Group #134983 — subgroups

Subgroups — #134983

Order 512.

254 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 108 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	#33474	C₂	1	1	yes	maximal, characteristic
2	#12909	C₂	1	1	yes	maximal, characteristic
3	C₂×#1371	C₂	1	1	yes	maximal, characteristic
4	#8685	C₂	1	2	yes	maximal
5	#7281	C₂	1	2	yes	maximal
6	#2960	C₂²	1	1	yes	characteristic
7	#2183	—	2	1	no	—
8	#2183	C₄	1	2	yes	—
9	Q₈×C₁₆	—	2	1	no	—
10	Q₈×C₁₆	C₄	1	2	yes	—
11	C₂×#368	—	2	1	no	—
12	C₂×#368	—	2	1	no	—
13	#2151	—	2	1	no	—
14	#2150	C₂²	1	1	yes	characteristic
15	#2108	—	2	1	no	—
16	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
17	#1848	C₂²	1	2	yes	—
18	C₂×#335	C₂²	1	2	yes	—
19	#1550	—	2	2	no	—
20	#1371	—	2	1	no	—
21	#1371	—	2	1	no	—
22	#1292	—	2	2	no	—
23	#533	D₄	1	1	yes	characteristic
24	C₂×M₃₂	—	2	1	no	—
25	C₂²×C₁₆	D₄	1	1	yes	characteristic
26	C₂²×C₁₆	D₄	1	1	yes	characteristic
27	#446	—	2	1	no	—
28	#446	Q₈	1	2	yes	—
29	C₈×Q₈	—	2	1	no	—
30	C₈×Q₈	C₂×C₄	1	2	yes	—
31	C₂×#108	—	2	2	no	—
32	C₂×#108	—	2	2	no	—
33	#424	—	2	1	no	—
34	#423	—	2	2	no	—
35	#423	D₄	1	1	yes	characteristic
36	C₂×C₄⋊C₈	—	2	2	no	—
37	#405	—	2	1	no	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#368	—	4	2	no	—
40	#368	—	4	2	no	—
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄⋊C₁₆	—	2	2	no	—
43	C₄⋊C₁₆	—	2	1	no	—
44	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
45	#346	—	2	2	no	—
46	C₄×C₁₆	—	2	2	no	—
47	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
48	C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
49	C₄×C₁₆	D₄	1	1	yes	characteristic
50	C₄×C₁₆	D₄	1	1	yes	characteristic
51	#335	—	2	2	no	—
52	#335	—	2	2	no	—
53	#333	—	4	2	no	—
54	#332	—	4	2	no	—
55	C₂×M₁₆	—	2	1	no	—
56	C₂×M₁₆	—	4	2	no	—
57	C₂²×C₈	—	4	2	no	—
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₈	#41	1	1	yes	characteristic
60	C₄⋊Q₈	D₈	1	1	yes	characteristic
61	C₄⋊Q₈	D₈	1	1	yes	characteristic
62	#125	—	2	1	no	—
63	C₂²⋊Q₈	—	2	1	no	—
64	C₂²⋊Q₈	Q₁₆	1	2	yes	—
65	C₄×Q₈	—	2	1	no	—
66	C₄×Q₈	C₄⋊C₄	1	2	yes	—
67	#117	—	2	1	no	—
68	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
69	M₃₂	—	4	1	no	—
70	M₃₂	—	4	1	no	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	2	1	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₁₆	SD₁₆	1	2	yes	—
79	C₂×C₁₆	Q₁₆	1	2	yes	—
80	#108	—	4	2	no	—
81	#108	—	4	2	no	—
82	#108	—	8	2	no	—
83	C₄⋊C₈	—	2	2	no	—
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	2	1	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₄⋊C₈	—	4	2	no	—
89	C₄⋊C₈	—	4	2	no	—
90	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
91	#97	—	2	2	no	—
92	C₄×C₈	—	2	2	no	—
93	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
94	C₄×C₈	C₂×D₄	1	1	yes	characteristic
95	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₄×C₈	#41	1	1	yes	characteristic
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	#107	1	2	yes	—
99	C₂²×C₄	#121	1	1	yes	characteristic
100	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
101	C₂²×C₄	#135	1	1	yes	characteristic
102	M₁₆	—	4	1	no	—
103	M₁₆	—	4	1	no	—
104	M₁₆	—	8	2	no	—
105	M₁₆	—	8	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	8	2	no	—
110	C₂×C₈	—	8	2	no	—
111	C₂×C₈	—	16	2	no	—
112	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₈	#117	1	1	yes	characteristic
114	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
115	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
117	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
118	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₈	#137	1	1	yes	characteristic
120	C₄⋊C₄	—	2	2	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₄⋊C₄	—	2	1	no	—
124	C₄⋊C₄	—	2	2	no	—
125	C₄⋊C₄	#107	1	2	yes	—
126	C₂²⋊C₄	—	2	2	no	—
127	C₄²	—	2	2	no	—
128	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₄²	#121	1	1	yes	characteristic
130	C₄²	C₂×D₈	1	1	yes	characteristic
131	C₄²	#135	1	1	yes	characteristic
132	C₁₆	—	2	1	no	—
133	C₁₆	—	2	1	no	—
134	C₁₆	—	4	2	no	—
135	C₁₆	#103	1	2	yes	—
136	C₁₆	#103	1	2	yes	—
137	C₂³	#467	1	1	yes	characteristic
138	Q₈	—	4	2	no	—
139	Q₈	—	4	1	no	—
140	Q₈	—	4	1	no	—
141	C₂×C₄	—	2	1	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#380	1	1	yes	characteristic
147	C₂×C₄	C₂×#107	1	1	yes	characteristic
148	C₂×C₄	#427	1	1	yes	characteristic
149	C₂×C₄	#460	1	1	yes	characteristic
150	C₂×C₄	#462	1	1	yes	characteristic
151	C₂×C₄	#466	1	1	yes	characteristic
152	C₂×C₄	#466	1	1	yes	characteristic
153	C₂×C₄	#467	1	1	yes	characteristic
154	C₂×C₄	#472	1	1	yes	characteristic
155	C₂×C₄	#473	1	1	yes	characteristic
156	C₈	—	2	1	no	—
157	C₈	—	2	1	no	—
158	C₈	—	4	2	no	—
159	C₈	—	16	2	no	—
160	C₈	—	16	2	no	—
161	C₈	C₂×#103	1	1	yes	characteristic
162	C₈	#417	1	1	yes	characteristic
163	C₈	#475	1	1	yes	characteristic
164	C₈	#477	1	1	yes	characteristic
165	C₂²	—	2	1	no	—
166	C₂²	—	2	1	no	—
167	C₂²	#1670	1	1	yes	characteristic
168	C₂²	#1676	1	1	yes	characteristic
169	C₂²	#1808	1	1	yes	characteristic
170	C₄	—	2	1	no	—
171	C₄	—	2	1	no	—
172	C₄	—	2	1	no	—
173	C₄	—	4	2	no	—
174	C₄	—	4	2	no	—
175	C₄	#1668	1	1	yes	characteristic
176	C₄	#1674	1	1	yes	characteristic
177	C₄	#1803	1	1	yes	characteristic
178	C₄	#1816	1	1	yes	characteristic
179	C₄	#2221	1	1	yes	characteristic
180	C₄	#2222	1	1	yes	characteristic
181	C₂	—	2	1	no	—
182	C₂	—	2	1	no	—
183	C₂	#8645	1	1	yes	characteristic
184	C₂	#12840	1	1	yes	characteristic
185	C₂	#12842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33474

C₂

yes

maximal, characteristic

#12909

C₂

yes

maximal, characteristic

C₂×#1371

C₂

yes

maximal, characteristic

#8685

C₂

yes

maximal

#7281

C₂

yes

maximal

#2960

C₂²

yes

characteristic

#2183

—

#2183

C₄

yes

—

Q₈×C₁₆

—

Q₈×C₁₆

C₄

yes

—

—

—

—

C₂²

yes

characteristic

#2108

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

—

C₂×#335

C₂²

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

#446

—

#446

Q₈

yes

—

C₈×Q₈

—

C₈×Q₈

C₂×C₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×C₄⋊C₈

—

#405

—

C₂×C₄×C₈

C₂³

yes

characteristic

#368

—

#368

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

C₂×C₄

yes

characteristic

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

D₈

yes

characteristic

C₄⋊Q₈

D₈

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Q₁₆

yes

—

C₄×Q₈

—

C₄×Q₈

C₄⋊C₄

yes

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₄⋊C₄

yes

—

#97

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

#107

yes

—

C₂²×C₄

#121

yes

characteristic

100

C₂²×C₄

C₂×Q₁₆

yes

characteristic

101

C₂²×C₄

#135

yes

characteristic

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

113

C₂×C₈

#117

yes

characteristic

114

C₂×C₈

C₂²⋊Q₈

yes

characteristic

115

C₂×C₈

C₂²⋊Q₈

yes

characteristic

116

C₂×C₈

C₂×SD₁₆

yes

characteristic

117

C₂×C₈

C₂×Q₁₆

yes

characteristic

118

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

119

C₂×C₈

#137

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

#107

yes

—

126

C₂²⋊C₄

—

127

C₄²

—

128

C₄²

C₂×C₄⋊C₄

yes

characteristic

129

C₄²

#121

yes

characteristic

130

C₄²

C₂×D₈

yes

characteristic

131

C₄²

#135

yes

characteristic

132

C₁₆

—

133

C₁₆

—

134

C₁₆

—

135

C₁₆

#103

yes

—

136

C₁₆

#103

yes

—

137

C₂³

#467

yes

characteristic

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

#380

yes

characteristic

147

C₂×C₄

C₂×#107

yes

characteristic

148

C₂×C₄

#427

yes

characteristic

149

C₂×C₄

#460

yes

characteristic

150

C₂×C₄

#462

yes

characteristic

151

C₂×C₄

#466

yes

characteristic

152

C₂×C₄

#466

yes

characteristic

153

C₂×C₄

#467

yes

characteristic

154

C₂×C₄

#472

yes

characteristic

155

C₂×C₄

#473

yes

characteristic

156

C₈

—

157

C₈

—

158

C₈

—

159

C₈

—

160

C₈

—

161

C₈

C₂×#103

yes

characteristic

162

C₈

#417

yes

characteristic

163

C₈

#475

yes

characteristic

164

C₈

#477

yes

characteristic

165

C₂²

—

166

C₂²

—

167

C₂²

#1670

yes

characteristic

168

C₂²

#1676

yes

characteristic

169

C₂²

#1808

yes

characteristic

170

C₄

—

171

C₄

—

172

C₄

—

173

C₄

—

174

C₄

—

175

C₄

#1668

yes

characteristic

176

C₄

#1674

yes

characteristic

177

C₄

#1803

yes

characteristic

178

C₄

#1816

yes

characteristic

179

C₄

#2221

yes

characteristic

180

C₄

#2222

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

#8645

yes

characteristic

184

C₂

#12840

yes

characteristic

185

C₂

#12842

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#467
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1808

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#467

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ 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
#33474	2
#12909	2
C₂×#1371	2
#8685	2
#7281	2

subgroup

index

Subgroups — #134983

Conjugacy classes of subgroups

Maximal subgroups