Group #10694004 — subgroups

Subgroups — #10694004

Order 768.

558 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 167 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×C₄×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#4535	C₂	1	4	yes	maximal
3	#69392	C₂	1	1	yes	maximal, characteristic
4	C₄×#803	C₂	1	1	yes	maximal, characteristic
5	#9647	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄×Q₈	C₂²	1	2	yes	—
8	C₃×#391	C₄	1	2	yes	—
9	C₃×#389	—	2	2	no	—
10	C₃×#384	C₄	1	1	yes	characteristic
11	C₃×C₄×C₄⋊C₄	—	2	2	no	—
12	C₃×C₄³	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×#803	—	2	2	no	—
15	C₂×C₄×C₃⋊C₈	C₂²	1	2	yes	—
16	#4535	—	2	8	no	—
17	#4524	—	2	2	no	—
18	C₄×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	#1831	—	3	1	no	—
20	C₄×#97	—	3	1	no	—
21	C₂×Q₈⋊C₈	—	3	4	no	—
22	C₂²×C₃×Q₈	C₂×C₄	1	2	yes	—
23	C₃×C₄⋊Q₈	—	2	4	no	—
24	C₃×C₄×Q₈	—	2	4	no	—
25	C₃×C₄×Q₈	—	2	4	no	—
26	C₂×C₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₃×C₄⋊C₄	—	2	2	no	—
28	C₂×C₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₃×C₄²	—	2	2	no	—
31	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₄²	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₄²	D₄	1	2	yes	—
34	C₂×C₃×C₄²	D₄	1	2	yes	—
35	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
36	C₂²×C₃⋊C₈	—	2	2	no	—
37	C₂²×C₃⋊C₈	—	2	2	no	—
38	C₃×#95	—	2	4	no	—
39	C₄⋊(C₃⋊C₈)	—	2	4	no	—
40	#803	—	2	8	no	—
41	C₄×C₃⋊C₈	—	2	2	no	—
42	C₄×C₃⋊C₈	D₄	1	4	yes	—
43	C₂×C₄⋊Q₈	Dic₃	1	1	yes	characteristic
44	C₂×C₄×Q₈	C₂×D₃	1	2	yes	—
45	C₂×C₄⋊C₈	—	3	2	no	—
46	C₂×#97	—	6	2	no	—
47	C₂×C₄×C₈	—	3	2	no	—
48	#391	Dic₃	1	2	yes	—
49	#389	—	2	2	no	—
50	#384	Dic₃	1	1	yes	characteristic
51	C₄×C₄⋊C₄	—	2	2	no	—
52	C₄³	C₂×D₃	1	1	yes	characteristic
53	#336	—	6	2	no	—
54	Q₈⋊C₈	—	6	8	no	—
55	C₂×C₃×Q₈	—	2	2	no	—
56	C₂×C₃×Q₈	—	2	4	no	—
57	C₂×C₃×Q₈	—	2	4	no	—
58	C₂×C₃×Q₈	M₁₆	1	4	yes	—
59	C₂²×C₃×C₄	—	2	2	no	—
60	C₂²×C₃×C₄	—	2	2	no	—
61	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₃×C₄	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
66	C₃×C₄⋊C₄	—	2	8	no	—
67	C₃×C₄⋊C₄	—	2	4	no	—
68	C₃×C₄⋊C₄	—	4	4	no	—
69	C₃×C₄⋊C₄	—	4	4	no	—
70	C₃×C₄²	—	2	4	no	—
71	C₃×C₄²	—	2	2	no	—
72	C₃×C₄²	—	2	2	no	—
73	C₃×C₄²	—	2	4	no	—
74	C₃×C₄²	—	4	2	no	—
75	C₃×C₄²	—	4	2	no	—
76	C₃×C₄²	C₂×D₄	1	2	yes	—
77	C₃×C₄²	#41	1	2	yes	—
78	C₂×C₃⋊C₈	—	2	8	no	—
79	C₂×C₃⋊C₈	—	2	4	no	—
80	C₂×C₃⋊C₈	—	4	2	no	—
81	C₂×C₃⋊C₈	—	4	4	no	—
82	C₂²×Q₈	C₂×Dic₃	1	2	yes	—
83	C₂²×C₈	—	6	2	no	—
84	C₂²×C₈	—	6	2	no	—
85	C₄⋊Q₈	—	2	4	no	—
86	C₄×Q₈	—	2	4	no	—
87	C₄×Q₈	—	2	4	no	—
88	C₂×C₄⋊C₄	—	2	2	no	—
89	C₂×C₄⋊C₄	—	2	2	no	—
90	C₂×C₄⋊C₄	—	2	2	no	—
91	C₂×C₄⋊C₄	C₂×Dic₃	1	2	yes	—
92	C₂×C₄²	—	2	2	no	—
93	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
94	C₂×C₄²	C₂×Dic₃	1	1	yes	characteristic
95	C₂×C₄²	#67	1	2	yes	—
96	C₂×C₄²	#67	1	2	yes	—
97	C₂×C₄²	C₂²×D₃	1	1	yes	characteristic
98	C₄⋊C₈	—	6	4	no	—
99	#97	—	6	8	no	—
100	C₄×C₈	—	3	4	no	—
101	C₄×C₈	—	6	2	no	—
102	#95	—	2	4	no	—
103	C₂³×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₃×Q₈	—	2	8	no	—
105	C₃×Q₈	—	4	4	no	—
106	C₂×C₃×C₄	—	2	2	no	—
107	C₂×C₃×C₄	—	2	2	no	—
108	C₂×C₃×C₄	—	2	2	no	—
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	—	2	2	no	—
111	C₂×C₃×C₄	—	2	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	4	2	no	—
114	C₂×C₃×C₄	—	4	2	no	—
115	C₂×C₃×C₄	—	4	2	no	—
116	C₂×C₃×C₄	—	4	4	no	—
117	C₂×C₃×C₄	#104	1	4	yes	—
118	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
119	C₂×C₃×C₄	#121	1	2	yes	—
120	C₂×C₃×C₄	#124	1	1	yes	characteristic
121	C₂×C₃×C₄	#127	1	1	yes	characteristic
122	C₂×C₃×C₄	C₂×M₁₆	1	2	yes	—
123	C₂×C₃×C₄	#131	1	2	yes	—
124	C₂×C₃×C₄	#137	1	4	yes	—
125	C₃⋊C₈	—	2	8	no	—
126	C₃⋊C₈	—	8	2	no	—
127	C₂×Q₈	—	2	2	no	—
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	—	2	4	no	—
130	C₂×Q₈	#208	1	4	yes	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	#217	1	2	yes	—
134	C₂²×C₄	#217	1	2	yes	—
135	C₂²×C₄	C₂²×Dic₃	1	1	yes	characteristic
136	C₂²×C₄	C₂×#67	1	1	yes	characteristic
137	C₂²×C₄	C₂×#67	1	1	yes	characteristic
138	C₂×C₈	—	6	8	no	—
139	C₂×C₈	—	6	4	no	—
140	C₂×C₈	—	12	2	no	—
141	C₂×C₈	—	12	4	no	—
142	C₄⋊C₄	—	2	8	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄⋊C₄	—	4	4	no	—
146	C₄²	—	2	4	no	—
147	C₄²	—	2	2	no	—
148	C₄²	—	2	4	no	—
149	C₄²	—	2	2	no	—
150	C₄²	—	4	2	no	—
151	C₄²	—	4	2	no	—
152	C₄²	D₃×D₄	1	2	yes	—
153	C₄²	#237	1	2	yes	—
154	C₂²×C₃	#390	1	1	yes	characteristic
155	C₂²×C₃	#407	1	1	yes	characteristic
156	C₂²×C₃	#408	1	1	yes	characteristic
157	C₂²×C₃	#419	1	2	yes	—
158	C₂²×C₃	C₂×#104	1	2	yes	—
159	C₃×C₄	—	2	2	no	—
160	C₃×C₄	—	2	4	no	—
161	C₃×C₄	—	4	4	no	—
162	C₃×C₄	—	4	2	no	—
163	C₃×C₄	—	8	2	no	—
164	C₃×C₄	#435	1	4	yes	—
165	C₃×C₄	#491	1	2	yes	—
166	C₃×C₄	#497	1	2	yes	—
167	C₂³	C₂×#217	1	1	yes	characteristic
168	Q₈	—	2	8	no	—
169	Q₈	—	4	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	4	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	#837	1	4	yes	—
182	C₂×C₄	C₂×#208	1	2	yes	—
183	C₂×C₄	D₄×Dic₃	1	2	yes	—
184	C₂×C₄	#936	1	1	yes	characteristic
185	C₂×C₄	#938	1	2	yes	—
186	C₂×C₄	#940	1	1	yes	characteristic
187	C₂×C₄	#942	1	4	yes	—
188	C₂×C₄	#948	1	2	yes	—
189	C₈	—	6	8	no	—
190	C₈	—	24	2	no	—
191	C₂×C₃	#1485	1	2	yes	—
192	C₂×C₃	#1491	1	2	yes	—
193	C₂×C₃	#1955	1	1	yes	characteristic
194	C₂×C₃	#1960	1	1	yes	characteristic
195	C₂×C₃	#1962	1	1	yes	characteristic
196	C₂²	#5207	1	1	yes	characteristic
197	C₂²	#5221	1	1	yes	characteristic
198	C₂²	#5225	1	2	yes	—
199	C₂²	#5234	1	1	yes	characteristic
200	C₂²	C₂×#837	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	4	no	—
203	C₄	—	4	2	no	—
204	C₄	—	4	4	no	—
205	C₄	—	8	2	no	—
206	C₄	#5011	1	4	yes	—
207	C₄	#5082	1	2	yes	—
208	C₄	#5086	1	2	yes	—
209	C₃	#9647	1	1	yes	characteristic, complemented, Sylow
210	C₂	#71615	1	1	yes	characteristic
211	C₂	#71623	1	2	yes	—
212	C₂	#71786	1	1	yes	characteristic
213	C₂	#71815	1	1	yes	characteristic
214	C₂	#71825	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#4535

C₂

yes

maximal

#69392

C₂

yes

maximal, characteristic

C₄×#803

C₂

yes

maximal, characteristic

#9647

—

maximal, Sylow

C₂×C₃×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₃×C₄×Q₈

C₂²

yes

—

C₃×#391

C₄

yes

—

C₃×#389

—

C₃×#384

C₄

yes

characteristic

C₃×C₄×C₄⋊C₄

—

C₃×C₄³

C₂²

yes

characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₂×#803

—

C₂×C₄×C₃⋊C₈

C₂²

yes

—

#4535

—

#4524

—

C₄×C₄⋊Q₈

D₃

yes

characteristic

#1831

—

C₄×#97

—

C₂×Q₈⋊C₈

—

C₂²×C₃×Q₈

C₂×C₄

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₂×C₃×C₄²

C₂×C₄

yes

characteristic

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

D₄

yes

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₂²×C₃⋊C₈

—

C₂²×C₃⋊C₈

—

C₃×#95

—

C₄⋊(C₃⋊C₈)

—

#803

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

D₄

yes

—

C₂×C₄⋊Q₈

Dic₃

yes

characteristic

C₂×C₄×Q₈

C₂×D₃

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#391

Dic₃

yes

—

#389

—

#384

Dic₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄³

C₂×D₃

yes

characteristic

#336

—

Q₈⋊C₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

M₁₆

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

—

C₂²×C₃×C₄

C₂²⋊C₄

yes

—

C₂²×C₃×C₄

C₂²×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×D₄

yes

—

C₃×C₄²

#41

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

C₂×Dic₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×Dic₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₃

yes

characteristic

C₂×C₄²

C₂×Dic₃

yes

characteristic

C₂×C₄²

#67

yes

—

C₂×C₄²

#67

yes

—

C₂×C₄²

C₂²×D₃

yes

characteristic

C₄⋊C₈

—

#97

—

100

C₄×C₈

—

101

C₄×C₈

—

102

#95

—

103

C₂³×C₃

C₂×C₂²⋊C₄

yes

characteristic

104

C₃×Q₈

—

105

C₃×Q₈

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

#104

yes

—

118

C₂×C₃×C₄

C₄×D₄

yes

—

119

C₂×C₃×C₄

#121

yes

—

120

C₂×C₃×C₄

#124

yes

characteristic

121

C₂×C₃×C₄

#127

yes

characteristic

122

C₂×C₃×C₄

C₂×M₁₆

yes

—

123

C₂×C₃×C₄

#131

yes

—

124

C₂×C₃×C₄

#137

yes

—

125

C₃⋊C₈

—

126

C₃⋊C₈

—

127

C₂×Q₈

—

128

C₂×Q₈

—

129

C₂×Q₈

—

130

C₂×Q₈

#208

yes

—

131

C₂²×C₄

—

132

C₂²×C₄

—

133

C₂²×C₄

#217

yes

—

134

C₂²×C₄

#217

yes

—

135

C₂²×C₄

C₂²×Dic₃

yes

characteristic

136

C₂²×C₄

C₂×#67

yes

characteristic

137

C₂²×C₄

C₂×#67

yes

characteristic

138

C₂×C₈

—

139

C₂×C₈

—

140

C₂×C₈

—

141

C₂×C₈

—

142

C₄⋊C₄

—

143

C₄⋊C₄

—

144

C₄⋊C₄

—

145

C₄⋊C₄

—

146

C₄²

—

147

C₄²

—

148

C₄²

—

149

C₄²

—

150

C₄²

—

151

C₄²

—

152

C₄²

D₃×D₄

yes

—

153

C₄²

#237

yes

—

154

C₂²×C₃

#390

yes

characteristic

155

C₂²×C₃

#407

yes

characteristic

156

C₂²×C₃

#408

yes

characteristic

157

C₂²×C₃

#419

yes

—

158

C₂²×C₃

C₂×#104

yes

—

159

C₃×C₄

—

160

C₃×C₄

—

161

C₃×C₄

—

162

C₃×C₄

—

163

C₃×C₄

—

164

C₃×C₄

#435

yes

—

165

C₃×C₄

#491

yes

—

166

C₃×C₄

#497

yes

—

167

C₂³

C₂×#217

yes

characteristic

168

Q₈

—

169

Q₈

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

—

181

C₂×C₄

#837

yes

—

182

C₂×C₄

C₂×#208

yes

—

183

C₂×C₄

D₄×Dic₃

yes

—

184

C₂×C₄

#936

yes

characteristic

185

C₂×C₄

#938

yes

—

186

C₂×C₄

#940

yes

characteristic

187

C₂×C₄

#942

yes

—

188

C₂×C₄

#948

yes

—

189

C₈

—

190

C₈

—

191

C₂×C₃

#1485

yes

—

192

C₂×C₃

#1491

yes

—

193

C₂×C₃

#1955

yes

characteristic

194

C₂×C₃

#1960

yes

characteristic

195

C₂×C₃

#1962

yes

characteristic

196

C₂²

#5207

yes

characteristic

197

C₂²

#5221

yes

characteristic

198

C₂²

#5225

yes

—

199

C₂²

#5234

yes

characteristic

200

C₂²

C₂×#837

yes

—

201

C₄

—

202

C₄

—

203

C₄

—

204

C₄

—

205

C₄

—

206

C₄

#5011

yes

—

207

C₄

#5082

yes

—

208

C₄

#5086

yes

—

209

C₃

#9647

yes

characteristic, complemented, Sylow

210

C₂

#71615

yes

characteristic

211

C₂

#71623

yes

—

212

C₂

#71786

yes

characteristic

213

C₂

#71815

yes

characteristic

214

C₂

#71825

yes

—

	subgroup	G/H
center	C₂²×C₄	C₂×#67
commutator	C₂²×C₃×C₄	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₃
Fitting	C₃×C₄×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₃	C₂×C₂²⋊C₄
cosocle	C₂×C₃×C₄²	C₂³
hypercenter	C₄×C₄⋊Q₈	D₃
hypocenter	C₃	#9647
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

subgroup

G/H

center

C₂²×C₄

C₂×#67

commutator

C₂²×C₃×C₄

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₃

Fitting

C₃×C₄×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₃

C₂×C₂²⋊C₄

cosocle

C₂×C₃×C₄²

C₂³

hypercenter

C₄×C₄⋊Q₈

D₃

hypocenter

C₃

#9647

LCS

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

N	H ≅ G/N	type
209. C₃	#9647	semidirect

H ≅ G/N

type

209. C₃

#9647

semidirect

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

subgroup	index
C₃×C₄×C₄⋊Q₈	2
C₂×#4535	2
#69392	2
C₄×#803	2
#9647	3

subgroup

index

C₃×C₄×C₄⋊Q₈

Subgroups — #10694004

Conjugacy classes of subgroups

Complements

Maximal subgroups