Group #134643 — subgroups

Subgroups — #134643

Order 512.

350 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 92 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	#21698	C₂	1	1	yes	maximal, characteristic
2	#12884	C₂	1	1	yes	maximal, characteristic
3	#11932	C₂	1	2	yes	maximal
4	#8780	C₂	1	1	yes	maximal, characteristic
5	#7279	C₂	1	2	yes	maximal
6	C₂×#494	C₂²	1	1	yes	characteristic
7	C₂×#441	C₄	1	2	yes	—
8	C₂×C₄×Q₁₆	—	2	1	no	—
9	#2327	—	2	1	no	—
10	#2150	C₂²	1	2	yes	—
11	#1971	C₄	1	2	yes	—
12	#1969	—	2	1	no	—
13	#1960	—	2	1	no	—
14	#1846	C₂²	1	1	yes	characteristic
15	#1726	C₂²	1	1	yes	characteristic
16	#1564	C₂²	1	2	yes	—
17	#1376	—	2	1	no	—
18	#1376	—	2	1	no	—
19	#1362	—	2	4	no	—
20	#1281	—	2	1	no	—
21	#1274	—	2	1	no	—
22	C₂²×Q₁₆	—	2	1	no	—
23	C₂²×Q₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊Q₈	—	2	1	no	—
25	C₂×C₄×Q₈	—	2	2	no	—
26	C₂×M₃₂	—	2	2	no	—
27	#494	—	2	1	no	—
28	#494	—	2	1	no	—
29	#494	D₄	1	2	yes	—
30	#494	Q₈	1	2	yes	—
31	#441	—	4	4	no	—
32	C₄×Q₁₆	—	4	4	no	—
33	#431	—	2	2	no	—
34	C₂×#107	—	2	1	no	—
35	C₂×#106	C₂×C₄	1	2	yes	—
36	C₂×#103	—	2	2	no	—
37	C₂×#103	—	2	2	no	—
38	C₂×#97	C₂×C₄	1	1	yes	characteristic
39	C₂×#97	C₂×C₄	1	1	yes	characteristic
40	C₂×#97	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#391	—	2	2	no	—
45	#387	—	4	4	no	—
46	#382	—	2	1	no	—
47	#376	D₄	1	1	yes	characteristic
48	C₄⋊C₁₆	—	2	2	no	—
49	C₄⋊C₁₆	—	2	2	no	—
50	#335	—	2	2	no	—
51	#334	—	2	2	no	—
52	C₂²×Q₈	—	2	2	no	—
53	C₂×Q₁₆	—	2	2	no	—
54	C₂×Q₁₆	—	2	2	no	—
55	C₂×Q₁₆	—	2	2	no	—
56	C₂×Q₁₆	—	4	2	no	—
57	C₂×Q₁₆	—	4	2	no	—
58	C₂×Q₁₆	—	4	2	no	—
59	C₂×Q₁₆	—	4	2	no	—
60	C₂×Q₁₆	C₄⋊C₄	1	4	yes	—
61	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	—	4	2	no	—
68	C₄×Q₈	—	4	4	no	—
69	C₄×Q₈	—	4	4	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	4	2	no	—
72	C₂×C₄⋊C₄	—	4	2	no	—
73	C₂×C₄²	—	4	2	no	—
74	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
77	M₃₂	—	4	2	no	—
78	M₃₂	—	4	2	no	—
79	C₂×C₁₆	—	4	2	no	—
80	#107	—	4	2	no	—
81	#106	—	4	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	#103	—	4	4	no	—
84	#103	—	4	4	no	—
85	C₂²⋊C₈	—	4	2	no	—
86	#97	—	2	2	no	—
87	#97	—	2	2	no	—
88	#97	—	4	1	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	4	2	no	—
92	C₄×C₈	C₂×D₄	1	1	yes	characteristic
93	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
94	C₄×C₈	#41	1	1	yes	characteristic
95	C₄×C₈	#41	1	1	yes	characteristic
96	#95	—	2	2	no	—
97	#95	—	4	4	no	—
98	#95	—	4	2	no	—
99	C₂×Q₈	—	2	4	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂×Q₈	—	4	2	no	—
102	C₂×Q₈	—	4	2	no	—
103	C₂×Q₈	—	4	2	no	—
104	C₂×Q₈	—	4	2	no	—
105	C₂²×C₄	—	2	1	no	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	4	2	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
111	Q₁₆	—	4	4	no	—
112	Q₁₆	—	4	4	no	—
113	Q₁₆	—	4	4	no	—
114	Q₁₆	—	4	4	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	2	1	no	—
118	C₂×C₈	—	2	1	no	—
119	C₂×C₈	—	2	1	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	2	1	no	—
122	C₂×C₈	—	2	1	no	—
123	C₂×C₈	—	8	2	no	—
124	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₈	#117	1	1	yes	characteristic
126	C₂×C₈	#121	1	1	yes	characteristic
127	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₈	#135	1	2	yes	—
129	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
130	C₄⋊C₄	—	4	4	no	—
131	C₄⋊C₄	—	4	4	no	—
132	C₄⋊C₄	—	4	2	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₄⋊C₄	—	8	4	no	—
135	C₄²	—	2	1	no	—
136	C₄²	—	2	1	no	—
137	C₄²	—	2	1	no	—
138	C₄²	—	8	4	no	—
139	C₄²	#100	1	2	yes	—
140	C₄²	#121	1	1	yes	characteristic
141	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
142	C₄²	#135	1	2	yes	—
143	C₁₆	—	8	2	no	—
144	C₂³	#418	1	1	yes	characteristic
145	Q₈	—	4	4	no	—
146	Q₈	—	4	4	no	—
147	Q₈	—	8	2	no	—
148	Q₈	—	8	2	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	4	2	no	—
151	C₂×C₄	—	4	2	no	—
152	C₂×C₄	—	4	2	no	—
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	4	1	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	8	4	no	—
158	C₂×C₄	—	8	2	no	—
159	C₂×C₄	#380	1	1	yes	characteristic
160	C₂×C₄	C₂×#100	1	1	yes	characteristic
161	C₂×C₄	#413	1	1	yes	characteristic
162	C₂×C₄	#416	1	1	yes	characteristic
163	C₂×C₄	#416	1	1	yes	characteristic
164	C₂×C₄	#418	1	1	yes	characteristic
165	C₂×C₄	#463	1	2	yes	—
166	C₂×C₄	#478	1	2	yes	—
167	C₈	—	2	1	no	—
168	C₈	—	2	1	no	—
169	C₈	—	4	1	no	—
170	C₈	—	4	1	no	—
171	C₈	—	16	2	no	—
172	C₈	#463	1	2	yes	—
173	C₈	#478	1	2	yes	—
174	C₂²	—	2	2	no	—
175	C₂²	#1553	1	1	yes	characteristic
176	C₂²	#1555	1	1	yes	characteristic
177	C₂²	#1804	1	1	yes	characteristic
178	C₄	—	2	2	no	—
179	C₄	—	2	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	8	1	no	—
182	C₄	—	8	4	no	—
183	C₄	—	16	2	no	—
184	C₄	#1543	1	1	yes	characteristic
185	C₄	#1553	1	1	yes	characteristic
186	C₄	#1800	1	1	yes	characteristic
187	C₄	#1804	1	1	yes	characteristic
188	C₂	—	2	1	no	—
189	C₂	—	2	1	no	—
190	C₂	#8592	1	1	yes	characteristic
191	C₂	#12827	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21698

C₂

yes

maximal, characteristic

#12884

C₂

yes

maximal, characteristic

#11932

C₂

yes

maximal

#8780

C₂

yes

maximal, characteristic

#7279

C₂

yes

maximal

C₂×#494

C₂²

yes

characteristic

C₂×#441

C₄

yes

—

C₂×C₄×Q₁₆

—

#2327

—

#2150

C₂²

yes

—

#1971

C₄

yes

—

#1969

—

#1960

—

#1846

C₂²

yes

characteristic

#1726

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

—

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂×C₄

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

—

C₂×M₃₂

—

#494

—

#494

—

#494

D₄

yes

—

#494

Q₈

yes

—

#441

—

C₄×Q₁₆

—

#431

—

C₂×#107

—

C₂×#106

C₂×C₄

yes

—

C₂×#103

—

C₂×#103

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#335

—

#334

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₄⋊C₄

yes

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#107

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

—

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

110

C₂²×C₄

AGL(1,ℤ/8)

yes

—

111

Q₁₆

—

112

Q₁₆

—

113

Q₁₆

—

114

Q₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

125

C₂×C₈

#117

yes

characteristic

126

C₂×C₈

#121

yes

characteristic

127

C₂×C₈

C₂²⋊Q₈

yes

characteristic

128

C₂×C₈

#135

yes

—

129

C₂×C₈

AGL(1,ℤ/8)

yes

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

—

138

C₄²

—

139

C₄²

#100

yes

—

140

C₄²

#121

yes

characteristic

141

C₄²

C₂²⋊Q₈

yes

characteristic

142

C₄²

#135

yes

—

143

C₁₆

—

144

C₂³

#418

yes

characteristic

145

Q₈

—

146

Q₈

—

147

Q₈

—

148

Q₈

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

—

152

C₂×C₄

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

—

159

C₂×C₄

#380

yes

characteristic

160

C₂×C₄

C₂×#100

yes

characteristic

161

C₂×C₄

#413

yes

characteristic

162

C₂×C₄

#416

yes

characteristic

163

C₂×C₄

#416

yes

characteristic

164

C₂×C₄

#418

yes

characteristic

165

C₂×C₄

#463

yes

—

166

C₂×C₄

#478

yes

—

167

C₈

—

168

C₈

—

169

C₈

—

170

C₈

—

171

C₈

—

172

C₈

#463

yes

—

173

C₈

#478

yes

—

174

C₂²

—

175

C₂²

#1553

yes

characteristic

176

C₂²

#1555

yes

characteristic

177

C₂²

#1804

yes

characteristic

178

C₄

—

179

C₄

—

180

C₄

—

181

C₄

—

182

C₄

—

183

C₄

—

184

C₄

#1543

yes

characteristic

185

C₄

#1553

yes

characteristic

186

C₄

#1800

yes

characteristic

187

C₄

#1804

yes

characteristic

188

C₂

—

189

C₂

—

190

C₂

#8592

yes

characteristic

191

C₂

#12827

yes

—

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1804

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#418

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#21698	2
#12884	2
#11932	2
#8780	2
#7279	2

subgroup

index

Subgroups — #134643

Conjugacy classes of subgroups

Maximal subgroups