Group #142120 — subgroups

Subgroups — #142120

Order 512.

302 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 118 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	#33257	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#8805	C₂	1	1	yes	maximal, characteristic
4	#7275	C₂	1	2	yes	maximal
5	#7275	C₂	1	2	yes	maximal
6	#2864	—	2	1	no	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	—	2	1	no	—
9	#1942	C₄	1	2	yes	—
10	#1942	C₄	1	2	yes	—
11	#1941	—	2	1	no	—
12	#1941	—	2	1	no	—
13	#1846	C₂²	1	1	yes	characteristic
14	#1846	C₂²	1	1	yes	characteristic
15	#1842	—	2	1	no	—
16	#1835	—	2	1	no	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	C₂×#334	—	2	1	no	—
19	C₂×#334	C₂²	1	2	yes	—
20	#1374	—	2	1	no	—
21	C₂²×C₁₆	—	2	2	no	—
22	C₂×#107	C₂×C₄	1	2	yes	—
23	C₂×#107	C₂×C₄	1	2	yes	—
24	C₂×#106	—	2	1	no	—
25	C₂×#106	—	2	1	no	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×C₄⋊C₈	—	4	2	no	—
28	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	#389	—	2	2	no	—
35	#389	—	2	2	no	—
36	#387	—	4	2	no	—
37	#387	—	4	2	no	—
38	#382	—	2	1	no	—
39	#382	—	2	1	no	—
40	C₄³	D₄	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄⋊C₁₆	Q₈	1	4	yes	—
43	C₄×C₁₆	—	4	2	no	—
44	#340	—	2	2	no	—
45	#340	—	2	2	no	—
46	#334	—	2	4	no	—
47	#334	—	4	2	no	—
48	C₂²×C₈	—	4	2	no	—
49	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄⋊C₄	—	2	2	no	—
55	C₂×C₄⋊C₄	—	4	2	no	—
56	C₂×C₄⋊C₄	—	4	2	no	—
57	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄²	D₈	1	2	yes	—
60	C₂×C₄²	SD₁₆	1	2	yes	—
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₆	—	2	4	no	—
63	C₂×C₁₆	—	4	2	no	—
64	C₂×C₁₆	—	4	2	no	—
65	#107	—	4	2	no	—
66	#107	—	4	2	no	—
67	#106	—	4	2	no	—
68	#106	—	4	1	no	—
69	#106	—	4	1	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄⋊C₈	—	4	2	no	—
72	C₄⋊C₈	—	8	4	no	—
73	C₄×C₈	—	2	2	no	—
74	C₄×C₈	—	2	1	no	—
75	C₄×C₈	—	2	1	no	—
76	C₄×C₈	—	4	1	no	—
77	C₄×C₈	—	4	1	no	—
78	C₄×C₈	SD₁₆	1	2	yes	—
79	C₄×C₈	Q₁₆	1	2	yes	—
80	C₄×C₈	C₂×Q₈	1	2	yes	—
81	C₄×C₈	#41	1	2	yes	—
82	#95	—	4	2	no	—
83	#95	—	4	2	no	—
84	#95	—	4	1	no	—
85	#95	—	4	1	no	—
86	#95	—	4	1	no	—
87	#95	—	4	1	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	—	4	2	no	—
90	C₂²×C₄	#102	1	2	yes	—
91	C₂²×C₄	#102	1	2	yes	—
92	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
94	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	8	4	no	—
102	C₂×C₈	—	8	2	no	—
103	C₂×C₈	#103	1	2	yes	—
104	C₂×C₈	#103	1	2	yes	—
105	C₂×C₈	C₄×Q₈	1	2	yes	—
106	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₈	#135	1	2	yes	—
108	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
109	C₂×C₈	#137	1	1	yes	characteristic
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	8	4	no	—
115	C₄⋊C₄	—	8	4	no	—
116	C₄²	—	2	4	no	—
117	C₄²	—	2	4	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	#124	1	1	yes	characteristic
123	C₄²	C₄⋊Q₈	1	1	yes	characteristic
124	C₄²	C₂×SD₁₆	1	1	yes	characteristic
125	C₄²	C₂×Q₁₆	1	1	yes	characteristic
126	C₁₆	—	4	4	no	—
127	C₂³	C₂×#102	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	#359	1	4	yes	—
141	C₂×C₄	#360	1	4	yes	—
142	C₂×C₄	#391	1	1	yes	characteristic
143	C₂×C₄	#413	1	1	yes	characteristic
144	C₂×C₄	#413	1	1	yes	characteristic
145	C₂×C₄	C₂×#103	1	1	yes	characteristic
146	C₂×C₄	#416	1	1	yes	characteristic
147	C₂×C₄	#417	1	1	yes	characteristic
148	C₂×C₄	#486	1	1	yes	characteristic
149	C₂×C₄	#487	1	1	yes	characteristic
150	C₂×C₄	#498	1	1	yes	characteristic
151	C₂×C₄	#500	1	1	yes	characteristic
152	C₈	—	2	2	no	—
153	C₈	—	2	2	no	—
154	C₈	—	2	2	no	—
155	C₈	—	16	2	no	—
156	C₈	#478	1	2	yes	—
157	C₈	#479	1	2	yes	—
158	C₂²	#1540	1	1	yes	characteristic
159	C₂²	#1547	1	1	yes	characteristic
160	C₂²	#1990	1	1	yes	characteristic
161	C₂²	C₂×#359	1	2	yes	—
162	C₂²	C₂×#360	1	2	yes	—
163	C₄	—	2	4	no	—
164	C₄	—	2	4	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	16	2	no	—
170	C₄	—	16	2	no	—
171	C₄	#1542	1	1	yes	characteristic
172	C₄	#1550	1	1	yes	characteristic
173	C₄	#1989	1	1	yes	characteristic
174	C₄	#1998	1	1	yes	characteristic
175	C₂	#9900	1	1	yes	characteristic
176	C₂	#11944	1	2	yes	—
177	C₂	#11977	1	2	yes	—
178	C₂	#13285	1	1	yes	characteristic
179	C₂	#13287	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33257

C₂

yes

maximal, characteristic

C₄×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#8805

C₂

yes

maximal, characteristic

#7275

C₂

yes

maximal

#7275

C₂

yes

maximal

#2864

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

—

#1942

C₄

yes

—

#1942

C₄

yes

—

#1941

—

#1941

—

#1846

C₂²

yes

characteristic

#1846

C₂²

yes

characteristic

#1842

—

#1835

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#334

—

C₂×#334

C₂²

yes

—

#1374

—

C₂²×C₁₆

—

C₂×#107

C₂×C₄

yes

—

C₂×#107

C₂×C₄

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

Q₈

yes

—

C₄×C₁₆

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

—

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

—

—

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#102

yes

—

C₂²×C₄

#102

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

#103

yes

—

104

C₂×C₈

#103

yes

—

105

C₂×C₈

C₄×Q₈

yes

—

106

C₂×C₈

C₂²⋊Q₈

yes

—

107

C₂×C₈

#135

yes

—

108

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

109

C₂×C₈

#137

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

#124

yes

characteristic

123

C₄²

C₄⋊Q₈

yes

characteristic

124

C₄²

C₂×SD₁₆

yes

characteristic

125

C₄²

C₂×Q₁₆

yes

characteristic

126

C₁₆

—

127

C₂³

C₂×#102

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#359

yes

—

141

C₂×C₄

#360

yes

—

142

C₂×C₄

#391

yes

characteristic

143

C₂×C₄

#413

yes

characteristic

144

C₂×C₄

#413

yes

characteristic

145

C₂×C₄

C₂×#103

yes

characteristic

146

C₂×C₄

#416

yes

characteristic

147

C₂×C₄

#417

yes

characteristic

148

C₂×C₄

#486

yes

characteristic

149

C₂×C₄

#487

yes

characteristic

150

C₂×C₄

#498

yes

characteristic

151

C₂×C₄

#500

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

#478

yes

—

157

C₈

#479

yes

—

158

C₂²

#1540

yes

characteristic

159

C₂²

#1547

yes

characteristic

160

C₂²

#1990

yes

characteristic

161

C₂²

C₂×#359

yes

—

162

C₂²

C₂×#360

yes

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₄

#1542

yes

characteristic

172

C₄

#1550

yes

characteristic

173

C₄

#1989

yes

characteristic

174

C₄

#1998

yes

characteristic

175

C₂

#9900

yes

characteristic

176

C₂

#11944

yes

—

177

C₂

#11977

yes

—

178

C₂

#13285

yes

characteristic

179

C₂

#13287

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#102

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ 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
#33257	2
C₄×C₄⋊C₁₆	2
#8805	2
#7275	2

subgroup

index

#33257

C₄×C₄⋊C₁₆

#8805

#7275

Subgroups — #142120

Conjugacy classes of subgroups

Maximal subgroups