Group #10628463 — subgroups

Subgroups — C₃×#9856 #10628463

Order 768.

530 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 174 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×#3147	C₂	1	1	yes	maximal, characteristic
2	C₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#1492	C₂	1	4	yes	maximal, complemented
4	C₂×C₃×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
5	#9856	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×#135	—	2	2	no	—
7	C₃×#532	C₂²	1	2	yes	—
8	C₃×#495	—	2	2	no	—
9	C₃×#494	C₄	1	1	yes	characteristic
10	C₃×#493	C₄	1	1	yes	characteristic
11	C₃×#492	C₄	1	2	yes	—
12	C₂×C₃×#108	—	2	2	no	—
13	C₃×#423	C₂²	1	2	yes	—
14	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
15	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×D₄⋊C₈	—	2	8	no	—
17	C₃×C₈²	—	2	2	no	—
18	#3147	C₂×C₃	1	1	yes	characteristic
19	#1848	C₂×C₃	1	1	yes	characteristic, complemented
20	#1492	C₂×C₃	1	4	yes	complemented
21	C₂×C₈²	C₂×C₃	1	1	yes	characteristic, complemented
22	C₂×C₃×#41	—	2	2	no	—
23	C₃×#135	—	4	4	no	—
24	C₃×#135	—	4	4	no	—
25	C₂×C₃×Q₁₆	—	2	2	no	—
26	C₂×C₃×SD₁₆	—	2	4	no	—
27	C₂×C₃×D₈	—	2	2	no	—
28	C₂×C₃×M₁₆	—	2	2	no	—
29	C₂²×C₃×C₈	—	2	2	no	—
30	C₂²×C₃×C₈	D₄	1	2	yes	—
31	C₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₃×#127	C₂×C₄	1	2	yes	—
33	C₃×#124	—	2	2	no	—
34	C₃×#121	—	2	4	no	—
35	C₃×C₄×D₄	—	2	4	no	—
36	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
37	C₃×#108	—	4	4	no	—
38	C₃×C₄⋊C₈	—	2	4	no	—
39	C₃×C₄×C₈	—	2	2	no	—
40	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
42	C₃×C₄×C₈	D₄	1	4	yes	—
43	C₃×C₄×C₈	D₄	1	2	yes	—
44	C₂×#135	—	2	2	no	—
45	#532	C₂²×C₃	1	2	yes	—
46	#495	—	2	2	no	—
47	#494	C₃×C₄	1	1	yes	characteristic
48	#493	C₃×C₄	1	1	yes	characteristic
49	#492	C₃×C₄	1	2	yes	—
50	C₂×#108	—	2	2	no	—
51	#423	C₂²×C₃	1	2	yes	—
52	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
53	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
54	D₄⋊C₈	—	2	8	no	—
55	C₈²	—	2	2	no	—
56	C₃×#41	—	4	4	no	—
57	C₃×#41	—	4	4	no	—
58	C₂×C₃×Q₈	—	2	2	no	—
59	C₂×C₃×D₄	—	2	2	no	—
60	C₂×C₃×D₄	—	2	4	no	—
61	C₂×C₃×D₄	—	4	2	no	—
62	C₂²×C₃×C₄	—	4	2	no	—
63	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×C₄	#41	1	2	yes	—
65	C₃×Q₁₆	—	4	4	no	—
66	C₃×SD₁₆	—	4	8	no	—
67	C₃×D₈	—	4	4	no	—
68	C₃×M₁₆	—	4	4	no	—
69	C₂×C₃×C₈	—	2	8	no	—
70	C₂×C₃×C₈	—	2	4	no	—
71	C₂×C₃×C₈	—	4	2	no	—
72	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
73	C₂×C₃×C₈	C₂²⋊C₄	1	2	yes	—
74	C₂×C₃×C₈	D₈	1	4	yes	—
75	C₂×C₃×C₈	SD₁₆	1	4	yes	—
76	C₃×C₄⋊C₄	—	2	4	no	—
77	C₃×C₂²⋊C₄	—	4	4	no	—
78	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
79	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
80	C₃×C₄²	C₂×D₄	1	2	yes	—
81	C₂×#41	—	2	2	no	—
82	#135	—	4	4	no	—
83	#135	—	4	4	no	—
84	C₂×Q₁₆	—	2	2	no	—
85	C₂×SD₁₆	—	2	4	no	—
86	C₂×D₈	—	2	2	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂²×C₈	—	2	2	no	—
89	C₂²×C₈	C₃×D₄	1	2	yes	—
90	C₄⋊Q₈	C₂×C₃×C₄	1	2	yes	—
91	#127	C₂×C₃×C₄	1	2	yes	—
92	#124	—	2	2	no	—
93	#121	—	2	4	no	—
94	C₄×D₄	—	2	4	no	—
95	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
96	#108	—	4	4	no	—
97	C₄⋊C₈	—	2	4	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
100	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
101	C₄×C₈	C₃×D₄	1	4	yes	—
102	C₄×C₈	C₃×D₄	1	2	yes	—
103	C₂³×C₃	—	4	2	no	—
104	C₂³×C₃	#124	1	1	yes	characteristic
105	C₃×Q₈	—	4	4	no	—
106	C₃×D₄	—	4	4	no	—
107	C₃×D₄	—	4	8	no	—
108	C₃×D₄	—	8	4	no	—
109	C₂×C₃×C₄	—	2	4	no	—
110	C₂×C₃×C₄	—	4	2	no	—
111	C₂×C₃×C₄	—	8	2	no	—
112	C₂×C₃×C₄	—	8	2	no	—
113	C₂×C₃×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₃×C₄	C₄×D₄	1	2	yes	—
115	C₂×C₃×C₄	#121	1	2	yes	—
116	C₂×C₃×C₄	#127	1	1	yes	characteristic
117	C₂×C₃×C₄	C₂×D₈	1	2	yes	—
118	C₂×C₃×C₄	C₂×SD₁₆	1	2	yes	—
119	C₃×C₈	—	2	8	no	—
120	C₃×C₈	—	8	2	no	—
121	C₃×C₈	#102	1	4	yes	—
122	C₃×C₈	#102	1	4	yes	—
123	#41	—	4	4	no	—
124	#41	—	4	4	no	—
125	C₂×Q₈	—	2	2	no	—
126	C₂×D₄	—	2	2	no	—
127	C₂×D₄	—	2	4	no	—
128	C₂×D₄	—	4	2	no	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
131	C₂²×C₄	C₃×#41	1	2	yes	—
132	Q₁₆	—	4	4	no	—
133	SD₁₆	—	4	8	no	—
134	D₈	—	4	4	no	—
135	M₁₆	—	4	4	no	—
136	C₂×C₈	—	2	8	no	—
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
140	C₂×C₈	C₃×C₂²⋊C₄	1	2	yes	—
141	C₂×C₈	C₃×D₈	1	4	yes	—
142	C₂×C₈	C₃×SD₁₆	1	4	yes	—
143	C₄⋊C₄	—	2	4	no	—
144	C₂²⋊C₄	—	4	4	no	—
145	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
147	C₄²	C₂×C₃×D₄	1	2	yes	—
148	C₂²×C₃	—	2	2	no	—
149	C₂²×C₃	—	8	2	no	—
150	C₂²×C₃	—	8	4	no	—
151	C₂²×C₃	#390	1	1	yes	characteristic
152	C₂²×C₃	#486	1	2	yes	—
153	C₃×C₄	—	2	2	no	—
154	C₃×C₄	—	8	2	no	—
155	C₃×C₄	C₂×#102	1	2	yes	—
156	C₃×C₄	#443	1	4	yes	—
157	C₃×C₄	#492	1	1	yes	characteristic
158	C₃×C₄	#493	1	1	yes	characteristic
159	C₂³	—	4	2	no	—
160	C₂³	C₃×#124	1	1	yes	characteristic
161	Q₈	—	4	4	no	—
162	D₄	—	4	4	no	—
163	D₄	—	4	8	no	—
164	D₄	—	8	4	no	—
165	C₂×C₄	—	2	4	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	8	2	no	—
168	C₂×C₄	—	8	2	no	—
169	C₂×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄	C₃×C₄×D₄	1	2	yes	—
171	C₂×C₄	C₃×#121	1	2	yes	—
172	C₂×C₄	C₃×#127	1	1	yes	characteristic
173	C₂×C₄	C₂×C₃×D₈	1	2	yes	—
174	C₂×C₄	C₂×C₃×SD₁₆	1	2	yes	—
175	C₈	—	2	8	no	—
176	C₈	—	8	2	no	—
177	C₈	C₃×#102	1	4	yes	—
178	C₈	C₃×#102	1	4	yes	—
179	C₂×C₃	—	2	2	no	—
180	C₂×C₃	—	16	2	no	—
181	C₂×C₃	#1957	1	1	yes	characteristic
182	C₂×C₃	#1974	1	2	yes	—
183	C₂²	—	2	2	no	—
184	C₂²	—	8	2	no	—
185	C₂²	—	8	4	no	—
186	C₂²	C₃×#390	1	1	yes	characteristic
187	C₂²	C₃×#486	1	2	yes	—
188	C₄	—	2	2	no	—
189	C₄	—	8	2	no	—
190	C₄	C₂×C₃×#102	1	2	yes	—
191	C₄	C₃×#443	1	4	yes	—
192	C₄	C₃×#492	1	1	yes	characteristic
193	C₄	C₃×#493	1	1	yes	characteristic
194	C₃	#9856	1	1	yes	characteristic, direct product factor, Sylow
195	C₂	—	2	2	no	—
196	C₂	—	16	2	no	—
197	C₂	C₃×#1957	1	1	yes	characteristic
198	C₂	C₃×#1974	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#3147

C₂

yes

maximal, characteristic

C₃×#1848

C₂

yes

maximal, characteristic, complemented

C₃×#1492

C₂

yes

maximal, complemented

C₂×C₃×C₈²

C₂

yes

maximal, characteristic, complemented

#9856

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃×#135

—

C₃×#532

C₂²

yes

—

C₃×#495

—

C₃×#494

C₄

yes

characteristic

C₃×#493

C₄

yes

characteristic

C₃×#492

C₄

yes

—

C₂×C₃×#108

—

C₃×#423

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

—

C₂×C₃×C₄×C₈

C₂²

yes

characteristic

C₃×D₄⋊C₈

—

C₃×C₈²

—

#3147

C₂×C₃

yes

characteristic

#1848

C₂×C₃

yes

characteristic, complemented

#1492

C₂×C₃

yes

complemented

C₂×C₈²

C₂×C₃

yes

characteristic, complemented

C₂×C₃×#41

—

C₃×#135

—

C₃×#135

—

C₂×C₃×Q₁₆

—

C₂×C₃×SD₁₆

—

C₂×C₃×D₈

—

C₂×C₃×M₁₆

—

C₂²×C₃×C₈

—

C₂²×C₃×C₈

D₄

yes

—

C₃×C₄⋊Q₈

C₂×C₄

yes

—

C₃×#127

C₂×C₄

yes

—

C₃×#124

—

C₃×#121

—

C₃×C₄×D₄

—

C₂×C₃×C₄²

C₂³

yes

characteristic

C₃×#108

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

C₂×C₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

D₄

yes

—

C₂×#135

—

#532

C₂²×C₃

yes

—

#495

—

#494

C₃×C₄

yes

characteristic

#493

C₃×C₄

yes

characteristic

#492

C₃×C₄

yes

—

C₂×#108

—

#423

C₂²×C₃

yes

—

C₂×C₄×C₈

C₂²×C₃

yes

—

C₂×C₄×C₈

C₂²×C₃

yes

characteristic

D₄⋊C₈

—

C₈²

—

C₃×#41

—

C₃×#41

—

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×D₄

yes

characteristic

C₂²×C₃×C₄

#41

yes

—

C₃×Q₁₆

—

C₃×SD₁₆

—

C₃×D₈

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

C₂²⋊C₄

yes

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

SD₁₆

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

C₂²×C₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₄²

C₂×D₄

yes

—

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₃×D₄

yes

—

C₄⋊Q₈

C₂×C₃×C₄

yes

—

#127

C₂×C₃×C₄

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄²

C₂³×C₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₃×C₄

yes

characteristic

100

C₄×C₈

C₂×C₃×C₄

yes

characteristic

101

C₄×C₈

C₃×D₄

yes

—

102

C₄×C₈

C₃×D₄

yes

—

103

C₂³×C₃

—

104

C₂³×C₃

#124

yes

characteristic

105

C₃×Q₈

—

106

C₃×D₄

—

107

C₃×D₄

—

108

C₃×D₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

C₂×C₂²⋊C₄

yes

characteristic

114

C₂×C₃×C₄

C₄×D₄

yes

—

115

C₂×C₃×C₄

#121

yes

—

116

C₂×C₃×C₄

#127

yes

characteristic

117

C₂×C₃×C₄

C₂×D₈

yes

—

118

C₂×C₃×C₄

C₂×SD₁₆

yes

—

119

C₃×C₈

—

120

C₃×C₈

—

121

C₃×C₈

#102

yes

—

122

C₃×C₈

#102

yes

—

123

#41

—

124

#41

—

125

C₂×Q₈

—

126

C₂×D₄

—

127

C₂×D₄

—

128

C₂×D₄

—

129

C₂²×C₄

—

130

C₂²×C₄

C₂×C₃×D₄

yes

characteristic

131

C₂²×C₄

C₃×#41

yes

—

132

Q₁₆

—

133

SD₁₆

—

134

D₈

—

135

M₁₆

—

136

C₂×C₈

—

137

C₂×C₈

—

138

C₂×C₈

—

139

C₂×C₈

C₃×C₂²⋊C₄

yes

—

140

C₂×C₈

C₃×C₂²⋊C₄

yes

—

141

C₂×C₈

C₃×D₈

yes

—

142

C₂×C₈

C₃×SD₁₆

yes

—

143

C₄⋊C₄

—

144

C₂²⋊C₄

—

145

C₄²

C₂²×C₃×C₄

yes

characteristic

146

C₄²

C₂×C₃×D₄

yes

characteristic

147

C₄²

C₂×C₃×D₄

yes

—

148

C₂²×C₃

—

149

C₂²×C₃

—

150

C₂²×C₃

—

151

C₂²×C₃

#390

yes

characteristic

152

C₂²×C₃

#486

yes

—

153

C₃×C₄

—

154

C₃×C₄

—

155

C₃×C₄

C₂×#102

yes

—

156

C₃×C₄

#443

yes

—

157

C₃×C₄

#492

yes

characteristic

158

C₃×C₄

#493

yes

characteristic

159

C₂³

—

160

C₂³

C₃×#124

yes

characteristic

161

Q₈

—

162

D₄

—

163

D₄

—

164

D₄

—

165

C₂×C₄

—

166

C₂×C₄

—

167

C₂×C₄

—

168

C₂×C₄

—

169

C₂×C₄

C₂×C₃×C₂²⋊C₄

yes

characteristic

170

C₂×C₄

C₃×C₄×D₄

yes

—

171

C₂×C₄

C₃×#121

yes

—

172

C₂×C₄

C₃×#127

yes

characteristic

173

C₂×C₄

C₂×C₃×D₈

yes

—

174

C₂×C₄

C₂×C₃×SD₁₆

yes

—

175

C₈

—

176

C₈

—

177

C₈

C₃×#102

yes

—

178

C₈

C₃×#102

yes

—

179

C₂×C₃

—

180

C₂×C₃

—

181

C₂×C₃

#1957

yes

characteristic

182

C₂×C₃

#1974

yes

—

183

C₂²

—

184

C₂²

—

185

C₂²

—

186

C₂²

C₃×#390

yes

characteristic

187

C₂²

C₃×#486

yes

—

188

C₄

—

189

C₄

—

190

C₄

C₂×C₃×#102

yes

—

191

C₄

C₃×#443

yes

—

192

C₄

C₃×#492

yes

characteristic

193

C₄

C₃×#493

yes

characteristic

194

C₃

#9856

yes

characteristic, direct product factor, Sylow

195

C₂

—

196

C₂

—

197

C₂

C₃×#1957

yes

characteristic

198

C₂

C₃×#1974

yes

—

	subgroup	G/H
center	C₂×C₃×C₄	#127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#390
prime radical	C₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

subgroup

G/H

center

C₂×C₃×C₄

#127

commutator

C₄²

C₂²×C₃×C₄

Frattini, cosocle

C₂×C₄²

C₂³×C₃

C₂²×C₃

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

N	H ≅ G/N	type
2. C₃×#1848	C₂	semidirect
3. C₃×#1492	C₂	semidirect
4. C₂×C₃×C₈²	C₂	semidirect
5. #9856	C₃	direct (N × H)
19. #1848	C₂×C₃	semidirect
20. #1492	C₂×C₃	semidirect
21. C₂×C₈²	C₂×C₃	semidirect
194. C₃	#9856	direct (N × H)

H ≅ G/N

type

2. C₃×#1848

C₂

semidirect

3. C₃×#1492

C₂

semidirect

4. C₂×C₃×C₈²

C₂

semidirect

5. #9856

C₃

direct (N × H)

19. #1848

C₂×C₃

semidirect

20. #1492

C₂×C₃

semidirect

21. C₂×C₈²

C₂×C₃

semidirect

194. C₃

#9856

direct (N × H)

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

subgroup	index
C₃×#3147	2
C₃×#1848	2
C₃×#1492	2
C₂×C₃×C₈²	2
#9856	3

subgroup

index

C₂×C₃×C₈²

Subgroups — C₃×#9856 #10628463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#9856 #10628463

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#9856 #10628463