Group #134635 — subgroups

Subgroups — #134635

Order 512.

277 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 100 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	#21733	C₂	1	1	yes	maximal, characteristic
2	#12899	C₂	1	1	yes	maximal, characteristic
3	#11939	C₂	1	2	yes	maximal
4	#8780	C₂	1	1	yes	maximal, characteristic
5	#7276	C₂	1	2	yes	maximal
6	#3160	C₂²	1	1	yes	characteristic
7	#2961	—	2	1	no	—
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
11	#1995	—	2	1	no	—
12	#1946	C₄	1	2	yes	—
13	#1945	—	2	1	no	—
14	#1848	C₂²	1	1	yes	characteristic
15	#1773	C₄	1	2	yes	—
16	#1772	—	2	1	no	—
17	#1726	C₂²	1	1	yes	characteristic
18	#1564	C₂²	1	2	yes	—
19	#1365	—	2	2	no	—
20	#1364	—	2	2	no	—
21	#1281	—	2	1	no	—
22	#1274	—	2	1	no	—
23	#533	—	2	1	no	—
24	C₂²×C₁₆	—	2	2	no	—
25	#500	D₄	1	2	yes	—
26	#499	—	2	1	no	—
27	#499	Q₈	1	2	yes	—
28	#498	—	2	1	no	—
29	#446	—	4	4	no	—
30	#431	—	2	2	no	—
31	C₂×#108	—	2	1	no	—
32	C₂×#108	C₂×C₄	1	2	yes	—
33	#427	—	2	1	no	—
34	#427	C₂×C₄	1	2	yes	—
35	#424	—	2	2	no	—
36	#423	—	2	1	no	—
37	#405	—	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	#376	D₄	1	1	yes	characteristic
45	C₄⋊C₁₆	—	2	2	no	—
46	C₄⋊C₁₆	—	2	2	no	—
47	C₄⋊C₁₆	—	4	2	no	—
48	C₄⋊C₁₆	—	4	2	no	—
49	#339	—	2	2	no	—
50	#339	—	2	2	no	—
51	#335	—	2	2	no	—
52	#334	—	2	2	no	—
53	C₂×M₁₆	—	2	2	no	—
54	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₄⋊Q₈	—	2	2	no	—
59	#125	—	2	2	no	—
60	C₂²⋊Q₈	—	4	2	no	—
61	C₄×Q₈	—	4	2	no	—
62	#117	—	2	2	no	—
63	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₁₆	—	4	2	no	—
67	C₂×C₁₆	—	4	2	no	—
68	C₂×C₁₆	—	4	2	no	—
69	#108	—	4	2	no	—
70	#108	—	4	1	no	—
71	#108	—	4	1	no	—
72	#107	—	2	2	no	—
73	#107	C₄⋊C₄	1	4	yes	—
74	#106	—	2	2	no	—
75	#106	—	2	2	no	—
76	C₄⋊C₈	—	4	4	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₄⋊C₈	—	4	2	no	—
79	C₂²⋊C₈	—	4	2	no	—
80	#97	—	2	2	no	—
81	#97	—	2	2	no	—
82	#97	—	4	1	no	—
83	#97	—	4	2	no	—
84	C₄×C₈	—	2	1	no	—
85	C₄×C₈	—	2	1	no	—
86	C₄×C₈	—	4	2	no	—
87	C₄×C₈	—	4	2	no	—
88	C₄×C₈	SD₁₆	1	2	yes	—
89	C₄×C₈	Q₁₆	1	2	yes	—
90	C₄×C₈	C₂×D₄	1	1	yes	characteristic
91	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
92	C₄×C₈	#41	1	1	yes	characteristic
93	C₄×C₈	#41	1	1	yes	characteristic
94	#95	—	2	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂²×C₄	#135	1	2	yes	—
100	M₁₆	—	4	2	no	—
101	M₁₆	—	4	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	8	2	no	—
111	C₂×C₈	#103	1	2	yes	—
112	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₈	#117	1	1	yes	characteristic
114	C₂×C₈	#121	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	4	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₂²⋊C₄	—	4	2	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	4	2	no	—
127	C₄²	#101	1	2	yes	—
128	C₄²	#103	1	2	yes	—
129	C₄²	#121	1	1	yes	characteristic
130	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
131	C₄²	AGL(1,ℤ/8)	1	2	yes	—
132	C₁₆	—	8	2	no	—
133	C₂³	#416	1	1	yes	characteristic
134	Q₈	—	16	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	#380	1	1	yes	characteristic
142	C₂×C₄	C₂×#101	1	1	yes	characteristic
143	C₂×C₄	#413	1	1	yes	characteristic
144	C₂×C₄	C₂×#103	1	1	yes	characteristic
145	C₂×C₄	#417	1	1	yes	characteristic
146	C₂×C₄	#418	1	1	yes	characteristic
147	C₂×C₄	#463	1	2	yes	—
148	C₂×C₄	#478	1	2	yes	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	2	2	no	—
152	C₈	—	4	1	no	—
153	C₈	—	8	2	no	—
154	C₈	—	16	2	no	—
155	C₈	#460	1	1	yes	characteristic
156	C₈	#462	1	1	yes	characteristic
157	C₈	#475	1	1	yes	characteristic
158	C₈	#477	1	1	yes	characteristic
159	C₂²	—	2	2	no	—
160	C₂²	#1542	1	1	yes	characteristic
161	C₂²	#1552	1	1	yes	characteristic
162	C₂²	#1804	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	2	2	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	8	4	no	—
168	C₄	#1549	1	1	yes	characteristic
169	C₄	#1557	1	1	yes	characteristic
170	C₄	#1801	1	1	yes	characteristic
171	C₄	#1803	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#8591	1	1	yes	characteristic
175	C₂	#12829	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21733

C₂

yes

maximal, characteristic

#12899

C₂

yes

maximal, characteristic

#11939

C₂

yes

maximal

#8780

C₂

yes

maximal, characteristic

#7276

C₂

yes

maximal

#3160

C₂²

yes

characteristic

#2961

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1995

—

#1946

C₄

yes

—

#1945

—

#1848

C₂²

yes

characteristic

#1773

C₄

yes

—

#1772

—

#1726

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

—

—

C₂²×C₁₆

—

#500

D₄

yes

—

#499

—

#499

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

#427

—

#427

C₂×C₄

yes

—

#424

—

#423

—

#405

—

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

characteristic

#376

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

C₂×M₁₆

—

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

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₄⋊C₄

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

#135

yes

—

100

M₁₆

—

101

M₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#103

yes

—

112

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

113

C₂×C₈

#117

yes

characteristic

114

C₂×C₈

#121

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

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#101

yes

—

128

C₄²

#103

yes

—

129

C₄²

#121

yes

characteristic

130

C₄²

C₂²⋊Q₈

yes

characteristic

131

C₄²

AGL(1,ℤ/8)

yes

—

132

C₁₆

—

133

C₂³

#416

yes

characteristic

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

#380

yes

characteristic

142

C₂×C₄

C₂×#101

yes

characteristic

143

C₂×C₄

#413

yes

characteristic

144

C₂×C₄

C₂×#103

yes

characteristic

145

C₂×C₄

#417

yes

characteristic

146

C₂×C₄

#418

yes

characteristic

147

C₂×C₄

#463

yes

—

148

C₂×C₄

#478

yes

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

#460

yes

characteristic

156

C₈

#462

yes

characteristic

157

C₈

#475

yes

characteristic

158

C₈

#477

yes

characteristic

159

C₂²

—

160

C₂²

#1542

yes

characteristic

161

C₂²

#1552

yes

characteristic

162

C₂²

#1804

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

#1549

yes

characteristic

169

C₄

#1557

yes

characteristic

170

C₄

#1801

yes

characteristic

171

C₄

#1803

yes

characteristic

172

C₂

—

173

C₂

—

174

C₂

#8591

yes

characteristic

175

C₂

#12829

yes

—

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1804
prime radical	C₂³	#416
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#418

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ 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
#21733	2
#12899	2
#11939	2
#8780	2
#7276	2

subgroup

index

Subgroups — #134635

Conjugacy classes of subgroups

Maximal subgroups