Group #135126 — subgroups

Subgroups — #135126

Order 512.

261 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 261 conjugacy classes of subgroups — too many to draw. Only its 94 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	#17958	C₂	1	1	yes	maximal, characteristic
2	C₂×#1367	C₂	1	1	yes	maximal, characteristic
3	#11983	C₂	1	2	yes	maximal
4	#8824	C₂	1	1	yes	maximal, characteristic
5	#7241	C₂	1	2	yes	maximal
6	#3170	C₂²	1	1	yes	characteristic
7	#2960	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	#1837	—	2	1	no	—
10	#1591	—	2	1	no	—
11	#1591	C₄	1	2	yes	—
12	C₈×Q₁₆	—	2	1	no	—
13	C₈×Q₁₆	C₄	1	2	yes	—
14	#1567	C₂²	1	2	yes	—
15	#1567	C₂²	1	1	yes	characteristic
16	#1566	—	2	1	no	—
17	#1493	—	2	1	no	—
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1367	—	2	1	no	—
20	#1367	—	2	1	no	—
21	#1365	—	2	2	no	—
22	#1362	—	2	2	no	—
23	#533	—	2	1	no	—
24	C₂²×C₁₆	—	2	2	no	—
25	#500	D₄	1	1	yes	characteristic
26	#494	D₄	1	1	yes	characteristic
27	#487	—	2	1	no	—
28	#467	—	2	1	no	—
29	#467	Q₈	1	2	yes	—
30	#446	—	2	2	no	—
31	#446	—	4	1	no	—
32	C₈×Q₈	—	2	2	no	—
33	C₈×Q₈	—	4	1	no	—
34	C₄×Q₁₆	—	2	1	no	—
35	C₄×Q₁₆	C₂×C₄	1	2	yes	—
36	#424	—	4	1	no	—
37	#423	—	2	1	no	—
38	#423	—	2	2	no	—
39	#405	—	4	1	no	—
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	C₄⋊C₁₆	—	2	2	no	—
44	C₄⋊C₁₆	—	2	2	no	—
45	#341	—	4	1	no	—
46	#341	—	4	1	no	—
47	#335	—	2	1	no	—
48	#335	—	2	2	no	—
49	#335	—	2	2	no	—
50	#335	C₂×C₄	1	2	yes	—
51	#334	—	4	2	no	—
52	Q₈⋊C₈	—	2	2	no	—
53	Q₈⋊C₈	—	2	2	no	—
54	C₈²	C₂×C₄	1	1	yes	characteristic
55	C₈²	C₂×C₄	1	1	yes	characteristic
56	C₈²	D₄	1	1	yes	characteristic
57	C₈²	D₄	1	1	yes	characteristic
58	C₂×Q₁₆	—	2	1	no	—
59	C₂×Q₁₆	C₄⋊C₄	1	2	yes	—
60	C₂×M₁₆	—	4	1	no	—
61	C₂×M₁₆	—	4	2	no	—
62	C₂²×C₈	—	2	1	no	—
63	C₂²×C₈	D₈	1	1	yes	characteristic
64	C₂²×C₈	D₈	1	1	yes	characteristic
65	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₈	#41	1	1	yes	characteristic
67	C₄⋊Q₈	—	2	1	no	—
68	C₄⋊Q₈	—	2	1	no	—
69	#125	—	4	1	no	—
70	C₂²⋊Q₈	—	2	2	no	—
71	C₂²⋊Q₈	—	4	1	no	—
72	C₄×Q₈	—	2	2	no	—
73	C₄×Q₈	—	4	1	no	—
74	#117	—	4	1	no	—
75	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
76	C₂×C₁₆	—	4	2	no	—
77	C₂×C₁₆	—	8	2	no	—
78	#107	—	2	1	no	—
79	#107	C₄⋊C₄	1	2	yes	—
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	4	2	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄⋊C₈	—	4	2	no	—
84	C₄⋊C₈	—	4	1	no	—
85	C₄⋊C₈	—	4	2	no	—
86	#103	—	2	2	no	—
87	#103	—	2	2	no	—
88	#97	—	4	2	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	4	2	no	—
91	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
94	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
95	C₄×C₈	SD₁₆	1	1	yes	characteristic
96	C₄×C₈	SD₁₆	1	1	yes	characteristic
97	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
98	C₄×C₈	C₂×D₄	1	1	yes	characteristic
99	C₄×C₈	C₂×D₄	1	1	yes	characteristic
100	C₄×C₈	#41	1	1	yes	characteristic
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	4	1	no	—
103	C₂²×C₄	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
105	C₂²×C₄	#137	1	1	yes	characteristic
106	Q₁₆	—	4	2	no	—
107	Q₁₆	—	4	2	no	—
108	M₁₆	—	8	2	no	—
109	M₁₆	—	8	2	no	—
110	M₁₆	—	8	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	4	1	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	4	2	no	—
119	C₂×C₈	—	8	2	no	—
120	C₂×C₈	#102	1	1	yes	characteristic
121	C₂×C₈	#102	1	1	yes	characteristic
122	C₂×C₈	#102	1	1	yes	characteristic
123	C₂×C₈	#102	1	1	yes	characteristic
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	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	4	2	no	—
130	C₄⋊C₄	—	4	1	no	—
131	C₄⋊C₄	—	4	1	no	—
132	C₄⋊C₄	—	4	1	no	—
133	C₄⋊C₄	—	4	2	no	—
134	C₂²⋊C₄	—	4	2	no	—
135	C₄²	—	4	2	no	—
136	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₄²	#121	1	1	yes	characteristic
138	C₄²	C₂×SD₁₆	1	1	yes	characteristic
139	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₁₆	—	8	2	no	—
141	C₂³	#474	1	1	yes	characteristic
142	Q₈	—	4	2	no	—
143	Q₈	—	4	2	no	—
144	Q₈	—	8	2	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	4	1	no	—
148	C₂×C₄	—	4	2	no	—
149	C₂×C₄	—	4	2	no	—
150	C₂×C₄	#380	1	1	yes	characteristic
151	C₂×C₄	C₂×#102	1	1	yes	characteristic
152	C₂×C₄	#417	1	1	yes	characteristic
153	C₂×C₄	#459	1	1	yes	characteristic
154	C₂×C₄	#461	1	1	yes	characteristic
155	C₂×C₄	#476	1	1	yes	characteristic
156	C₂×C₄	#508	1	2	yes	—
157	C₂×C₄	#510	1	2	yes	—
158	C₈	—	2	2	no	—
159	C₈	—	2	1	no	—
160	C₈	—	2	1	no	—
161	C₈	—	2	1	no	—
162	C₈	—	2	1	no	—
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	2	1	no	—
166	C₈	—	4	1	no	—
167	C₈	—	8	2	no	—
168	C₈	—	16	2	no	—
169	C₈	#471	1	2	yes	—
170	C₂²	—	2	1	no	—
171	C₂²	—	2	1	no	—
172	C₂²	#1802	1	1	yes	characteristic
173	C₂²	#2229	1	2	yes	—
174	C₄	—	2	1	no	—
175	C₄	—	2	1	no	—
176	C₄	—	2	1	no	—
177	C₄	—	8	2	no	—
178	C₄	—	8	2	no	—
179	C₄	#1813	1	1	yes	characteristic
180	C₄	#1815	1	1	yes	characteristic
181	C₄	#2138	1	1	yes	characteristic
182	C₄	#2141	1	1	yes	characteristic
183	C₄	#2211	1	2	yes	—
184	C₂	—	2	2	no	—
185	C₂	#8682	1	1	yes	characteristic
186	C₂	#12814	1	1	yes	characteristic
187	C₂	#12825	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17958

C₂

yes

maximal, characteristic

C₂×#1367

C₂

yes

maximal, characteristic

#11983

C₂

yes

maximal

#8824

C₂

yes

maximal, characteristic

#7241

C₂

yes

maximal

#3170

C₂²

yes

characteristic

#2960

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1837

—

#1591

—

#1591

C₄

yes

—

C₈×Q₁₆

—

C₈×Q₁₆

C₄

yes

—

#1567

C₂²

yes

—

#1567

C₂²

yes

characteristic

#1566

—

#1493

—

C₂×C₈²

C₂²

yes

—

—

—

—

—

C₂²×C₁₆

—

#500

D₄

yes

characteristic

#494

D₄

yes

—

—

Q₈

yes

—

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

C₂×C₄

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

—

C₂×C₄

yes

—

#334

—

Q₈⋊C₈

—

Q₈⋊C₈

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

C₄⋊C₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

C₄⋊C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

100

C₄×C₈

#41

yes

characteristic

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂²×C₄

C₂²⋊Q₈

yes

characteristic

104

C₂²×C₄

C₂×D₈

yes

characteristic

105

C₂²×C₄

#137

yes

characteristic

106

Q₁₆

—

107

Q₁₆

—

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

#102

yes

characteristic

121

C₂×C₈

#102

yes

characteristic

122

C₂×C₈

#102

yes

characteristic

123

C₂×C₈

#102

yes

characteristic

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

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄⋊C₄

—

133

C₄⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

C₂×C₂²⋊C₄

yes

characteristic

137

C₄²

#121

yes

characteristic

138

C₄²

C₂×SD₁₆

yes

characteristic

139

C₄²

AGL(1,ℤ/8)

yes

characteristic

140

C₁₆

—

141

C₂³

#474

yes

characteristic

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

#380

yes

characteristic

151

C₂×C₄

C₂×#102

yes

characteristic

152

C₂×C₄

#417

yes

characteristic

153

C₂×C₄

#459

yes

characteristic

154

C₂×C₄

#461

yes

characteristic

155

C₂×C₄

#476

yes

characteristic

156

C₂×C₄

#508

yes

—

157

C₂×C₄

#510

yes

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

—

162

C₈

—

163

C₈

—

164

C₈

—

165

C₈

—

166

C₈

—

167

C₈

—

168

C₈

—

169

C₈

#471

yes

—

170

C₂²

—

171

C₂²

—

172

C₂²

#1802

yes

characteristic

173

C₂²

#2229

yes

—

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

—

178

C₄

—

179

C₄

#1813

yes

characteristic

180

C₄

#1815

yes

characteristic

181

C₄

#2138

yes

characteristic

182

C₄

#2141

yes

characteristic

183

C₄

#2211

yes

—

184

C₂

—

185

C₂

#8682

yes

characteristic

186

C₂

#12814

yes

characteristic

187

C₂

#12825

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1802
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#474
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1802

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#474

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
#17958	2
C₂×#1367	2
#11983	2
#8824	2
#7241	2

subgroup

index

Subgroups — #135126

Conjugacy classes of subgroups

Maximal subgroups