Group #11766747 — subgroups

Subgroups — C₉×#2026 #11766747

Order 1152.

387 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 387 conjugacy classes of subgroups — too many to draw. Only its 150 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	C₂×C₉×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₉×#103	C₂	1	2	yes	maximal
3	C₉×#391	C₂	1	1	yes	maximal, characteristic
4	C₉×#340	C₂	1	2	yes	maximal
5	C₉×#336	C₂	1	1	yes	maximal, characteristic
6	C₃×#2026	C₃	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₉	C₂²	1	2	yes	—
9	C₉×#125	—	2	4	no	—
10	C₂×C₉×C₄⋊C₄	—	2	2	no	—
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₉×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
14	C₉×#103	—	2	4	no	—
15	C₉×#95	—	2	2	no	—
16	C₂×C₃×#125	C₂×C₃	1	1	yes	characteristic
17	C₂×C₃×#103	C₂×C₃	1	2	yes	—
18	C₃×#391	C₂×C₃	1	1	yes	characteristic
19	C₃×#340	C₂×C₃	1	2	yes	—
20	C₃×#336	C₂×C₃	1	1	yes	characteristic
21	C₂×Q₈×C₉	—	2	1	no	—
22	C₂×Q₈×C₉	—	2	2	no	—
23	C₂×Q₈×C₉	—	2	2	no	—
24	C₂×Q₈×C₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₉	—	2	2	no	—
26	C₂²×C₄×C₉	—	2	1	no	—
27	C₂²×C₄×C₉	—	2	1	no	—
28	C₂²×C₄×C₉	D₄	1	2	yes	—
29	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₉	—	2	2	no	—
31	C₂×C₈×C₉	—	2	2	no	—
32	C₂×C₈×C₉	—	2	2	no	—
33	C₉×C₄⋊C₄	—	2	8	no	—
34	C₉×C₄⋊C₄	—	2	2	no	—
35	C₉×C₄⋊C₄	—	2	2	no	—
36	C₉×C₄⋊C₄	D₄	1	4	yes	—
37	C₄²×C₉	—	2	2	no	—
38	#2026	C₉	1	1	yes	characteristic, direct product factor, Sylow
39	C₂²×C₃×Q₈	C₂²×C₃	1	1	yes	characteristic
40	C₂²×C₃×C₈	C₂²×C₃	1	2	yes	—
41	C₃×#125	—	2	4	no	—
42	C₂×C₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
44	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
45	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
46	C₃×#103	—	2	4	no	—
47	C₃×#95	—	2	2	no	—
48	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
49	Q₈×C₉	—	2	4	no	—
50	Q₈×C₉	—	4	2	no	—
51	C₂×C₄×C₉	—	2	2	no	—
52	C₂×C₄×C₉	—	2	4	no	—
53	C₂×C₄×C₉	—	2	2	no	—
54	C₂×C₄×C₉	—	2	2	no	—
55	C₂×C₄×C₉	—	2	2	no	—
56	C₂×C₄×C₉	—	4	1	no	—
57	C₂×C₄×C₉	—	4	1	no	—
58	C₂×C₄×C₉	—	4	2	no	—
59	C₂×C₄×C₉	—	4	2	no	—
60	C₂×C₄×C₉	—	4	1	no	—
61	C₂×C₄×C₉	—	4	2	no	—
62	C₂×C₄×C₉	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₉	#41	1	2	yes	—
65	C₂×C₄×C₉	#41	1	1	yes	characteristic
66	C₈×C₉	—	4	2	no	—
67	C₂×#125	C₂×C₉	1	1	yes	characteristic
68	C₂×#103	C₂×C₉	1	2	yes	—
69	#391	C₂×C₉	1	1	yes	characteristic
70	#340	C₂×C₉	1	2	yes	—
71	#336	C₂×C₉	1	1	yes	characteristic
72	C₂×C₃×Q₈	—	2	1	no	—
73	C₂×C₃×Q₈	—	2	2	no	—
74	C₂×C₃×Q₈	—	2	2	no	—
75	C₂×C₃×Q₈	C₃×Q₈	1	2	yes	—
76	C₂²×C₃×C₄	—	2	1	no	—
77	C₂²×C₃×C₄	—	2	2	no	—
78	C₂²×C₃×C₄	—	2	1	no	—
79	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
80	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
81	C₂×C₃×C₈	—	2	2	no	—
82	C₂×C₃×C₈	—	2	2	no	—
83	C₂×C₃×C₈	—	2	2	no	—
84	C₃×C₄⋊C₄	—	2	8	no	—
85	C₃×C₄⋊C₄	—	2	2	no	—
86	C₃×C₄⋊C₄	—	2	2	no	—
87	C₃×C₄⋊C₄	C₃×D₄	1	4	yes	—
88	C₃×C₄²	—	2	2	no	—
89	C₂²×C₉	#120	1	1	yes	characteristic
90	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂²×C₉	#124	1	1	yes	characteristic
92	C₂²×C₉	#135	1	2	yes	—
93	C₂²×C₉	#137	1	2	yes	—
94	C₄×C₉	—	4	2	no	—
95	C₄×C₉	—	4	2	no	—
96	C₄×C₉	—	4	4	no	—
97	C₄×C₉	—	8	1	no	—
98	C₄×C₉	C₂²⋊Q₈	1	2	yes	—
99	C₄×C₉	#124	1	2	yes	—
100	C₂²×Q₈	C₂²×C₉	1	1	yes	characteristic
101	C₂²×C₈	C₂²×C₉	1	2	yes	—
102	#125	—	2	4	no	—
103	C₂×C₄⋊C₄	—	2	2	no	—
104	C₂×C₄⋊C₄	C₂²×C₉	1	1	yes	characteristic
105	C₂×C₄⋊C₄	C₂²×C₉	1	2	yes	—
106	C₂×C₄²	C₂²×C₉	1	1	yes	characteristic
107	#103	—	2	4	no	—
108	#95	—	2	2	no	—
109	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
110	C₃×Q₈	—	2	4	no	—
111	C₃×Q₈	—	4	2	no	—
112	C₂×C₃×C₄	—	2	2	no	—
113	C₂×C₃×C₄	—	2	4	no	—
114	C₂×C₃×C₄	—	2	2	no	—
115	C₂×C₃×C₄	—	2	2	no	—
116	C₂×C₃×C₄	—	2	2	no	—
117	C₂×C₃×C₄	—	4	1	no	—
118	C₂×C₃×C₄	—	4	1	no	—
119	C₂×C₃×C₄	—	4	2	no	—
120	C₂×C₃×C₄	—	4	2	no	—
121	C₂×C₃×C₄	—	4	1	no	—
122	C₂×C₃×C₄	—	4	2	no	—
123	C₂×C₃×C₄	C₂×C₃×D₄	1	2	yes	—
124	C₂×C₃×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
125	C₂×C₃×C₄	C₃×#41	1	2	yes	—
126	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
127	C₃×C₈	—	4	2	no	—
128	C₂×C₉	#393	1	1	yes	characteristic
129	C₂×C₉	#452	1	2	yes	—
130	C₂×C₉	#479	1	2	yes	—
131	C₂×C₉	#488	1	1	yes	characteristic
132	C₂×C₉	#491	1	1	yes	characteristic
133	C₂×Q₈	—	2	1	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂×Q₈	—	2	2	no	—
136	C₂×Q₈	Q₈×C₉	1	2	yes	—
137	C₂²×C₄	—	2	1	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	1	no	—
140	C₂²×C₄	D₄×C₉	1	2	yes	—
141	C₂²×C₄	C₂³×C₉	1	1	yes	characteristic
142	C₂×C₈	—	2	2	no	—
143	C₂×C₈	—	2	2	no	—
144	C₂×C₈	—	2	2	no	—
145	C₄⋊C₄	—	2	8	no	—
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	D₄×C₉	1	4	yes	—
149	C₄²	—	2	2	no	—
150	C₂²×C₃	C₃×#120	1	1	yes	characteristic
151	C₂²×C₃	C₃×C₂²⋊Q₈	1	1	yes	characteristic
152	C₂²×C₃	C₃×#124	1	1	yes	characteristic
153	C₂²×C₃	C₃×#135	1	2	yes	—
154	C₂²×C₃	C₃×#137	1	2	yes	—
155	C₃×C₄	—	4	2	no	—
156	C₃×C₄	—	4	2	no	—
157	C₃×C₄	—	4	4	no	—
158	C₃×C₄	—	8	1	no	—
159	C₃×C₄	C₃×C₂²⋊Q₈	1	2	yes	—
160	C₃×C₄	C₃×#124	1	2	yes	—
161	C₉	#2026	1	1	yes	characteristic, direct product factor, Sylow
162	C₂³	C₂×D₄×C₉	1	1	yes	characteristic
163	Q₈	—	2	4	no	—
164	Q₈	—	4	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	2	no	—
170	C₂×C₄	—	4	1	no	—
171	C₂×C₄	—	4	1	no	—
172	C₂×C₄	—	4	2	no	—
173	C₂×C₄	—	4	2	no	—
174	C₂×C₄	—	4	1	no	—
175	C₂×C₄	—	4	2	no	—
176	C₂×C₄	C₂×D₄×C₉	1	2	yes	—
177	C₂×C₄	C₂×Q₈×C₉	1	1	yes	characteristic
178	C₂×C₄	C₉×#41	1	2	yes	—
179	C₂×C₄	C₉×#41	1	1	yes	characteristic
180	C₈	—	4	2	no	—
181	C₂×C₃	C₃×#393	1	1	yes	characteristic
182	C₂×C₃	C₃×#452	1	2	yes	—
183	C₂×C₃	C₃×#479	1	2	yes	—
184	C₂×C₃	C₃×#488	1	1	yes	characteristic
185	C₂×C₃	C₃×#491	1	1	yes	characteristic
186	C₂²	C₉×#120	1	1	yes	characteristic
187	C₂²	C₉×C₂²⋊Q₈	1	1	yes	characteristic
188	C₂²	C₉×#124	1	1	yes	characteristic
189	C₂²	C₉×#135	1	2	yes	—
190	C₂²	C₉×#137	1	2	yes	—
191	C₄	—	4	2	no	—
192	C₄	—	4	2	no	—
193	C₄	—	4	4	no	—
194	C₄	—	8	1	no	—
195	C₄	C₉×C₂²⋊Q₈	1	2	yes	—
196	C₄	C₉×#124	1	2	yes	—
197	C₃	C₃×#2026	1	1	yes	characteristic
198	C₂	C₉×#393	1	1	yes	characteristic
199	C₂	C₉×#452	1	2	yes	—
200	C₂	C₉×#479	1	2	yes	—
201	C₂	C₉×#488	1	1	yes	characteristic
202	C₂	C₉×#491	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₉×#125

C₂

yes

maximal, characteristic

C₂×C₉×#103

C₂

yes

maximal

C₉×#391

C₂

yes

maximal, characteristic

C₉×#340

C₂

yes

maximal

C₉×#336

C₂

yes

maximal, characteristic

C₃×#2026

C₃

yes

maximal, characteristic

C₂²×Q₈×C₉

C₂²

yes

characteristic

C₂²×C₈×C₉

C₂²

yes

—

C₉×#125

—

C₂×C₉×C₄⋊C₄

—

C₂×C₉×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₉×C₄⋊C₄

C₂²

yes

—

C₂×C₄²×C₉

C₂²

yes

characteristic

C₉×#103

—

C₉×#95

—

C₂×C₃×#125

C₂×C₃

yes

characteristic

C₂×C₃×#103

C₂×C₃

yes

—

C₃×#391

C₂×C₃

yes

characteristic

C₃×#340

C₂×C₃

yes

—

C₃×#336

C₂×C₃

yes

characteristic

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

Q₈

yes

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

D₄

yes

—

C₂²×C₄×C₉

C₂³

yes

characteristic

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

D₄

yes

—

C₄²×C₉

—

#2026

C₉

yes

characteristic, direct product factor, Sylow

C₂²×C₃×Q₈

C₂²×C₃

yes

characteristic

C₂²×C₃×C₈

C₂²×C₃

yes

—

C₃×#125

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂²×C₃

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²×C₃

yes

—

C₂×C₃×C₄²

C₂²×C₃

yes

characteristic

C₃×#103

—

C₃×#95

—

C₂³×C₉

C₂×D₄

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×D₄

yes

—

C₂×C₄×C₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₉

#41

yes

—

C₂×C₄×C₉

#41

yes

characteristic

C₈×C₉

—

C₂×#125

C₂×C₉

yes

characteristic

C₂×#103

C₂×C₉

yes

—

#391

C₂×C₉

yes

characteristic

#340

C₂×C₉

yes

—

#336

C₂×C₉

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃×Q₈

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃×D₄

yes

—

C₂²×C₃×C₄

C₂³×C₃

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₃×D₄

yes

—

C₃×C₄²

—

C₂²×C₉

#120

yes

characteristic

C₂²×C₉

C₂²⋊Q₈

yes

C₂²×C₉

yes

C₂²×C₉

yes

—

C₂²×C₉

#137

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂²⋊Q₈

yes

—

C₄×C₉

#124

yes

—

100

C₂²×Q₈

C₂²×C₉

yes

characteristic

101

C₂²×C₈

C₂²×C₉

yes

—

102

#125

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

C₂²×C₉

yes

characteristic

105

C₂×C₄⋊C₄

C₂²×C₉

yes

—

106

C₂×C₄²

C₂²×C₉

yes

characteristic

107

#103

—

108

#95

—

109

C₂³×C₃

C₂×C₃×D₄

yes

characteristic

110

C₃×Q₈

—

111

C₃×Q₈

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

—

118

C₂×C₃×C₄

—

119

C₂×C₃×C₄

—

120

C₂×C₃×C₄

—

121

C₂×C₃×C₄

—

122

C₂×C₃×C₄

—

123

C₂×C₃×C₄

C₂×C₃×D₄

yes

—

124

C₂×C₃×C₄

C₂×C₃×Q₈

yes

characteristic

125

C₂×C₃×C₄

C₃×#41

yes

—

126

C₂×C₃×C₄

C₃×#41

yes

characteristic

127

C₃×C₈

—

128

C₂×C₉

#393

yes

characteristic

129

C₂×C₉

#452

yes

—

130

C₂×C₉

#479

yes

—

131

C₂×C₉

#488

yes

characteristic

132

C₂×C₉

#491

yes

characteristic

133

C₂×Q₈

—

134

C₂×Q₈

—

135

C₂×Q₈

—

136

C₂×Q₈

Q₈×C₉

yes

—

137

C₂²×C₄

—

138

C₂²×C₄

—

139

C₂²×C₄

—

140

C₂²×C₄

D₄×C₉

yes

—

141

C₂²×C₄

C₂³×C₉

yes

characteristic

142

C₂×C₈

—

143

C₂×C₈

—

144

C₂×C₈

—

145

C₄⋊C₄

—

146

C₄⋊C₄

—

147

C₄⋊C₄

—

148

C₄⋊C₄

D₄×C₉

yes

—

149

C₄²

—

150

C₂²×C₃

C₃×#120

yes

characteristic

151

C₂²×C₃

C₃×C₂²⋊Q₈

yes

characteristic

152

C₂²×C₃

C₃×#124

yes

characteristic

153

C₂²×C₃

C₃×#135

yes

—

154

C₂²×C₃

C₃×#137

yes

—

155

C₃×C₄

—

156

C₃×C₄

—

157

C₃×C₄

—

158

C₃×C₄

—

159

C₃×C₄

C₃×C₂²⋊Q₈

yes

—

160

C₃×C₄

C₃×#124

yes

—

161

C₉

#2026

yes

characteristic, direct product factor, Sylow

162

C₂³

C₂×D₄×C₉

yes

characteristic

163

Q₈

—

164

Q₈

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

—

170

C₂×C₄

—

171

C₂×C₄

—

172

C₂×C₄

—

173

C₂×C₄

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

C₂×D₄×C₉

yes

—

177

C₂×C₄

C₂×Q₈×C₉

yes

characteristic

178

C₂×C₄

C₉×#41

yes

—

179

C₂×C₄

C₉×#41

yes

characteristic

180

C₈

—

181

C₂×C₃

C₃×#393

yes

characteristic

182

C₂×C₃

C₃×#452

yes

—

183

C₂×C₃

C₃×#479

yes

—

184

C₂×C₃

C₃×#488

yes

characteristic

185

C₂×C₃

C₃×#491

yes

characteristic

186

C₂²

C₉×#120

yes

characteristic

187

C₂²

C₉×C₂²⋊Q₈

yes

characteristic

188

C₂²

C₉×#124

yes

characteristic

189

C₂²

C₉×#135

yes

—

190

C₂²

C₉×#137

yes

—

191

C₄

—

192

C₄

—

193

C₄

—

194

C₄

—

195

C₄

C₉×C₂²⋊Q₈

yes

—

196

C₄

C₉×#124

yes

—

197

C₃

C₃×#2026

yes

characteristic

198

C₂

C₉×#393

yes

characteristic

199

C₂

C₉×#452

yes

—

200

C₂

C₉×#479

yes

—

201

C₂

C₉×#488

yes

characteristic

202

C₂

C₉×#491

yes

characteristic

	subgroup	G/H
center	C₂³×C₉	C₂×D₄
commutator	C₂²×C₄	C₂³×C₉
Frattini, cosocle	C₂²×C₃×C₄	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂×C₃×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

subgroup

G/H

center

C₂³×C₉

C₂×D₄

commutator

C₂²×C₄

C₂³×C₉

Frattini, cosocle

C₂²×C₃×C₄

C₂³×C₃

socle, prime radical

C₂³×C₃

C₂×C₃×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₉ ◁ C₂×C₄²×C₉ ◁ G

N	H ≅ G/N	type
38. #2026	C₉	direct (N × H)
161. C₉	#2026	direct (N × H)

H ≅ G/N

type

38. #2026

C₉

direct (N × H)

161. C₉

#2026

direct (N × H)

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

subgroup	index
C₂×C₉×#125	2
C₂×C₉×#103	2
C₉×#391	2
C₉×#340	2
C₉×#336	2
C₃×#2026	3

subgroup

index

C₂×C₉×#125

C₂×C₉×#103

Subgroups — C₉×#2026 #11766747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#2026 #11766747

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#2026 #11766747