³

—

175

D₄

—

176

D₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

#63632

yes

characteristic

181

C₂×C₄

#63647

yes

characteristic

182

C₂×C₄

#63648

yes

characteristic

183

C₈

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

—

188

C₈

C₁₆×C₃⋊D₃

yes

characteristic

189

C₈

#63625

yes

characteristic

190

C₂×C₃

#67443

yes

—

191

C₂×C₃

#67503

yes

—

192

C₂×C₃

#67516

yes

—

193

D₃

—

194

C₂²

—

195

C₂²

—

196

C₂²

—

197

C₂²

#10589325

yes

characteristic

198

C₄

—

199

C₄

—

200

C₄

#10589373

yes

characteristic

201

C₄

#10589374

yes

characteristic

202

C₄

#10589382

yes

characteristic

203

C₄

#10589384

yes

characteristic

204

C₃

#10681843

yes

complemented

205

C₂

—

206

C₂

#11771532

yes

characteristic

207

C₂

#11771606

yes

characteristic

208

C₂

#11771661

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#63648
commutator	C₃²×C₈	C₂×C₁₆
Frattini	C₈²	C₂×C₃⋊D₃
Fitting	C₃²×#1312	C₂
socle	C₂²×C₃²	D₄⋊C₈
prime radical	C₂×#3745	C₈
cosocle	C₃²×C₈²	C₂²
hypercenter	#1312	C₃⋊D₃
hypocenter	C₃²	#7070
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂×C₄

#63648

commutator

C₃²×C₈

C₂×C₁₆

Frattini

C₈²

C₂×C₃⋊D₃

Fitting

C₃²×#1312

C₂

socle

C₂²×C₃²

D₄⋊C₈

prime radical

C₂×#3745

C₈

cosocle

C₃²×C₈²

C₂²

C₃⋊D₃

C₃²

G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1312

N	H ≅ G/N	type
2. #11772164	C₂	semidirect
3. C₃²×#1312	C₂	semidirect
13. C₃×#1312	D₃	semidirect
47. #3745	C₁₆	semidirect
53. #1312	C₃⋊D₃	semidirect
173. C₃²	#7070	semidirect
204. C₃	#10681843	semidirect

H ≅ G/N

type

2. #11772164

C₂

semidirect

3. C₃²×#1312

C₂

semidirect

13. C₃×#1312

D₃

semidirect

47. #3745

C₁₆

semidirect

53. #1312

C₃⋊D₃

semidirect

173. C₃²

#7070

semidirect

204. C₃

#10681843

semidirect

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

subgroup	index
C₈×#3745	2
#11772164	2
C₃²×#1312	2
#10681843	3

subgroup

index

C₈×#3745

#11772164

C₃²×#1312

#10681843

Subgroups — #72269806

Conjugacy classes of subgroups

Complements

Maximal subgroups