Group #140101 — subgroups

Subgroups — #140101

Order 512.

192 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 82 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	#13164	C₂	1	1	yes	maximal, characteristic
2	#12670	C₂	1	1	yes	maximal, characteristic
3	#12667	C₂	1	1	yes	maximal, characteristic
4	#7181	C₂	1	1	yes	maximal, characteristic
5	#7180	C₂	1	1	yes	maximal, characteristic
6	#7160	C₂	1	1	yes	maximal, characteristic
7	#7141	C₂	1	1	yes	maximal, characteristic
8	C₂×#368	C₄	1	1	yes	characteristic
9	#2158	C₄	1	1	yes	characteristic
10	#2153	—	2	1	no	—
11	C₂×#364	C₂²	1	1	yes	characteristic
12	C₂×#364	C₂²	1	1	yes	characteristic
13	#2110	—	2	1	no	—
14	C₄×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
16	#1772	C₂²	1	1	yes	characteristic
17	#1772	C₂²	1	1	yes	characteristic
18	#1450	C₂²	1	1	yes	characteristic
19	#1382	C₄	1	1	yes	characteristic
20	#1382	C₄	1	1	yes	characteristic
21	#1378	—	2	1	no	—
22	#1378	—	2	1	no	—
23	C₂×M₃₂	—	2	1	no	—
24	C₂×M₃₂	—	2	1	no	—
25	C₂×M₃₂	D₄	1	1	yes	characteristic
26	C₂×M₃₂	D₄	1	1	yes	characteristic
27	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
29	#431	—	2	1	no	—
30	C₂×#108	C₂×C₄	1	1	yes	characteristic
31	C₂×#108	C₂×C₄	1	1	yes	characteristic
32	#427	C₂×C₄	1	1	yes	characteristic
33	#427	C₂×C₄	1	1	yes	characteristic
34	#405	—	2	1	no	—
35	#405	—	2	1	no	—
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	#368	—	2	2	no	—
38	#367	—	2	1	no	—
39	#366	—	2	1	no	—
40	#364	—	2	1	no	—
41	#364	—	2	1	no	—
42	#364	—	2	1	no	—
43	#364	—	4	2	no	—
44	#364	Q₈	1	2	yes	—
45	#347	—	4	2	no	—
46	#346	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	—	2	1	no	—
50	#339	—	2	1	no	—
51	#339	—	2	1	no	—
52	#322	—	2	1	no	—
53	C₈²	—	2	1	no	—
54	C₂×M₁₆	—	2	1	no	—
55	C₂×M₁₆	—	2	1	no	—
56	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	#117	—	2	1	no	—
59	#117	—	2	1	no	—
60	C₂×C₄²	#41	1	1	yes	characteristic
61	M₃₂	—	2	1	no	—
62	M₃₂	—	2	2	no	—
63	M₃₂	—	2	1	no	—
64	M₃₂	—	2	1	no	—
65	M₃₂	—	4	1	no	—
66	M₃₂	—	4	1	no	—
67	M₃₂	—	4	1	no	—
68	M₃₂	Q₁₆	1	2	yes	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	1	no	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	2	1	no	—
73	C₂×C₁₆	—	2	1	no	—
74	C₂×C₁₆	—	2	1	no	—
75	C₂×C₁₆	—	2	1	no	—
76	C₂×C₁₆	—	4	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	M₁₆	1	1	yes	characteristic
79	C₂×C₁₆	M₁₆	1	1	yes	characteristic
80	#108	—	2	1	no	—
81	#108	—	2	1	no	—
82	#108	—	2	1	no	—
83	#108	M₁₆	1	2	yes	—
84	#107	—	2	1	no	—
85	#107	—	2	1	no	—
86	#106	—	2	1	no	—
87	#106	—	2	1	no	—
88	C₄⋊C₈	—	2	2	no	—
89	C₂²⋊C₈	—	4	1	no	—
90	#97	—	4	1	no	—
91	#97	—	4	1	no	—
92	C₄×C₈	—	4	1	no	—
93	C₄×C₈	—	4	1	no	—
94	C₄×C₈	—	4	1	no	—
95	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
96	C₄×C₈	#41	1	1	yes	characteristic
97	C₄×C₈	#41	1	1	yes	characteristic
98	C₄×C₈	#41	1	1	yes	characteristic
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	M₁₆	—	4	1	no	—
103	M₁₆	—	4	1	no	—
104	M₁₆	—	4	1	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	4	1	no	—
108	C₂×C₈	#117	1	1	yes	characteristic
109	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₈	#123	1	1	yes	characteristic
112	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
113	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₈	#131	1	1	yes	characteristic
115	C₂×C₈	#131	1	1	yes	characteristic
116	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
117	C₂×C₈	#135	1	1	yes	characteristic
118	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₄⋊C₄	—	4	1	no	—
121	C₄⋊C₄	—	4	1	no	—
122	C₂²⋊C₄	—	4	1	no	—
123	C₂²⋊C₄	—	4	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	4	1	no	—
126	C₄²	—	4	1	no	—
127	C₄²	#125	1	1	yes	characteristic
128	C₄²	#126	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	1	no	—
131	C₁₆	—	2	2	no	—
132	C₁₆	—	4	1	no	—
133	C₁₆	—	4	2	no	—
134	C₁₆	—	4	1	no	—
135	C₁₆	—	8	1	no	—
136	C₂³	#444	1	1	yes	characteristic
137	C₂×C₄	—	2	1	no	—
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	8	1	no	—
141	C₂×C₄	—	8	1	no	—
142	C₂×C₄	#387	1	1	yes	characteristic
143	C₂×C₄	#435	1	1	yes	characteristic
144	C₂×C₄	#436	1	1	yes	characteristic
145	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
146	C₂×C₄	#442	1	1	yes	characteristic
147	C₂×C₄	#443	1	1	yes	characteristic
148	C₈	—	4	2	no	—
149	C₈	—	8	2	no	—
150	C₈	—	8	1	no	—
151	C₈	#432	1	1	yes	characteristic
152	C₈	#433	1	1	yes	characteristic
153	C₈	#446	1	1	yes	characteristic
154	C₈	#446	1	1	yes	characteristic
155	C₈	#477	1	1	yes	characteristic
156	C₈	#478	1	1	yes	characteristic
157	C₈	#482	1	1	yes	characteristic
158	C₈	#484	1	1	yes	characteristic
159	C₂²	—	4	1	no	—
160	C₂²	#1584	1	1	yes	characteristic
161	C₂²	#1594	1	1	yes	characteristic
162	C₂²	#1922	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	—	4	1	no	—
165	C₄	—	16	1	no	—
166	C₄	—	16	1	no	—
167	C₄	#1586	1	1	yes	characteristic
168	C₄	#1591	1	1	yes	characteristic
169	C₄	#1917	1	1	yes	characteristic
170	C₄	#1921	1	1	yes	characteristic
171	C₂	—	2	1	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#9540	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13164

C₂

yes

maximal, characteristic

#12670

C₂

yes

maximal, characteristic

#12667

C₂

yes

maximal, characteristic

#7181

C₂

yes

maximal, characteristic

#7180

C₂

yes

maximal, characteristic

#7160

C₂

yes

maximal, characteristic

#7141

C₂

yes

maximal, characteristic

C₂×#368

C₄

yes

characteristic

#2158

C₄

yes

characteristic

#2153

—

C₂×#364

C₂²

yes

characteristic

C₂×#364

C₂²

yes

characteristic

#2110

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1772

C₂²

yes

characteristic

#1772

C₂²

yes

characteristic

#1450

C₂²

yes

characteristic

#1382

C₄

yes

characteristic

#1382

C₄

yes

characteristic

#1378

—

#1378

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂×M₃₂

D₄

yes

characteristic

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

—

C₂×#108

C₂×C₄

yes

characteristic

C₂×#108

C₂×C₄

yes

characteristic

#427

C₂×C₄

yes

characteristic

#427

C₂×C₄

yes

characteristic

#405

—

#405

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

Q₈

yes

—

#347

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#339

—

#339

—

#322

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

#117

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

M₁₆

yes

characteristic

C₂×C₁₆

M₁₆

yes

—

—

—

M₁₆

yes

—

—

—

—

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂²×C₄

—

100

C₂²×C₄

C₄×D₄

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

#117

yes

characteristic

109

C₂×C₈

C₄×Q₈

yes

characteristic

110

C₂×C₈

C₂²⋊Q₈

yes

characteristic

111

C₂×C₈

#123

yes

characteristic

112

C₂×C₈

C₂×M₁₆

yes

characteristic

113

C₂×C₈

C₂×M₁₆

yes

characteristic

114

C₂×C₈

#131

yes

characteristic

115

C₂×C₈

#131

yes

characteristic

116

C₂×C₈

C₂×Q₁₆

yes

characteristic

117

C₂×C₈

#135

yes

characteristic

118

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

119

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

#125

yes

characteristic

128

C₄²

#126

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₁₆

—

132

C₁₆

—

133

C₁₆

—

134

C₁₆

—

135

C₁₆

—

136

C₂³

#444

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

#387

yes

characteristic

143

C₂×C₄

#435

yes

characteristic

144

C₂×C₄

#436

yes

characteristic

145

C₂×C₄

C₄×Q₁₆

yes

characteristic

146

C₂×C₄

#442

yes

characteristic

147

C₂×C₄

#443

yes

characteristic

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

#432

yes

characteristic

152

C₈

#433

yes

characteristic

153

C₈

#446

yes

characteristic

154

C₈

#446

yes

characteristic

155

C₈

#477

yes

characteristic

156

C₈

#478

yes

characteristic

157

C₈

#482

yes

characteristic

158

C₈

#484

yes

characteristic

159

C₂²

—

160

C₂²

#1584

yes

characteristic

161

C₂²

#1594

yes

characteristic

162

C₂²

#1922

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#1586

yes

characteristic

168

C₄

#1591

yes

characteristic

169

C₄

#1917

yes

characteristic

170

C₄

#1921

yes

characteristic

171

C₂

—

172

C₂

—

173

C₂

—

174

C₂

#9540

yes

characteristic

	subgroup	G/H
center	C₈	#482
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9540
prime radical	C₂³	#444
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#482

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₂³

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ 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
#13164	2
#12670	2
#12667	2
#7181	2
#7180	2
#7160	2
#7141	2

subgroup

index

Subgroups — #140101

Conjugacy classes of subgroups

Maximal subgroups