Group #13143611 — subgroups

Subgroups — #13143611

Order 1728.

684 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 156 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	C₂²×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₂×AGL(1,ℤ/9))⋊C₄	C₂	1	3	yes	maximal
3	3₋¹⁺²⋊#95	C₂	1	3	yes	maximal, complemented
4	C₃×#5231	—	3	1	no	maximal
5	#10588911	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂³×AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
7	C₂×Q₈×3₋¹⁺²	—	2	6	no	—
8	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
9	C₂²×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
10	(C₂×AGL(1,ℤ/9))⋊C₄	—	2	6	no	—
11	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
12	C₂×C₃×(C₂×D₃)⋊C₄	—	3	3	no	—
13	C₂²×Q₈×C₉	—	2	1	no	—
14	C₂²×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃×C₃⋊#95	—	3	3	no	—
16	C₂×(C₂×D₉)⋊C₄	C₂×C₃	1	3	yes	—
17	C₉⋊#95	C₂×C₃	1	3	yes	complemented
18	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
19	C₂²×AGL(1,ℤ/9)	—	2	3	no	—
20	C₂²×AGL(1,ℤ/9)	—	2	3	no	—
21	C₂²×AGL(1,ℤ/9)	Q₈	1	2	yes	complemented
22	Q₈×3₋¹⁺²	—	2	8	no	—
23	C₂×C₄×3₋¹⁺²	—	2	6	no	—
24	C₂×C₄×3₋¹⁺²	D₄	1	6	yes	—
25	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
26	C₂×3₋¹⁺²⋊C₄	—	2	6	no	—
27	C₃×#393	—	9	1	no	—
28	#5231	—	3	1	no	—
29	C₂³×C₃×D₃	—	3	1	no	—
30	C₂×C₃²×Q₈	—	2	6	no	—
31	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
32	C₂²×C₃×Dic₃	—	3	3	no	—
33	C₂³×D₉	C₂²×C₃	1	1	yes	characteristic
34	C₃×(C₂×D₃)⋊C₄	—	6	6	no	—
35	C₂×Q₈×C₉	—	2	12	no	—
36	C₂×Q₈×C₉	—	2	6	no	—
37	C₂²×C₄×C₉	—	2	3	no	—
38	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
39	C₂²×Dic₉	C₂²×C₃	1	3	yes	—
40	(C₂×D₉)⋊C₄	—	2	6	no	—
41	C₂²×3₋¹⁺²	C₂×D₄	1	3	yes	—
42	C₂²×3₋¹⁺²	C₂×Q₈	1	1	yes	characteristic
43	C₂²×3₋¹⁺²	#41	1	3	yes	—
44	C₂×AGL(1,ℤ/9)	—	2	6	no	—
45	C₂×AGL(1,ℤ/9)	—	4	1	no	—
46	C₂×AGL(1,ℤ/9)	—	4	3	no	—
47	C₄×3₋¹⁺²	—	2	6	no	—
48	3₋¹⁺²⋊C₄	—	4	3	no	—
49	C₂²×C₃×Q₈	—	3	1	no	—
50	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
51	C₂×C₃×C₂²⋊C₄	—	9	3	no	—
52	C₂×(C₂×D₃)⋊C₄	—	3	3	no	—
53	C₃×#95	—	9	3	no	—
54	C₃⋊#95	—	3	3	no	—
55	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
56	C₂²×C₃×D₃	—	3	2	no	—
57	C₂²×C₃×D₃	—	6	3	no	—
58	C₂²×C₃×D₃	—	6	3	no	—
59	C₃²×Q₈	—	2	8	no	—
60	C₂×C₃²×C₄	—	2	6	no	—
61	C₂×C₃²×C₄	#67	1	6	yes	—
62	C₂×C₃×Dic₃	—	6	6	no	—
63	C₂×C₃×Dic₃	—	6	3	no	—
64	C₂³×C₉	—	2	1	no	—
65	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
66	C₂²×D₉	—	2	3	no	—
67	C₂²×D₉	—	2	3	no	—
68	C₂²×D₉	C₃×Q₈	1	2	yes	complemented
69	Q₈×C₉	—	2	16	no	—
70	Q₈×C₉	—	2	8	no	—
71	C₂×C₄×C₉	—	2	12	no	—
72	C₂×C₄×C₉	—	2	6	no	—
73	C₂×C₄×C₉	—	2	6	no	—
74	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
75	C₂×Dic₉	—	2	3	no	—
76	C₂×Dic₉	—	2	6	no	—
77	#393	—	9	1	no	Sylow
78	C₂×3₋¹⁺²	#120	1	1	yes	characteristic
79	C₂×3₋¹⁺²	C₂²⋊Q₈	1	3	yes	—
80	C₂×3₋¹⁺²	#124	1	3	yes	—
81	AGL(1,ℤ/9)	—	4	2	no	—
82	C₂⁴×C₃	—	9	1	no	—
83	C₂³×D₃	—	3	1	no	—
84	C₂×C₃×Q₈	—	2	6	no	—
85	C₂×C₃×Q₈	—	6	6	no	—
86	C₂²×C₃×C₄	—	3	3	no	—
87	C₂²×C₃×C₄	—	9	3	no	—
88	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
89	C₂²×Dic₃	—	3	3	no	—
90	C₃×C₂²⋊C₄	—	18	6	no	—
91	(C₂×D₃)⋊C₄	—	6	6	no	—
92	C₂²×C₃²	D₃×Q₈	1	1	yes	characteristic
93	C₂²×C₃²	#239	1	3	yes	—
94	C₂²×C₃²	C₂×#67	1	3	yes	—
95	C₂×C₃×D₃	—	6	6	no	—
96	C₂×C₃×D₃	—	12	3	no	—
97	C₂×C₃×D₃	—	12	1	no	—
98	C₃²×C₄	—	2	6	no	—
99	C₃×Dic₃	—	12	3	no	—
100	C₂²×C₉	—	2	1	no	—
101	C₂²×C₉	—	2	3	no	—
102	C₂²×C₉	—	2	3	no	—
103	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
104	C₂²×C₉	C₂×C₃×Q₈	1	1	yes	characteristic
105	C₂²×C₉	C₃×#41	1	3	yes	—
106	C₂×D₉	—	2	6	no	—
107	C₂×D₉	—	4	1	no	—
108	C₂×D₉	—	4	3	no	—
109	C₄×C₉	—	2	12	no	—
110	C₄×C₉	—	2	6	no	—
111	Dic₉	—	4	3	no	—
112	C₂²×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
113	C₂×C₂²⋊C₄	—	9	3	no	—
114	#95	—	9	3	no	—
115	3₋¹⁺²	#393	1	1	yes	characteristic, complemented, Sylow
116	C₂³×C₃	—	3	1	no	—
117	C₂³×C₃	—	9	2	no	—
118	C₂³×C₃	—	18	3	no	—
119	C₂³×C₃	—	18	3	no	—
120	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
121	C₂²×D₃	—	3	2	no	—
122	C₂²×D₃	—	6	3	no	—
123	C₂²×D₃	—	6	3	no	—
124	C₃×Q₈	—	2	8	no	—
125	C₃×Q₈	—	6	8	no	—
126	C₂×C₃×C₄	—	2	6	no	—
127	C₂×C₃×C₄	—	3	6	no	—
128	C₂×C₃×C₄	—	6	6	no	—
129	C₂×C₃×C₄	—	18	3	no	—
130	C₂×C₃×C₄	—	18	6	no	—
131	C₂×C₃×C₄	C₃×#67	1	6	yes	—
132	C₂×Dic₃	—	6	6	no	—
133	C₂×Dic₃	—	6	3	no	—
134	C₂×C₃²	#946	1	3	yes	—
135	C₂×C₃²	#947	1	3	yes	—
136	C₂×C₃²	#953	1	1	yes	characteristic
137	C₃×D₃	—	12	2	no	—
138	C₂×C₉	—	2	3	no	—
139	C₂×C₉	—	2	3	no	—
140	C₂×C₉	—	2	1	no	—
141	C₂×C₉	C₃×#120	1	1	yes	characteristic
142	C₂×C₉	C₃×C₂²⋊Q₈	1	3	yes	—
143	C₂×C₉	C₃×#124	1	3	yes	—
144	D₉	—	4	2	no	—
145	C₂⁴	—	9	1	no	—
146	C₂×Q₈	—	2	6	no	—
147	C₂²×C₄	—	9	3	no	—
148	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
149	C₂²⋊C₄	—	18	6	no	—
150	C₂²×C₃	—	3	1	no	—
151	C₂²×C₃	—	3	3	no	—
152	C₂²×C₃	—	3	3	no	—
153	C₂²×C₃	—	18	6	no	—
154	C₂²×C₃	—	36	3	no	—
155	C₂²×C₃	—	36	1	no	—
156	C₂²×C₃	C₃×D₃×Q₈	1	1	yes	characteristic
157	C₂²×C₃	C₃×#239	1	3	yes	—
158	C₂²×C₃	C₂×C₃×#67	1	3	yes	—
159	C₂×D₃	—	6	6	no	—
160	C₂×D₃	—	12	3	no	—
161	C₂×D₃	—	12	1	no	—
162	C₃×C₄	—	2	6	no	—
163	C₃×C₄	—	6	6	no	—
164	C₃×C₄	—	36	3	no	—
165	Dic₃	—	12	3	no	—
166	C₃²	#5231	1	1	yes	characteristic
167	C₉	—	2	1	no	—
168	C₉	C₃×#393	1	1	yes	characteristic, complemented
169	C₂³	—	9	2	no	—
170	C₂³	—	18	3	no	—
171	C₂³	—	18	3	no	—
172	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
173	Q₈	—	2	8	no	—
174	C₂×C₄	—	2	6	no	—
175	C₂×C₄	—	18	6	no	—
176	C₂×C₄	—	18	3	no	—
177	C₂×C₄	#611⋊C₃	1	6	yes	—
178	C₂×C₃	—	3	3	no	—
179	C₂×C₃	—	3	3	no	—
180	C₂×C₃	—	3	1	no	—
181	C₂×C₃	—	36	2	no	—
182	C₂×C₃	C₃×#946	1	3	yes	—
183	C₂×C₃	C₃×#947	1	3	yes	—
184	C₂×C₃	C₃×#953	1	1	yes	characteristic
185	D₃	—	12	2	no	—
186	C₂²	—	18	6	no	—
187	C₂²	—	36	3	no	—
188	C₂²	—	36	1	no	—
189	C₂²	Q₈×AGL(1,ℤ/9)	1	1	yes	characteristic
190	C₂²	#88724	1	3	yes	—
191	C₂²	C₂×#611⋊C₃	1	3	yes	—
192	C₄	—	2	6	no	—
193	C₄	—	36	3	no	—
194	C₃	—	3	1	no	—
195	C₃	C₃×#5231	1	1	yes	characteristic
196	C₂	—	36	2	no	—
197	C₂	#63508⋊C₃	1	3	yes	—
198	C₂	#63509⋊C₃	1	3	yes	—
199	C₂	#11721704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₈×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

C₂×(C₂×AGL(1,ℤ/9))⋊C₄

C₂

yes

maximal

3₋¹⁺²⋊#95

C₂

yes

maximal, complemented

C₃×#5231

—

maximal

#10588911

C₃

yes

maximal, characteristic, complemented

C₂³×AGL(1,ℤ/9)

C₂²

yes

characteristic

C₂×Q₈×3₋¹⁺²

—

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

—

(C₂×AGL(1,ℤ/9))⋊C₄

—

C₂²×C₃²×Q₈

D₃

yes

characteristic

C₂×C₃×(C₂×D₃)⋊C₄

—

C₂²×Q₈×C₉

—

C₂²×Q₈×C₉

C₂×C₃

yes

characteristic, complemented

C₃×C₃⋊#95

—

C₂×(C₂×D₉)⋊C₄

C₂×C₃

yes

—

C₉⋊#95

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

C₂³

yes

—

—

Q₈

yes

complemented

Q₈×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

—

C₃×#393

—

#5231

—

C₂³×C₃×D₃

—

C₂×C₃²×Q₈

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₂³×D₉

C₂²×C₃

yes

characteristic

C₃×(C₂×D₃)⋊C₄

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×Dic₉

C₂²×C₃

yes

—

(C₂×D₉)⋊C₄

—

C₂²×3₋¹⁺²

C₂×D₄

yes

—

C₂²×3₋¹⁺²

C₂×Q₈

yes

characteristic

C₂²×3₋¹⁺²

#41

yes

—

C₂×AGL(1,ℤ/9)

—

C₂×AGL(1,ℤ/9)

—

C₂×AGL(1,ℤ/9)

—

C₄×3₋¹⁺²

—

3₋¹⁺²⋊C₄

—

C₂²×C₃×Q₈

—

C₂²×C₃×Q₈

C₃×D₃

yes

characteristic

C₂×C₃×C₂²⋊C₄

—

C₂×(C₂×D₃)⋊C₄

—

C₃×#95

—

C₃⋊#95

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₂²×C₃×D₃

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₂³×C₃

yes

characteristic

C₂²×D₉

—

C₂²×D₉

—

C₂²×D₉

C₃×Q₈

yes

complemented

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

—

C₂×3₋¹⁺²

yes

C₂×3₋¹⁺²

C₂²⋊Q₈

yes

—

C₂×3₋¹⁺²

#124

yes

—

AGL(1,ℤ/9)

—

C₂⁴×C₃

—

C₂³×D₃

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×Dic₃

—

C₃×C₂²⋊C₄

—

(C₂×D₃)⋊C₄

—

C₂²×C₃²

D₃×Q₈

yes

characteristic

C₂²×C₃²

#239

yes

—

C₂²×C₃²

C₂×#67

yes

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃×Dic₃

—

100

C₂²×C₉

—

101

C₂²×C₉

—

102

C₂²×C₉

—

103

C₂²×C₉

C₂×C₃×D₄

yes

—

104

C₂²×C₉

C₂×C₃×Q₈

yes

characteristic

105

C₂²×C₉

C₃×#41

yes

—

106

C₂×D₉

—

107

C₂×D₉

—

108

C₂×D₉

—

109

C₄×C₉

—

110

C₄×C₉

—

111

Dic₉

—

112

C₂²×Q₈

AGL(1,ℤ/9)

yes

characteristic, complemented

113

C₂×C₂²⋊C₄

—

114

#95

—

115

3₋¹⁺²

#393

yes

characteristic, complemented, Sylow

116

C₂³×C₃

—

117

C₂³×C₃

—

118

C₂³×C₃

—

119

C₂³×C₃

—

120

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

121

C₂²×D₃

—

122

C₂²×D₃

—

123

C₂²×D₃

—

124

C₃×Q₈

—

125

C₃×Q₈

—

126

C₂×C₃×C₄

—

127

C₂×C₃×C₄

—

128

C₂×C₃×C₄

—

129

C₂×C₃×C₄

—

130

C₂×C₃×C₄

—

131

C₂×C₃×C₄

C₃×#67

yes

—

132

C₂×Dic₃

—

133

C₂×Dic₃

—

134

C₂×C₃²

#946

yes

—

135

C₂×C₃²

#947

yes

—

136

C₂×C₃²

#953

yes

characteristic

137

C₃×D₃

—

138

C₂×C₉

—

139

C₂×C₉

—

140

C₂×C₉

—

141

C₂×C₉

C₃×#120

yes

characteristic

142

C₂×C₉

C₃×C₂²⋊Q₈

yes

—

143

C₂×C₉

C₃×#124

yes

—

144

D₉

—

145

C₂⁴

—

146

C₂×Q₈

—

147

C₂²×C₄

—

148

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

149

C₂²⋊C₄

—

150

C₂²×C₃

—

151

C₂²×C₃

—

152

C₂²×C₃

—

153

C₂²×C₃

—

154

C₂²×C₃

—

155

C₂²×C₃

—

156

C₂²×C₃

C₃×D₃×Q₈

yes

characteristic

157

C₂²×C₃

C₃×#239

yes

—

158

C₂²×C₃

C₂×C₃×#67

yes

—

159

C₂×D₃

—

160

C₂×D₃

—

161

C₂×D₃

—

162

C₃×C₄

—

163

C₃×C₄

—

164

C₃×C₄

—

165

Dic₃

—

166

C₃²

#5231

yes

characteristic

167

C₉

—

168

C₉

C₃×#393

yes

characteristic, complemented

169

C₂³

—

170

C₂³

—

171

C₂³

—

172

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

173

Q₈

—

174

C₂×C₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

#611⋊C₃

yes

—

178

C₂×C₃

—

179

C₂×C₃

—

180

C₂×C₃

—

181

C₂×C₃

—

182

C₂×C₃

C₃×#946

yes

—

183

C₂×C₃

C₃×#947

yes

—

184

C₂×C₃

C₃×#953

yes

characteristic

185

D₃

—

186

C₂²

—

187

C₂²

—

188

C₂²

—

189

C₂²

Q₈×AGL(1,ℤ/9)

yes

characteristic

190

C₂²

#88724

yes

—

191

C₂²

C₂×#611⋊C₃

yes

—

192

C₄

—

193

C₄

—

194

C₃

—

195

C₃

C₃×#5231

yes

characteristic

196

C₂

—

197

C₂

#63508⋊C₃

yes

—

198

C₂

#63509⋊C₃

yes

—

199

C₂

#11721704

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×AGL(1,ℤ/9)
commutator, cosocle	C₂³×C₉	C₂³×C₃
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂²×Q₈×3₋¹⁺²	C₂
prime radical	C₂³×AGL(1,ℤ/9)	C₂²
hypercenter	C₂²×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#393
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center

C₂³

C₂²×AGL(1,ℤ/9)

commutator, cosocle

C₂³×C₉

C₂³×C₃

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

C₂²×Q₈×3₋¹⁺²

C₂

C₂²

C₂²×Q₈

C₉

G ▷ C₂³×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
1. C₂²×Q₈×3₋¹⁺²	C₂	semidirect
3. 3₋¹⁺²⋊#95	C₂	semidirect
5. #10588911	C₃	semidirect
14. C₂²×Q₈×C₉	C₂×C₃	semidirect
17. C₉⋊#95	C₂×C₃	semidirect
21. C₂²×AGL(1,ℤ/9)	Q₈	semidirect
68. C₂²×D₉	C₃×Q₈	semidirect
112. C₂²×Q₈	AGL(1,ℤ/9)	semidirect
115. 3₋¹⁺²	#393	semidirect
168. C₉	C₃×#393	semidirect

H ≅ G/N

type

1. C₂²×Q₈×3₋¹⁺²

C₂

semidirect

3. 3₋¹⁺²⋊#95

C₂

semidirect

5. #10588911

C₃

semidirect

14. C₂²×Q₈×C₉

C₂×C₃

semidirect

17. C₉⋊#95

C₂×C₃

semidirect

21. C₂²×AGL(1,ℤ/9)

Q₈

semidirect

68. C₂²×D₉

C₃×Q₈

semidirect

112. C₂²×Q₈

AGL(1,ℤ/9)

semidirect

115. 3₋¹⁺²

#393

semidirect

168. C₉

C₃×#393

semidirect

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

subgroup	index
C₂²×Q₈×3₋¹⁺²	2
C₂×(C₂×AGL(1,ℤ/9))⋊C₄	2
3₋¹⁺²⋊#95	2
C₃×#5231	3
#10588911	3

subgroup

index

C₂²×Q₈×3₋¹⁺²

C₂×(C₂×AGL(1,ℤ/9))⋊C₄

3₋¹⁺²⋊#95

C₃×#5231

#10588911

Subgroups — #13143611

Conjugacy classes of subgroups

Complements

Maximal subgroups