Group #13154277 — subgroups

Subgroups — #13154277

Order 1728.

1060 conjugacy classes of subgroups; only the 307 normal subgroups are shown.

normal subgroups only

This group has 1060 conjugacy classes of subgroups — too many to draw. Only its 307 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×C₉×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11721854	C₂	1	4	yes	maximal
3	#11721744	C₂	1	4	yes	maximal
4	#11721743	C₂	1	4	yes	maximal
5	C₄×#6226	C₂	1	2	yes	maximal
6	#10596193	—	3	1	no	maximal
7	#10589824	—	3	3	no	maximal
8	C₂×C₃×Q₈×C₉	C₂²	1	2	yes	—
9	C₂×#6226	C₂²	1	4	yes	—
10	C₂×#6226	C₂²	1	2	yes	—
11	C₃×C₉×C₄⋊C₄	C₂²	1	4	yes	—
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₄⋊(C₃⋊Dic₉)	C₂²	1	2	yes	—
14	#88534	C₂²	1	8	yes	—
15	#88534	C₂²	1	4	yes	—
16	C₄×C₃⋊Dic₉	C₂²	1	4	yes	—
17	C₄×C₃⋊Dic₉	C₂²	1	4	yes	—
18	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
19	#64153	—	3	4	no	—
20	#64097	—	3	4	no	—
21	#64096	—	3	4	no	—
22	C₄×#634	—	3	2	no	—
23	C₉×C₄⋊Q₈	D₃	1	3	yes	—
24	#63506	—	3	12	no	—
25	#63450	—	3	12	no	—
26	#63449	—	3	12	no	—
27	C₄×Dic₁₈	—	3	6	no	—
28	C₃×Q₈×C₉	—	4	2	no	—
29	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
32	C₂×C₃⋊Dic₉	C₂³	1	4	yes	—
33	C₂×C₃⋊Dic₉	C₂³	1	4	yes	—
34	#6226	—	4	4	no	—
35	#6226	Q₈	1	8	yes	complemented
36	#5722	—	9	3	no	—
37	#5722	—	9	1	no	—
38	C₂×C₃²×Q₈	C₂×D₃	1	2	yes	—
39	C₂×#634	—	3	2	no	—
40	C₂×#634	—	3	4	no	—
41	C₃²×C₄⋊C₄	C₂×D₃	1	4	yes	—
42	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
43	C₄⋊(C₃⋊Dic₃)	—	3	2	no	—
44	#3750	—	3	4	no	—
45	#3750	—	3	8	no	—
46	C₄×C₃⋊Dic₃	—	3	4	no	—
47	C₄×C₃⋊Dic₃	—	3	4	no	—
48	C₂×Q₈×C₉	C₂×D₃	1	6	yes	—
49	C₂×Dic₁₈	—	3	6	no	—
50	C₂×Dic₁₈	—	3	12	no	—
51	C₉×C₄⋊C₄	C₂×D₃	1	12	yes	—
52	C₄²×C₉	C₂×D₃	1	3	yes	—
53	C₄⋊Dic₉	—	3	6	no	—
54	Dic₉⋊C₄	—	3	12	no	—
55	Dic₉⋊C₄	—	3	24	no	—
56	C₄×Dic₉	—	3	12	no	—
57	C₄×Dic₉	—	3	12	no	—
58	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
59	C₃×C₄×C₉	—	2	4	no	—
60	C₃×C₄×C₉	—	2	1	no	—
61	C₃×C₄×C₉	C₂×Q₈	1	4	yes	—
62	C₃⋊Dic₉	—	2	4	no	—
63	C₃⋊Dic₉	C₂×Q₈	1	8	yes	—
64	C₃×C₄⋊Q₈	D₉	1	3	yes	—
65	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
66	#944	—	9	12	no	—
67	#944	—	9	4	no	—
68	#888	—	9	12	no	—
69	#888	—	9	4	no	—
70	#887	—	9	12	no	—
71	#887	—	9	4	no	—
72	C₄×Dic₆	—	9	6	no	—
73	C₄×Dic₆	—	9	2	no	—
74	C₃²×Q₈	—	4	2	no	—
75	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
76	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
77	C₂×C₃²×C₄	C₂²×D₃	1	2	yes	—
78	C₂×C₃⋊Dic₃	—	3	4	no	—
79	C₂×C₃⋊Dic₃	—	3	4	no	—
80	#634	—	3	8	no	—
81	#634	—	12	4	no	—
82	Q₈×C₉	—	4	6	no	—
83	C₂×C₄×C₉	C₂²×D₃	1	12	yes	—
84	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
85	C₂×C₄×C₉	C₂²×D₃	1	6	yes	—
86	C₂×Dic₉	—	3	12	no	—
87	C₂×Dic₉	—	3	12	no	—
88	Dic₁₈	—	3	24	no	—
89	Dic₁₈	—	12	12	no	—
90	Q₈²	—	27	1	no	Sylow
91	C₂×C₃×C₉	C₂²×Q₈	1	2	yes	—
92	C₂×C₃×C₉	H(5,𝔽₂)	1	1	yes	characteristic
93	C₂×C₃×Q₈	C₂×D₉	1	6	yes	—
94	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
95	C₂×Dic₆	—	9	6	no	—
96	C₂×Dic₆	—	9	2	no	—
97	C₂×Dic₆	—	9	12	no	—
98	C₂×Dic₆	—	9	4	no	—
99	C₃×C₄⋊C₄	C₂×D₉	1	12	yes	—
100	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	4	yes	—
101	C₃×C₄²	C₂×D₉	1	3	yes	—
102	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
103	C₄⋊Dic₃	—	9	6	no	—
104	C₄⋊Dic₃	—	9	2	no	—
105	Dic₃⋊C₄	—	9	12	no	—
106	Dic₃⋊C₄	—	9	24	no	—
107	Dic₃⋊C₄	—	9	4	no	—
108	Dic₃⋊C₄	—	9	8	no	—
109	C₄×Dic₃	—	9	12	no	—
110	C₄×Dic₃	—	9	12	no	—
111	C₄×Dic₃	—	9	4	no	—
112	C₄×Dic₃	—	9	4	no	—
113	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
114	C₃²×C₄	—	2	4	no	—
115	C₃²×C₄	—	2	1	no	—
116	C₃²×C₄	D₃×Q₈	1	4	yes	—
117	C₃⋊Dic₃	—	3	8	no	—
118	C₃⋊Dic₃	—	6	4	no	—
119	C₂²×C₉	C₂³×D₃	1	3	yes	—
120	C₄×C₉	—	2	12	no	—
121	C₄×C₉	—	2	3	no	—
122	C₄×C₉	D₃×Q₈	1	12	yes	—
123	Dic₉	—	3	24	no	—
124	Dic₉	—	6	12	no	—
125	C₄⋊Q₈	—	27	4	no	—
126	C₄⋊Q₈	—	27	4	no	—
127	C₄⋊Q₈	C₃⋊D₉	1	1	yes	characteristic
128	C₄×Q₈	—	27	4	no	—
129	C₄×Q₈	—	27	2	no	—
130	C₃×C₉	Q₈²	1	1	yes	characteristic, complemented, Sylow
131	C₃×Q₈	—	4	2	no	—
132	C₃×Q₈	—	4	6	no	—
133	C₂×C₃×C₄	C₂²×D₉	1	12	yes	—
134	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
135	C₂×C₃×C₄	C₂²×D₉	1	6	yes	—
136	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	4	yes	—
137	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
138	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
139	C₂×Dic₃	—	9	4	no	—
140	C₂×Dic₃	—	9	12	no	—
141	C₂×Dic₃	—	9	4	no	—
142	C₂×Dic₃	—	9	12	no	—
143	Dic₆	—	9	24	no	—
144	Dic₆	—	9	8	no	—
145	Dic₆	—	36	4	no	—
146	Dic₆	—	36	12	no	—
147	C₂×C₃²	#1004	1	1	yes	characteristic
148	C₂×C₃²	C₂×D₃×Q₈	1	2	yes	—
149	C₂×C₉	#1004	1	3	yes	—
150	C₂×C₉	C₂×D₃×Q₈	1	6	yes	—
151	C₂×Q₈	—	27	2	no	—
152	C₂×Q₈	—	27	4	no	—
153	C₂×Q₈	C₂×C₃⋊D₉	1	2	yes	—
154	C₄⋊C₄	—	27	8	no	—
155	C₄⋊C₄	—	27	4	no	—
156	C₄⋊C₄	—	27	2	no	—
157	C₄⋊C₄	C₂×C₃⋊D₉	1	4	yes	—
158	C₄²	—	27	4	no	—
159	C₄²	—	27	4	no	—
160	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
161	C₂²×C₃	C₂³×D₉	1	3	yes	—
162	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
163	C₃×C₄	—	2	4	no	—
164	C₃×C₄	—	2	1	no	—
165	C₃×C₄	—	2	12	no	—
166	C₃×C₄	—	2	3	no	—
167	C₃×C₄	Q₈×D₉	1	12	yes	—
168	C₃×C₄	Q₈×C₃⋊D₃	1	4	yes	—
169	Dic₃	—	9	8	no	—
170	Dic₃	—	9	24	no	—
171	Dic₃	—	18	4	no	—
172	Dic₃	—	18	12	no	—
173	C₃²	#5722	1	1	yes	characteristic
174	C₉	#5722	1	3	yes	complemented
175	Q₈	—	4	2	no	—
176	Q₈	—	27	8	no	—
177	Q₈	—	108	4	no	—
178	C₂×C₄	—	27	4	no	—
179	C₂×C₄	—	27	4	no	—
180	C₂×C₄	C₂²×C₃⋊D₉	1	4	yes	—
181	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
182	C₂×C₄	C₂²×C₃⋊D₉	1	2	yes	—
183	C₂×C₃	#63710	1	3	yes	—
184	C₂×C₃	C₂×Q₈×D₉	1	6	yes	—
185	C₂×C₃	#64361	1	1	yes	characteristic
186	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	2	yes	—
187	C₂²	C₂³×C₃⋊D₉	1	1	yes	characteristic
188	C₄	—	2	1	no	—
189	C₄	—	2	4	no	—
190	C₄	—	27	8	no	—
191	C₄	—	54	4	no	—
192	C₄	Q₈×C₃⋊D₉	1	4	yes	—
193	C₃	#10589824	1	3	yes	complemented
194	C₃	#10596193	1	1	yes	characteristic
195	C₂	#11722661	1	1	yes	characteristic
196	C₂	C₂×Q₈×C₃⋊D₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₉×C₄⋊Q₈

C₂

yes

maximal, characteristic

#11721854

C₂

yes

maximal

#11721744

C₂

yes

maximal

#11721743

C₂

yes

maximal

C₄×#6226

C₂

yes

—

—

C₂×C₃×Q₈×C₉

C₂²

yes

—

C₂×#6226

C₂²

yes

—

C₂×#6226

C₂²

yes

—

C₃×C₉×C₄⋊C₄

C₂²

yes

—

C₃×C₄²×C₉

C₂²

yes

characteristic

C₄⋊(C₃⋊Dic₉)

C₂²

yes

—

#88534

C₂²

yes

—

#88534

C₂²

yes

—

C₄×C₃⋊Dic₉

C₂²

yes

—

C₄×C₃⋊Dic₉

C₂²

yes

—

C₃²×C₄⋊Q₈

D₃

yes

—

—

—

—

C₉×C₄⋊Q₈

D₃

yes

—

#63506

—

#63450

—

#63449

—

C₄×Dic₁₈

—

C₃×Q₈×C₉

—

C₂×C₃×C₄×C₉

C₂³

yes

—

C₂×C₃×C₄×C₉

C₂³

yes

characteristic

C₂×C₃×C₄×C₉

C₂³

yes

—

C₂×C₃⋊Dic₉

C₂³

yes

—

C₂×C₃⋊Dic₉

C₂³

yes

—

#6226

—

#6226

Q₈

yes

complemented

#5722

—

#5722

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂×#634

—

C₂×#634

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₄⋊(C₃⋊Dic₃)

—

#3750

—

#3750

—

C₄×C₃⋊Dic₃

—

C₄×C₃⋊Dic₃

—

C₂×Q₈×C₉

C₂×D₃

yes

—

C₂×Dic₁₈

—

C₂×Dic₁₈

—

C₉×C₄⋊C₄

C₂×D₃

yes

—

C₄²×C₉

C₂×D₃

yes

—

C₄⋊Dic₉

—

Dic₉⋊C₄

—

Dic₉⋊C₄

—

C₄×Dic₉

—

C₄×Dic₉

—

C₂²×C₃×C₉

C₂⁴

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×Q₈

yes

—

C₃⋊Dic₉

—

C₃⋊Dic₉

C₂×Q₈

yes

—

C₃×C₄⋊Q₈

D₉

yes

—

C₃×C₄⋊Q₈

C₃⋊D₃

yes

—

—

—

—

—

—

C₄×Dic₆

—

C₄×Dic₆

—

C₃²×Q₈

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

—

#634

—

#634

—

Q₈×C₉

—

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×C₄×C₉

C₂²×D₃

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

Dic₁₈

—

Dic₁₈

—

Q₈²

—

Sylow

C₂×C₃×C₉

C₂²×Q₈

yes

—

C₂×C₃×C₉

H(5,𝔽₂)

yes

characteristic

C₂×C₃×Q₈

C₂×D₉

yes

—

C₂×C₃×Q₈

C₂×C₃⋊D₃

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

C₂×D₉

yes

—

100

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

101

C₃×C₄²

C₂×D₉

yes

—

102

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

103

C₄⋊Dic₃

—

104

C₄⋊Dic₃

—

105

Dic₃⋊C₄

—

106

Dic₃⋊C₄

—

107

Dic₃⋊C₄

—

108

Dic₃⋊C₄

—

109

C₄×Dic₃

—

110

C₄×Dic₃

—

111

C₄×Dic₃

—

112

C₄×Dic₃

—

113

C₂²×C₃²

C₂³×D₃

yes

characteristic

114

C₃²×C₄

—

115

C₃²×C₄

—

116

C₃²×C₄

D₃×Q₈

yes

—

117

C₃⋊Dic₃

—

118

C₃⋊Dic₃

—

119

C₂²×C₉

C₂³×D₃

yes

—

120

C₄×C₉

—

121

C₄×C₉

—

122

C₄×C₉

D₃×Q₈

yes

—

123

Dic₉

—

124

Dic₉

—

125

C₄⋊Q₈

—

126

C₄⋊Q₈

—

127

C₄⋊Q₈

C₃⋊D₉

yes

characteristic

128

C₄×Q₈

—

129

C₄×Q₈

—

130

C₃×C₉

Q₈²

yes

characteristic, complemented, Sylow

131

C₃×Q₈

—

132

C₃×Q₈

—

133

C₂×C₃×C₄

C₂²×D₉

yes

—

134

C₂×C₃×C₄

C₂²×D₉

yes

—

135

C₂×C₃×C₄

C₂²×D₉

yes

—

136

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

137

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

138

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

139

C₂×Dic₃

—

140

C₂×Dic₃

—

141

C₂×Dic₃

—

142

C₂×Dic₃

—

143

Dic₆

—

144

Dic₆

—

145

Dic₆

—

146

Dic₆

—

147

C₂×C₃²

#1004

yes

characteristic

148

C₂×C₃²

C₂×D₃×Q₈

yes

—

149

C₂×C₉

#1004

yes

—

150

C₂×C₉

C₂×D₃×Q₈

yes

—

151

C₂×Q₈

—

152

C₂×Q₈

—

153

C₂×Q₈

C₂×C₃⋊D₉

yes

—

154

C₄⋊C₄

—

155

C₄⋊C₄

—

156

C₄⋊C₄

—

157

C₄⋊C₄

C₂×C₃⋊D₉

yes

—

158

C₄²

—

159

C₄²

—

160

C₄²

C₂×C₃⋊D₉

yes

characteristic

161

C₂²×C₃

C₂³×D₉

yes

—

162

C₂²×C₃

C₂³×C₃⋊D₃

yes

characteristic

163

C₃×C₄

—

164

C₃×C₄

—

165

C₃×C₄

—

166

C₃×C₄

—

167

C₃×C₄

Q₈×D₉

yes

—

168

C₃×C₄

Q₈×C₃⋊D₃

yes

—

169

Dic₃

—

170

Dic₃

—

171

Dic₃

—

172

Dic₃

—

173

C₃²

#5722

yes

characteristic

174

C₉

#5722

yes

complemented

175

Q₈

—

176

Q₈

—

177

Q₈

—

108

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂²×C₃⋊D₉

yes

—

181

C₂×C₄

C₂²×C₃⋊D₉

yes

characteristic

182

C₂×C₄

C₂²×C₃⋊D₉

yes

—

183

C₂×C₃

#63710

yes

—

184

C₂×C₃

C₂×Q₈×D₉

yes

—

185

C₂×C₃

#64361

yes

characteristic

186

C₂×C₃

C₂×Q₈×C₃⋊D₃

yes

—

187

C₂²

C₂³×C₃⋊D₉

yes

characteristic

188

C₄

—

189

C₄

—

190

C₄

—

191

C₄

—

192

C₄

Q₈×C₃⋊D₉

yes

—

193

C₃

#10589824

yes

complemented

194

C₃

#10596193

yes

characteristic

195

C₂

#11722661

yes

characteristic

196

C₂

C₂×Q₈×C₃⋊D₉

yes

—

	subgroup	G/H
center	C₂²	C₂³×C₃⋊D₉
commutator, cosocle	C₂²×C₃×C₉	C₂⁴
Frattini	C₂²×C₃	C₂³×C₃⋊D₃
Fitting	C₃×C₉×C₄⋊Q₈	C₂
socle, prime radical	C₂²×C₃²	C₂³×D₃
hypercenter	C₄⋊Q₈	C₃⋊D₉
hypocenter	C₃×C₉	Q₈²
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂³×C₃⋊D₉

commutator, cosocle

C₂²×C₃×C₉

C₂⁴

Frattini

C₂²×C₃

C₂³×C₃⋊D₃

Fitting

C₃×C₉×C₄⋊Q₈

C₂

socle, prime radical

C₂²×C₃²

C₂³×D₃

hypercenter

C₄⋊Q₈

C₃⋊D₉

hypocenter

C₃×C₉

Q₈²

LCS

G ▷ C₂²×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
35. #6226	Q₈	semidirect
130. C₃×C₉	Q₈²	semidirect
174. C₉	#5722	semidirect
193. C₃	#10589824	semidirect

H ≅ G/N

type

35. #6226

Q₈

semidirect

130. C₃×C₉

Q₈²

semidirect

174. C₉

#5722

semidirect

193. C₃

#10589824

semidirect

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

subgroup	index
C₃×C₉×C₄⋊Q₈	2
#11721854	2
#11721744	2
#11721743	2
C₄×#6226	2
#10596193	3
#10589824	3

subgroup

index

C₃×C₉×C₄⋊Q₈

Subgroups — #13154277

Conjugacy classes of subgroups

Complements

Maximal subgroups