Group #146266 — subgroups

Subgroups — #146266

Order 512.

241 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 241 conjugacy classes of subgroups — too many to draw. Only its 88 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	#13290	C₂	1	1	yes	maximal, characteristic
2	#13283	C₂	1	1	yes	maximal, characteristic
3	#12910	C₂	1	1	yes	maximal, characteristic
4	#12904	C₂	1	1	yes	maximal, characteristic
5	#9771	C₂	1	1	yes	maximal, characteristic
6	#8790	C₂	1	1	yes	maximal, characteristic
7	#7140	C₂	1	1	yes	maximal, characteristic
8	C₂×#500	C₂²	1	1	yes	characteristic
9	C₂×#368	—	2	2	no	—
10	C₂×#365	—	2	2	no	—
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×#346	C₂²	1	1	yes	characteristic
13	#1848	C₂²	1	1	yes	characteristic
14	#1848	C₂²	1	1	yes	characteristic
15	#1846	C₂²	1	1	yes	characteristic
16	#1549	—	2	1	no	—
17	#1548	—	2	1	no	—
18	#1450	C₂²	1	1	yes	characteristic
19	#1387	—	2	2	no	—
20	#1380	—	2	2	no	—
21	#1286	—	2	1	no	—
22	#1286	—	2	1	no	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂×M₃₂	D₄	1	2	yes	—
25	C₂²×C₁₆	D₄	1	2	yes	—
26	#500	—	2	1	no	—
27	#500	—	2	1	no	—
28	#500	—	2	1	no	—
29	#500	Q₈	1	2	yes	—
30	#431	D₄	1	1	yes	characteristic
31	#431	D₄	1	1	yes	characteristic
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	C₂×#108	—	2	1	no	—
35	C₂×#108	—	2	1	no	—
36	C₂×#107	—	2	1	no	—
37	C₂×#107	—	2	1	no	—
38	C₂×#107	—	2	1	no	—
39	C₂×#106	—	2	1	no	—
40	#423	—	2	1	no	—
41	#423	—	2	1	no	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#382	—	2	1	no	—
44	#368	—	4	4	no	—
45	#365	—	4	4	no	—
46	#346	—	2	1	no	—
47	#346	—	2	1	no	—
48	#346	—	2	1	no	—
49	C₄×C₁₆	—	2	1	no	—
50	#333	—	4	2	no	—
51	#332	—	4	2	no	—
52	#322	—	2	1	no	—
53	C₈²	—	2	1	no	—
54	C₂²×Q₈	—	4	1	no	—
55	C₂×M₁₆	—	4	1	no	—
56	C₂×M₁₆	—	4	1	no	—
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₄⋊Q₈	—	2	2	no	—
60	C₄⋊Q₈	—	4	1	no	—
61	C₄⋊Q₈	—	4	1	no	—
62	C₄⋊Q₈	—	4	1	no	—
63	C₂×C₄⋊C₄	—	4	1	no	—
64	C₂×C₄⋊C₄	—	4	1	no	—
65	C₂×C₄⋊C₄	—	4	1	no	—
66	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
67	M₃₂	—	2	2	no	—
68	M₃₂	Q₁₆	1	4	yes	—
69	C₂×C₁₆	—	2	2	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	—	2	2	no	—
72	C₂×C₁₆	SD₁₆	1	4	yes	—
73	#108	—	4	2	no	—
74	#108	—	4	2	no	—
75	#108	—	4	2	no	—
76	#108	—	4	2	no	—
77	#107	—	2	2	no	—
78	#107	—	2	2	no	—
79	#107	—	4	1	no	—
80	#107	—	4	1	no	—
81	#107	—	4	2	no	—
82	#106	—	4	2	no	—
83	C₄⋊C₈	—	2	1	no	—
84	C₄⋊C₈	—	2	1	no	—
85	C₄⋊C₈	—	4	1	no	—
86	C₄⋊C₈	—	4	1	no	—
87	C₄⋊C₈	—	4	1	no	—
88	C₄⋊C₈	—	4	1	no	—
89	C₂²⋊C₈	—	2	1	no	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	—	2	1	no	—
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	C₄×C₈	#41	1	1	yes	characteristic
97	#95	—	4	1	no	—
98	#95	—	4	1	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	8	1	no	—
101	C₂×Q₈	—	16	2	no	—
102	C₂²×C₄	—	8	1	no	—
103	C₂²×C₄	—	8	1	no	—
104	C₂²×C₄	#127	1	1	yes	characteristic
105	C₂²×C₄	#135	1	1	yes	characteristic
106	C₂²×C₄	#135	1	1	yes	characteristic
107	M₁₆	—	8	1	no	—
108	M₁₆	—	8	1	no	—
109	M₁₆	—	8	1	no	—
110	M₁₆	—	8	1	no	—
111	C₂×C₈	—	4	1	no	—
112	C₂×C₈	—	4	1	no	—
113	C₂×C₈	—	8	1	no	—
114	C₂×C₈	—	8	1	no	—
115	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₈	#123	1	1	yes	characteristic
118	C₂×C₈	#123	1	1	yes	characteristic
119	C₂×C₈	C₂×SD₁₆	1	2	yes	—
120	C₂×C₈	C₂×Q₁₆	1	2	yes	—
121	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
122	C₂×C₈	#137	1	2	yes	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	8	1	no	—
126	C₄⋊C₄	—	8	1	no	—
127	C₄⋊C₄	—	8	1	no	—
128	C₄⋊C₄	—	8	1	no	—
129	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
130	C₄²	#123	1	1	yes	characteristic
131	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₁₆	—	2	4	no	—
134	C₁₆	—	2	4	no	—
135	C₂³	#495	1	1	yes	characteristic
136	Q₈	—	16	2	no	—
137	Q₈	—	16	2	no	—
138	C₂×C₄	—	4	1	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	16	2	no	—
142	C₂×C₄	—	16	1	no	—
143	C₂×C₄	—	16	1	no	—
144	C₂×C₄	—	16	1	no	—
145	C₂×C₄	#399	1	1	yes	characteristic
146	C₂×C₄	#478	1	1	yes	characteristic
147	C₂×C₄	#478	1	1	yes	characteristic
148	C₂×C₄	#482	1	1	yes	characteristic
149	C₂×C₄	#482	1	1	yes	characteristic
150	C₂×C₄	#492	1	1	yes	characteristic
151	C₂×C₄	#494	1	1	yes	characteristic
152	C₂×C₄	#496	1	1	yes	characteristic
153	C₂×C₄	#496	1	1	yes	characteristic
154	C₂×C₄	#497	1	1	yes	characteristic
155	C₈	—	8	1	no	—
156	C₈	—	8	1	no	—
157	C₈	—	16	1	no	—
158	C₈	—	16	1	no	—
159	C₈	#475	1	2	yes	—
160	C₈	#477	1	2	yes	—
161	C₈	#483	1	2	yes	—
162	C₈	#484	1	2	yes	—
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	#1707	1	1	yes	characteristic
166	C₂²	#1719	1	1	yes	characteristic
167	C₂²	#2092	1	1	yes	characteristic
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	16	2	no	—
173	C₄	—	32	1	no	—
174	C₄	#1715	1	1	yes	characteristic
175	C₄	#1723	1	1	yes	characteristic
176	C₄	#2095	1	1	yes	characteristic
177	C₄	#2096	1	1	yes	characteristic
178	C₂	—	2	1	no	—
179	C₂	—	2	1	no	—
180	C₂	#10680	1	1	yes	characteristic
181	C₂	#13523	1	1	yes	characteristic
182	C₂	#13524	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13290

C₂

yes

maximal, characteristic

#13283

C₂

yes

maximal, characteristic

#12910

C₂

yes

maximal, characteristic

#12904

C₂

yes

maximal, characteristic

#9771

C₂

yes

maximal, characteristic

#8790

C₂

yes

maximal, characteristic

#7140

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

characteristic

C₂×#368

—

C₂×#365

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1846

C₂²

yes

—

—

C₂²

yes

—

—

—

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

—

—

—

Q₈

yes

—

#431

D₄

yes

characteristic

#431

D₄

yes

—

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

C₄×C₁₆

—

#333

—

#332

—

#322

—

C₈²

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

—

—

—

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

#127

yes

characteristic

105

C₂²×C₄

#135

yes

characteristic

106

C₂²×C₄

#135

yes

characteristic

107

M₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²⋊Q₈

yes

characteristic

116

C₂×C₈

C₂²⋊Q₈

yes

characteristic

117

C₂×C₈

#123

yes

characteristic

118

C₂×C₈

#123

yes

characteristic

119

C₂×C₈

C₂×SD₁₆

yes

—

120

C₂×C₈

C₂×Q₁₆

yes

—

121

C₂×C₈

AGL(1,ℤ/8)

yes

—

122

C₂×C₈

#137

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄²

C₂²⋊Q₈

yes

characteristic

130

C₄²

#123

yes

characteristic

131

C₄²

AGL(1,ℤ/8)

yes

characteristic

132

C₄²

AGL(1,ℤ/8)

yes

characteristic

133

C₁₆

—

134

C₁₆

—

135

C₂³

#495

yes

characteristic

136

Q₈

—

137

Q₈

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#399

yes

characteristic

146

C₂×C₄

#478

yes

characteristic

147

C₂×C₄

#478

yes

characteristic

148

C₂×C₄

#482

yes

characteristic

149

C₂×C₄

#482

yes

characteristic

150

C₂×C₄

#492

yes

characteristic

151

C₂×C₄

#494

yes

characteristic

152

C₂×C₄

#496

yes

characteristic

153

C₂×C₄

#496

yes

characteristic

154

C₂×C₄

#497

yes

characteristic

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

#475

yes

—

160

C₈

#477

yes

—

161

C₈

#483

yes

—

162

C₈

#484

yes

—

163

C₂²

—

164

C₂²

—

165

C₂²

#1707

yes

characteristic

166

C₂²

#1719

yes

characteristic

167

C₂²

#2092

yes

characteristic

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

#1715

yes

characteristic

175

C₄

#1723

yes

characteristic

176

C₄

#2095

yes

characteristic

177

C₄

#2096

yes

characteristic

178

C₂

—

179

C₂

—

180

C₂

#10680

yes

characteristic

181

C₂

#13523

yes

characteristic

182

C₂

#13524

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2092

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#495

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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
#13290	2
#13283	2
#12910	2
#12904	2
#9771	2
#8790	2
#7140	2

subgroup

index

Subgroups — #146266

Conjugacy classes of subgroups

Maximal subgroups