Group #11895153 — subgroups

Subgroups — #11895153

Order 1152.

555 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 186 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	D₄×Q₈×C₉	C₂	1	1	yes	maximal, characteristic
2	#10589672	C₂	1	1	yes	maximal, characteristic, complemented
3	#10588767	C₂	1	3	yes	maximal, complemented
4	#10588766	C₂	1	3	yes	maximal, complemented
5	Q₈×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588695	C₂	1	3	yes	maximal
7	C₄×#3672	C₂	1	3	yes	maximal
8	#83568	—	3	1	no	maximal
9	C₂²×Q₈×C₉	—	2	1	no	—
10	C₉×C₄⋊Q₈	C₂²	1	3	yes	—
11	C₉×C₂²⋊Q₈	—	2	3	no	—
12	C₄×Q₈×C₉	C₂²	1	1	yes	characteristic
13	C₄×D₄×C₉	C₂²	1	3	yes	—
14	Q₈×Dic₉	—	2	1	no	—
15	C₂×#3672	C₂²	1	3	yes	—
16	#63452	—	2	3	no	—
17	#63431	C₂²	1	3	yes	—
18	C₄×Dic₁₈	C₂²	1	3	yes	—
19	#63392	C₂²	1	1	yes	characteristic
20	#63368	C₂²	1	3	yes	—
21	#63368	C₂²	1	6	yes	—
22	#63367	C₂²	1	3	yes	—
23	C₄⋊(C₉⋊C₈)	C₂²	1	3	yes	—
24	C₄×C₉⋊C₈	C₂²	1	3	yes	—
25	C₃×D₄×Q₈	D₃	1	1	yes	characteristic
26	#5570	—	3	1	no	—
27	#5094	—	3	3	no	—
28	#5093	—	3	3	no	—
29	Q₈×C₃⋊C₈	—	3	1	no	—
30	#5019	—	3	3	no	—
31	C₄×#214	—	3	3	no	—
32	C₂×Q₈×C₉	—	2	3	no	—
33	C₂×Q₈×C₉	—	4	4	no	—
34	C₂×Q₈×C₉	D₄	1	1	yes	characteristic
35	C₂×D₄×C₉	C₂³	1	1	yes	characteristic
36	C₂²×C₄×C₉	—	2	3	no	—
37	C₂×Dic₁₈	C₂³	1	3	yes	—
38	C₉×C₄⋊C₄	—	2	3	no	—
39	C₉×C₄⋊C₄	D₄	1	3	yes	—
40	C₉×C₄⋊C₄	C₂³	1	3	yes	—
41	C₉×C₂²⋊C₄	—	2	3	no	—
42	C₄²×C₉	C₂³	1	3	yes	—
43	#3672	—	2	6	no	—
44	C₄⋊Dic₉	—	2	3	no	—
45	C₄⋊Dic₉	—	2	3	no	—
46	C₄⋊Dic₉	C₂³	1	1	yes	characteristic
47	Dic₉⋊C₄	—	2	3	no	—
48	C₄×Dic₉	—	2	3	no	—
49	C₂×C₉⋊C₈	C₂³	1	3	yes	—
50	C₂×C₉⋊C₈	C₂³	1	1	yes	characteristic
51	#3359	—	9	1	no	Sylow
52	C₂²×C₃×Q₈	—	2	1	no	—
53	C₃×C₄⋊Q₈	C₂×D₃	1	3	yes	—
54	C₃×C₂²⋊Q₈	—	2	3	no	—
55	C₃×C₄×Q₈	C₂×D₃	1	1	yes	characteristic
56	C₃×C₄×D₄	C₂×D₃	1	3	yes	—
57	Q₈×Dic₃	—	6	1	no	—
58	C₂×#214	—	3	3	no	—
59	#890	—	6	3	no	—
60	#869	—	3	3	no	—
61	C₄×Dic₆	—	3	3	no	—
62	#832	—	3	1	no	—
63	#810	—	3	3	no	—
64	#810	—	3	6	no	—
65	#808	—	3	3	no	—
66	C₄⋊(C₃⋊C₈)	—	3	3	no	—
67	C₄×C₃⋊C₈	—	3	3	no	—
68	C₂³×C₉	—	2	1	no	—
69	Q₈×C₉	—	4	6	no	—
70	Q₈×C₉	SD₁₆	1	4	yes	complemented
71	D₄×C₉	—	2	1	no	—
72	D₄×C₉	#41	1	2	yes	—
73	C₂×C₄×C₉	—	2	3	no	—
74	C₂×C₄×C₉	—	4	6	no	—
75	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
76	C₂×C₄×C₉	C₂×D₄	1	3	yes	—
77	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
78	C₂×Dic₉	—	2	1	no	—
79	C₂×Dic₉	—	2	3	no	—
80	Dic₁₈	—	2	3	no	—
81	Dic₁₈	—	2	3	no	—
82	C₉⋊C₈	—	2	3	no	—
83	C₉⋊C₈	—	2	1	no	—
84	#557	—	9	1	no	—
85	D₄×Q₈	D₉	1	1	yes	characteristic
86	#487	—	9	3	no	—
87	#475	—	9	3	no	—
88	#461	—	9	3	no	—
89	C₈×Q₈	—	9	1	no	—
90	C₄×SD₁₆	—	9	3	no	—
91	C₂×C₃×Q₈	—	2	3	no	—
92	C₂×C₃×Q₈	—	4	4	no	—
93	C₂×C₃×Q₈	#67	1	1	yes	characteristic
94	C₂×C₃×D₄	C₂²×D₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	—	2	3	no	—
96	C₂×Dic₆	—	3	3	no	—
97	C₃×C₄⋊C₄	—	2	3	no	—
98	C₃×C₄⋊C₄	#67	1	3	yes	—
99	C₃×C₄⋊C₄	C₂²×D₃	1	3	yes	—
100	C₃×C₂²⋊C₄	—	2	3	no	—
101	C₃×C₄²	C₂²×D₃	1	3	yes	—
102	#214	—	6	6	no	—
103	C₄⋊Dic₃	—	3	1	no	—
104	C₄⋊Dic₃	—	6	3	no	—
105	C₄⋊Dic₃	—	6	3	no	—
106	Dic₃⋊C₄	—	6	3	no	—
107	C₄×Dic₃	—	6	3	no	—
108	C₂×C₃⋊C₈	—	3	1	no	—
109	C₂×C₃⋊C₈	—	3	3	no	—
110	C₂²×C₉	—	2	2	no	—
111	C₂²×C₉	—	4	1	no	—
112	C₂²×C₉	—	4	1	no	—
113	C₂²×C₉	C₂²×D₄	1	1	yes	characteristic
114	C₄×C₉	—	2	3	no	—
115	C₄×C₉	—	4	3	no	—
116	C₄×C₉	C₂×SD₁₆	1	6	yes	—
117	C₄×C₉	C₂×#41	1	1	yes	characteristic
118	C₄×C₉	2₋¹⁺⁴	1	1	yes	characteristic
119	Dic₉	—	4	3	no	—
120	Dic₉	—	4	1	no	—
121	C₂²×Q₈	—	2	1	no	—
122	C₂×SD₁₆	—	9	3	no	—
123	C₄⋊Q₈	—	9	3	no	—
124	C₄⋊Q₈	C₂×D₉	1	3	yes	—
125	#125	—	18	3	no	—
126	C₂²⋊Q₈	—	2	3	no	—
127	C₄×Q₈	—	9	3	no	—
128	C₄×Q₈	—	18	1	no	—
129	C₄×Q₈	C₂×D₉	1	1	yes	characteristic
130	C₄×D₄	C₂×D₉	1	3	yes	—
131	#106	—	9	3	no	—
132	C₄⋊C₈	—	9	3	no	—
133	#103	—	9	3	no	—
134	#103	—	9	6	no	—
135	#102	—	9	1	no	—
136	C₄×C₈	—	9	3	no	—
137	C₂³×C₃	—	2	1	no	—
138	C₃×Q₈	—	4	6	no	—
139	C₃×Q₈	#214	1	4	yes	—
140	C₃×D₄	—	2	1	no	—
141	C₃×D₄	#239	1	2	yes	—
142	C₂×C₃×C₄	—	2	3	no	—
143	C₂×C₃×C₄	—	4	6	no	—
144	C₂×C₃×C₄	C₂×#67	1	3	yes	—
145	C₂×C₃×C₄	C₂×#67	1	3	yes	—
146	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
147	C₂×Dic₃	—	6	1	no	—
148	C₂×Dic₃	—	6	3	no	—
149	Dic₆	—	6	3	no	—
150	Dic₆	—	6	3	no	—
151	C₃⋊C₈	—	6	3	no	—
152	C₃⋊C₈	—	6	1	no	—
153	C₂×C₉	—	4	2	no	—
154	C₂×C₉	#548	1	1	yes	characteristic
155	C₂×C₉	C₂²×SD₁₆	1	1	yes	characteristic
156	C₂×C₉	#578	1	1	yes	characteristic
157	C₂×Q₈	—	2	3	no	—
158	C₂×Q₈	—	4	4	no	—
159	C₂×Q₈	—	9	3	no	—
160	C₂×Q₈	#611	1	1	yes	characteristic
161	C₂×D₄	C₂²×D₉	1	1	yes	characteristic
162	C₂²×C₄	—	2	3	no	—
163	SD₁₆	—	18	6	no	—
164	C₂×C₈	—	9	1	no	—
165	C₂×C₈	—	9	3	no	—
166	C₄⋊C₄	—	2	3	no	—
167	C₄⋊C₄	—	9	1	no	—
168	C₄⋊C₄	—	18	3	no	—
169	C₄⋊C₄	—	18	3	no	—
170	C₄⋊C₄	—	18	3	no	—
171	C₄⋊C₄	#611	1	3	yes	—
172	C₄⋊C₄	C₂²×D₉	1	3	yes	—
173	C₂²⋊C₄	—	2	3	no	—
174	C₄²	—	18	3	no	—
175	C₄²	C₂²×D₉	1	3	yes	—
176	C₂²×C₃	—	2	2	no	—
177	C₂²×C₃	—	4	1	no	—
178	C₂²×C₃	—	4	1	no	—
179	C₂²×C₃	C₂²×#67	1	1	yes	characteristic
180	C₃×C₄	—	2	3	no	—
181	C₃×C₄	—	4	3	no	—
182	C₃×C₄	C₂×#214	1	6	yes	—
183	C₃×C₄	C₂×#239	1	1	yes	characteristic
184	C₃×C₄	#1007	1	1	yes	characteristic
185	Dic₃	—	12	3	no	—
186	Dic₃	—	12	1	no	—
187	C₉	#3359	1	1	yes	characteristic, complemented, Sylow
188	C₂³	—	2	1	no	—
189	Q₈	—	4	6	no	—
190	Q₈	—	18	3	no	—
191	Q₈	—	18	3	no	—
192	Q₈	#3672	1	4	yes	complemented
193	D₄	—	2	1	no	—
194	D₄	#3701	1	2	yes	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	4	6	no	—
197	C₂×C₄	—	18	1	no	—
198	C₂×C₄	—	18	3	no	—
199	C₂×C₄	C₂×#611	1	3	yes	—
200	C₂×C₄	C₂×#611	1	3	yes	—
201	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
202	C₈	—	18	3	no	—
203	C₈	—	18	1	no	—
204	C₂×C₃	—	4	2	no	—
205	C₂×C₃	C₂²×#214	1	1	yes	characteristic
206	C₂×C₃	#5817	1	1	yes	characteristic
207	C₂×C₃	#5838	1	1	yes	characteristic
208	C₂²	—	2	2	no	—
209	C₂²	—	4	1	no	—
210	C₂²	—	4	1	no	—
211	C₂²	C₂²×#611	1	1	yes	characteristic
212	C₄	—	2	3	no	—
213	C₄	—	4	3	no	—
214	C₄	—	36	3	no	—
215	C₄	—	36	1	no	—
216	C₄	C₂×#3672	1	6	yes	—
217	C₄	C₂×#3701	1	1	yes	characteristic
218	C₄	#63713	1	1	yes	characteristic
219	C₃	#83568	1	1	yes	characteristic
220	C₂	—	4	2	no	—
221	C₂	C₂²×#3672	1	1	yes	characteristic
222	C₂	#10589919	1	1	yes	characteristic
223	C₂	#10589940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×Q₈×C₉

C₂

yes

maximal, characteristic

#10589672

C₂

yes

maximal, characteristic, complemented

#10588767

C₂

yes

maximal, complemented

#10588766

C₂

yes

maximal, complemented

Q₈×C₉⋊C₈

C₂

yes

maximal, characteristic, complemented

#10588695

C₂

yes

maximal

C₄×#3672

C₂

yes

maximal

#83568

—

maximal

C₂²×Q₈×C₉

—

C₉×C₄⋊Q₈

C₂²

yes

—

C₉×C₂²⋊Q₈

—

C₄×Q₈×C₉

C₂²

yes

characteristic

C₄×D₄×C₉

C₂²

yes

—

Q₈×Dic₉

—

C₂×#3672

C₂²

yes

—

#63452

—

#63431

C₂²

yes

—

C₄×Dic₁₈

C₂²

yes

—

#63392

C₂²

yes

characteristic

#63368

C₂²

yes

—

#63368

C₂²

yes

—

#63367

C₂²

yes

—

C₄⋊(C₉⋊C₈)

C₂²

yes

—

C₄×C₉⋊C₈

C₂²

yes

—

C₃×D₄×Q₈

D₃

yes

—

—

—

Q₈×C₃⋊C₈

—

#5019

—

C₄×#214

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

D₄

yes

characteristic

C₂×D₄×C₉

C₂³

yes

characteristic

C₂²×C₄×C₉

—

C₂×Dic₁₈

C₂³

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

D₄

yes

—

C₉×C₄⋊C₄

C₂³

yes

—

C₉×C₂²⋊C₄

—

C₄²×C₉

C₂³

yes

—

#3672

—

C₄⋊Dic₉

—

C₄⋊Dic₉

—

C₄⋊Dic₉

C₂³

yes

characteristic

Dic₉⋊C₄

—

C₄×Dic₉

—

C₂×C₉⋊C₈

C₂³

yes

—

C₂×C₉⋊C₈

C₂³

yes

characteristic

#3359

—

Sylow

C₂²×C₃×Q₈

—

C₃×C₄⋊Q₈

C₂×D₃

yes

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

C₂×D₃

yes

characteristic

C₃×C₄×D₄

C₂×D₃

yes

—

Q₈×Dic₃

—

C₂×#214

—

#890

—

#869

—

C₄×Dic₆

—

—

—

—

—

C₄⋊(C₃⋊C₈)

—

C₄×C₃⋊C₈

—

C₂³×C₉

—

Q₈×C₉

—

Q₈×C₉

SD₁₆

yes

complemented

D₄×C₉

—

D₄×C₉

#41

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂⁴

yes

characteristic

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

C₉⋊C₈

—

C₉⋊C₈

—

#557

—

D₄×Q₈

D₉

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

#67

yes

characteristic

C₂×C₃×D₄

C₂²×D₃

yes

characteristic

C₂²×C₃×C₄

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

#67

yes

—

C₃×C₄⋊C₄

C₂²×D₃

yes

—

100

C₃×C₂²⋊C₄

—

101

C₃×C₄²

C₂²×D₃

yes

—

102

#214

—

103

C₄⋊Dic₃

—

104

C₄⋊Dic₃

—

105

C₄⋊Dic₃

—

106

Dic₃⋊C₄

—

107

C₄×Dic₃

—

108

C₂×C₃⋊C₈

—

109

C₂×C₃⋊C₈

—

110

C₂²×C₉

—

111

C₂²×C₉

—

112

C₂²×C₉

—

113

C₂²×C₉

C₂²×D₄

yes

characteristic

114

C₄×C₉

—

115

C₄×C₉

—

116

C₄×C₉

C₂×SD₁₆

yes

—

117

C₄×C₉

C₂×#41

yes

characteristic

118

C₄×C₉

2₋¹⁺⁴

yes

characteristic

119

Dic₉

—

120

Dic₉

—

121

C₂²×Q₈

—

122

C₂×SD₁₆

—

123

C₄⋊Q₈

—

124

C₄⋊Q₈

C₂×D₉

yes

—

125

#125

—

126

C₂²⋊Q₈

—

127

C₄×Q₈

—

128

C₄×Q₈

—

129

C₄×Q₈

C₂×D₉

yes

characteristic

130

C₄×D₄

C₂×D₉

yes

—

131

#106

—

132

C₄⋊C₈

—

133

#103

—

134

#103

—

135

#102

—

136

C₄×C₈

—

137

C₂³×C₃

—

138

C₃×Q₈

—

139

C₃×Q₈

#214

yes

—

140

C₃×D₄

—

141

C₃×D₄

#239

yes

—

142

C₂×C₃×C₄

—

143

C₂×C₃×C₄

—

144

C₂×C₃×C₄

C₂×#67

yes

—

145

C₂×C₃×C₄

C₂×#67

yes

—

146

C₂×C₃×C₄

C₂³×D₃

yes

characteristic

147

C₂×Dic₃

—

148

C₂×Dic₃

—

149

Dic₆

—

150

Dic₆

—

151

C₃⋊C₈

—

152

C₃⋊C₈

—

153

C₂×C₉

—

154

C₂×C₉

#548

yes

characteristic

155

C₂×C₉

C₂²×SD₁₆

yes

characteristic

156

C₂×C₉

#578

yes

characteristic

157

C₂×Q₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

#611

yes

characteristic

161

C₂×D₄

C₂²×D₉

yes

characteristic

162

C₂²×C₄

—

163

SD₁₆

—

164

C₂×C₈

—

165

C₂×C₈

—

166

C₄⋊C₄

—

167

C₄⋊C₄

—

168

C₄⋊C₄

—

169

C₄⋊C₄

—

170

C₄⋊C₄

—

171

C₄⋊C₄

#611

yes

—

172

C₄⋊C₄

C₂²×D₉

yes

—

173

C₂²⋊C₄

—

174

C₄²

—

175

C₄²

C₂²×D₉

yes

—

176

C₂²×C₃

—

177

C₂²×C₃

—

178

C₂²×C₃

—

179

C₂²×C₃

C₂²×#67

yes

characteristic

180

C₃×C₄

—

181

C₃×C₄

—

182

C₃×C₄

C₂×#214

yes

—

183

C₃×C₄

C₂×#239

yes

characteristic

184

C₃×C₄

#1007

yes

characteristic

185

Dic₃

—

186

Dic₃

—

187

C₉

#3359

yes

characteristic, complemented, Sylow

188

C₂³

—

189

Q₈

—

190

Q₈

—

191

Q₈

—

192

Q₈

#3672

yes

complemented

193

D₄

—

194

D₄

#3701

yes

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

C₂×#611

yes

—

200

C₂×C₄

C₂×#611

yes

—

201

C₂×C₄

C₂³×D₉

yes

characteristic

202

C₈

—

203

C₈

—

204

C₂×C₃

—

205

C₂×C₃

C₂²×#214

yes

characteristic

206

C₂×C₃

#5817

yes

characteristic

207

C₂×C₃

#5838

yes

characteristic

208

C₂²

—

209

C₂²

—

210

C₂²

—

211

C₂²

C₂²×#611

yes

characteristic

212

C₄

—

213

C₄

—

214

C₄

—

215

C₄

—

216

C₄

C₂×#3672

yes

—

217

C₄

C₂×#3701

yes

characteristic

218

C₄

#63713

yes

characteristic

219

C₃

#83568

yes

characteristic

220

C₂

—

221

C₂

C₂²×#3672

yes

characteristic

222

C₂

#10589919

yes

characteristic

223

C₂

#10589940

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#611
commutator, cosocle	C₂×C₄×C₉	C₂⁴
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	D₄×Q₈×C₉	C₂
socle	C₂²×C₃	C₂²×#67
prime radical	C₂×C₃×D₄	C₂²×D₃
hypercenter	D₄×Q₈	D₉
hypocenter	C₉	#3359
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×#611

commutator, cosocle

C₂×C₄×C₉

C₂⁴

Frattini

C₂×C₃×C₄

C₂³×D₃

Fitting

D₄×Q₈×C₉

C₂

socle

C₂²×C₃

C₂²×#67

prime radical

C₂×C₃×D₄

C₂²×D₃

hypercenter

D₄×Q₈

D₉

hypocenter

C₉

#3359

LCS

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
2. #10589672	C₂	semidirect
3. #10588767	C₂	semidirect
4. #10588766	C₂	semidirect
5. Q₈×C₉⋊C₈	C₂	semidirect
70. Q₈×C₉	SD₁₆	semidirect
187. C₉	#3359	semidirect
192. Q₈	#3672	semidirect

H ≅ G/N

type

2. #10589672

C₂

semidirect

3. #10588767

C₂

semidirect

4. #10588766

C₂

semidirect

5. Q₈×C₉⋊C₈

C₂

semidirect

70. Q₈×C₉

SD₁₆

semidirect

187. C₉

#3359

semidirect

192. Q₈

#3672

semidirect

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

subgroup	index
D₄×Q₈×C₉	2
#10589672	2
#10588767	2
#10588766	2
Q₈×C₉⋊C₈	2
#10588695	2
C₄×#3672	2
#83568	3

subgroup

index

D₄×Q₈×C₉

Q₈×C₉⋊C₈

Subgroups — #11895153

Conjugacy classes of subgroups

Complements

Maximal subgroups