Group #92867 — subgroups

Subgroups — #92867

Order 512.

197 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 68 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	#9823	C₂	1	1	yes	maximal, characteristic
2	#7151	C₂	1	1	yes	maximal, characteristic
3	#7151	C₂	1	1	yes	maximal, characteristic
4	C₂×#500	C₄	1	1	yes	characteristic
5	C₂×C₄⋊C₁₆	—	2	1	no	—
6	C₂×C₄⋊C₁₆	—	2	1	no	—
7	#1990	—	2	1	no	—
8	#1956	—	2	1	no	—
9	#1845	C₄	1	1	yes	characteristic
10	C₂×C₈²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈	—	2	1	no	—
12	C₂²×C₁₆	—	4	1	no	—
13	C₂²×C₁₆	—	4	1	no	—
14	#500	—	2	1	no	—
15	#500	—	4	1	no	—
16	#500	C₈	1	2	yes	—
17	C₂×#107	—	4	1	no	—
18	C₂×#107	—	4	1	no	—
19	C₂×#103	—	4	1	no	—
20	C₂×#103	—	4	1	no	—
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	#384	—	2	1	no	—
25	C₄⋊C₁₆	—	2	2	no	—
26	C₄⋊C₁₆	—	2	2	no	—
27	C₄⋊C₁₆	—	4	1	no	—
28	C₄⋊C₁₆	—	4	1	no	—
29	#340	—	4	1	no	—
30	#340	—	4	1	no	—
31	C₈²	—	4	1	no	—
32	C₂²×Q₈	—	4	1	no	—
33	C₂²×C₈	—	2	1	no	—
34	C₂²×C₈	D₈	1	1	yes	characteristic
35	C₂²×C₈	D₈	1	1	yes	characteristic
36	C₂²×C₈	SD₁₆	1	1	yes	characteristic
37	C₂²×C₈	SD₁₆	1	1	yes	characteristic
38	C₄⋊Q₈	—	2	2	no	—
39	C₄⋊Q₈	—	4	1	no	—
40	C₄⋊Q₈	—	4	1	no	—
41	C₄⋊Q₈	—	4	1	no	—
42	C₂×C₄⋊C₄	—	4	1	no	—
43	C₂×C₄⋊C₄	—	4	1	no	—
44	C₂×C₄⋊C₄	—	4	1	no	—
45	C₂×C₄⋊C₄	—	4	1	no	—
46	C₂×C₄⋊C₄	—	4	1	no	—
47	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₁₆	—	4	2	no	—
49	C₂×C₁₆	—	4	2	no	—
50	C₂×C₁₆	—	8	1	no	—
51	C₂×C₁₆	—	8	1	no	—
52	C₂×C₁₆	—	8	1	no	—
53	C₂×C₁₆	—	8	1	no	—
54	#107	—	4	2	no	—
55	#107	—	4	2	no	—
56	#107	—	8	2	no	—
57	#103	—	8	2	no	—
58	#103	—	8	1	no	—
59	#103	—	8	1	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	C₂×C₈	1	1	yes	characteristic
66	C₄×C₈	M₁₆	1	1	yes	characteristic
67	C₂×Q₈	—	4	2	no	—
68	C₂×Q₈	—	8	1	no	—
69	C₂×Q₈	—	8	1	no	—
70	C₂×Q₈	—	8	1	no	—
71	C₂×Q₈	—	8	1	no	—
72	C₂×Q₈	—	8	1	no	—
73	C₂²×C₄	—	8	1	no	—
74	C₂²×C₄	—	8	1	no	—
75	C₂²×C₄	#100	1	1	yes	characteristic
76	C₂²×C₄	#102	1	1	yes	characteristic
77	C₂²×C₄	#102	1	1	yes	characteristic
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	D₁₆	1	1	yes	characteristic
93	C₂×C₈	SD₃₂	1	1	yes	characteristic
94	C₂×C₈	SD₃₂	1	1	yes	characteristic
95	C₂×C₈	Q₃₂	1	1	yes	characteristic
96	C₄⋊C₄	—	4	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	8	2	no	—
99	C₄⋊C₄	—	8	1	no	—
100	C₄⋊C₄	—	8	1	no	—
101	C₄⋊C₄	—	8	1	no	—
102	C₄⋊C₄	—	8	1	no	—
103	C₄⋊C₄	—	8	1	no	—
104	C₄⋊C₄	—	8	1	no	—
105	C₄²	C₂²⋊C₈	1	1	yes	characteristic
106	C₄²	#99	1	1	yes	characteristic
107	C₄²	#104	1	1	yes	characteristic
108	C₄²	#104	1	1	yes	characteristic
109	C₁₆	—	8	2	no	—
110	C₁₆	—	8	2	no	—
111	C₂³	#331	1	1	yes	characteristic
112	Q₈	—	8	2	no	—
113	Q₈	—	8	2	no	—
114	Q₈	—	16	1	no	—
115	Q₈	—	16	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	8	2	no	—
119	C₂×C₄	—	16	1	no	—
120	C₂×C₄	—	16	1	no	—
121	C₂×C₄	—	16	1	no	—
122	C₂×C₄	—	16	1	no	—
123	C₂×C₄	—	16	1	no	—
124	C₂×C₄	#323	1	1	yes	characteristic
125	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
126	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
127	C₂×C₄	#327	1	1	yes	characteristic
128	C₂×C₄	#327	1	1	yes	characteristic
129	C₂×C₄	#329	1	1	yes	characteristic
130	C₂×C₄	#357	1	1	yes	characteristic
131	C₂×C₄	#358	1	1	yes	characteristic
132	C₂×C₄	#359	1	1	yes	characteristic
133	C₂×C₄	#359	1	1	yes	characteristic
134	C₂×C₄	#360	1	1	yes	characteristic
135	C₂×C₄	#360	1	1	yes	characteristic
136	C₂×C₄	#361	1	1	yes	characteristic
137	C₂×C₄	#362	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	4	1	no	—
145	C₈	—	4	1	no	—
146	C₂²	#1270	1	1	yes	characteristic
147	C₂²	#1358	1	1	yes	characteristic
148	C₂²	#1359	1	1	yes	characteristic
149	C₂²	#1360	1	1	yes	characteristic
150	C₂²	#1360	1	1	yes	characteristic
151	C₂²	#1361	1	1	yes	characteristic
152	C₂²	#1362	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	16	2	no	—
156	C₄	—	32	1	no	—
157	C₄	#1331	1	1	yes	characteristic
158	C₄	#1332	1	1	yes	characteristic
159	C₄	#1333	1	1	yes	characteristic
160	C₄	#1334	1	1	yes	characteristic
161	C₄	#1347	1	1	yes	characteristic
162	C₄	#1348	1	1	yes	characteristic
163	C₄	#1349	1	1	yes	characteristic
164	C₄	#1350	1	1	yes	characteristic
165	C₂	#7093	1	1	yes	characteristic
166	C₂	#7094	1	1	yes	characteristic
167	C₂	#7095	1	1	yes	characteristic
168	C₂	#7097	1	1	yes	characteristic
169	C₂	#7108	1	1	yes	characteristic
170	C₂	#7112	1	1	yes	characteristic
171	C₂	#7113	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#9823

C₂

yes

maximal, characteristic

#7151

C₂

yes

maximal, characteristic

#7151

C₂

yes

maximal, characteristic

C₂×#500

C₄

yes

characteristic

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

—

#1990

—

#1956

—

#1845

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

C₂²×C₁₆

—

#500

—

#500

—

#500

C₈

yes

—

—

—

—

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

#384

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#340

—

#340

—

C₈²

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₆

yes

characteristic

C₂×C₈

SD₃₂

yes

characteristic

C₂×C₈

SD₃₂

yes

characteristic

C₂×C₈

Q₃₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

C₂²⋊C₈

yes

characteristic

106

C₄²

#99

yes

characteristic

107

C₄²

#104

yes

characteristic

108

C₄²

#104

yes

characteristic

109

C₁₆

—

110

C₁₆

—

111

C₂³

#331

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

#323

yes

characteristic

125

C₂×C₄

D₄⋊C₈

yes

characteristic

126

C₂×C₄

D₄⋊C₈

yes

characteristic

127

C₂×C₄

#327

yes

characteristic

128

C₂×C₄

#327

yes

characteristic

129

C₂×C₄

#329

yes

characteristic

130

C₂×C₄

#357

yes

characteristic

131

C₂×C₄

#358

yes

characteristic

132

C₂×C₄

#359

yes

characteristic

133

C₂×C₄

#359

yes

characteristic

134

C₂×C₄

#360

yes

characteristic

135

C₂×C₄

#360

yes

characteristic

136

C₂×C₄

#361

yes

characteristic

137

C₂×C₄

#362

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₂²

#1270

yes

characteristic

147

C₂²

#1358

yes

characteristic

148

C₂²

#1359

yes

characteristic

149

C₂²

#1360

yes

characteristic

150

C₂²

#1360

yes

characteristic

151

C₂²

#1361

yes

characteristic

152

C₂²

#1362

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#1331

yes

characteristic

158

C₄

#1332

yes

characteristic

159

C₄

#1333

yes

characteristic

160

C₄

#1334

yes

characteristic

161

C₄

#1347

yes

characteristic

162

C₄

#1348

yes

characteristic

163

C₄

#1349

yes

characteristic

164

C₄

#1350

yes

characteristic

165

C₂

#7093

yes

characteristic

166

C₂

#7094

yes

characteristic

167

C₂

#7095

yes

characteristic

168

C₂

#7097

yes

characteristic

169

C₂

#7108

yes

characteristic

170

C₂

#7112

yes

characteristic

171

C₂

#7113

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#331
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	C₂×C₈²	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#331

commutator

C₄×C₈

C₂×C₈

Frattini, cosocle

C₂×C₈²

C₂²

LCS

G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₈² ▷ C₂×C₄² ▷ C₂² ▷ C₁

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#9823	2
#7151	2

subgroup

index

#9823

#7151

Subgroups — #92867

Conjugacy classes of subgroups

Maximal subgroups