Group #93010 — subgroups

Subgroups — #93010

Order 512.

193 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 60 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	#13264	C₂	1	1	yes	maximal, characteristic
2	#11493	C₂	1	1	yes	maximal, characteristic
3	C₈×#101	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
5	#2162	—	2	1	no	—
6	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
7	C₂×#346	C₄	1	1	yes	characteristic
8	C₄×#101	C₄	1	1	yes	characteristic
9	#1461	C₄	1	1	yes	characteristic
10	C₈×M₁₆	—	2	1	no	—
11	#1374	—	2	1	no	—
12	#1374	—	2	1	no	—
13	#1322	—	2	1	no	—
14	#1322	C₄	1	2	yes	—
15	#1283	—	2	1	no	—
16	C₂×C₄×Q₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₁₆	—	2	1	no	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₈×Q₈	—	4	1	no	—
20	C₈×Q₈	—	4	1	no	—
21	#431	—	2	1	no	—
22	C₂×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
24	C₂×C₄⋊C₈	Q₈	1	1	yes	characteristic
25	C₂×#101	—	2	1	no	—
26	C₄×M₁₆	—	2	1	no	—
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	C₄⋊C₁₆	—	4	1	no	—
31	C₄⋊C₁₆	—	4	1	no	—
32	C₂²⋊C₁₆	—	2	2	no	—
33	#346	—	4	1	no	—
34	C₄×C₁₆	—	4	1	no	—
35	#343	C₈	1	2	yes	—
36	#324	—	2	1	no	—
37	#324	C₈	1	2	yes	—
38	#322	—	4	1	no	—
39	C₈²	—	4	1	no	—
40	C₂²×Q₈	M₁₆	1	1	yes	characteristic
41	C₂×M₁₆	—	2	2	no	—
42	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₄×Q₈	—	4	1	no	—
47	C₄×Q₈	—	4	1	no	—
48	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄⋊C₄	C₂×C₈	1	1	yes	characteristic
50	C₂×C₄⋊C₄	M₁₆	1	1	yes	characteristic
51	C₂×C₄²	C₄²	1	1	yes	characteristic
52	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	4	1	no	—
56	C₂×C₁₆	—	4	1	no	—
57	C₂×C₁₆	—	4	2	no	—
58	C₂×C₁₆	—	4	2	no	—
59	C₂×C₁₆	—	4	1	no	—
60	C₄⋊C₈	—	4	1	no	—
61	C₄⋊C₈	—	4	1	no	—
62	C₄⋊C₈	—	4	1	no	—
63	C₄⋊C₈	—	4	1	no	—
64	#101	—	2	4	no	—
65	C₂²⋊C₈	—	2	2	no	—
66	#97	—	4	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	—	4	1	no	—
74	C₄×C₈	—	4	1	no	—
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	4	1	no	—
78	C₂²×C₄	#95	1	1	yes	characteristic
79	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
80	C₂²×C₄	#97	1	1	yes	characteristic
81	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
82	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
83	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
84	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
85	M₁₆	—	2	4	no	—
86	M₁₆	—	4	2	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	—	4	2	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	#99	1	1	yes	characteristic
98	C₂×C₈	#99	1	1	yes	characteristic
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	2	1	no	—
108	C₁₆	—	4	2	no	—
109	C₁₆	—	8	2	no	—
110	C₁₆	—	8	1	no	—
111	C₂³	#336	1	1	yes	characteristic
112	Q₈	—	4	4	no	—
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#342	1	1	yes	characteristic
129	C₂×C₄	#344	1	1	yes	characteristic
130	C₂×C₄	#350	1	2	yes	—
131	C₂×C₄	#364	1	2	yes	—
132	C₈	—	2	1	no	—
133	C₈	—	4	4	no	—
134	C₈	—	4	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	4	1	no	—
137	C₈	—	8	2	no	—
138	C₈	#355	1	1	yes	characteristic
139	C₈	#356	1	1	yes	characteristic
140	C₂²	—	2	1	no	—
141	C₂²	—	2	1	no	—
142	C₂²	#1282	1	1	yes	characteristic
143	C₂²	#1376	1	1	yes	characteristic
144	C₂²	#1379	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	4	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	4	1	no	—
151	C₄	—	4	1	no	—
152	C₄	#1394	1	1	yes	characteristic
153	C₄	#1395	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#7202	1	1	yes	characteristic
157	C₂	#7206	1	1	yes	characteristic
158	C₂	#7207	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13264

C₂

yes

maximal, characteristic

#11493

C₂

yes

maximal, characteristic

C₈×#101

C₂

yes

maximal, characteristic

C₂×C₈×Q₈

C₂²

yes

characteristic

#2162

—

C₂×C₄⋊C₁₆

C₄

yes

characteristic

C₂×#346

C₄

yes

characteristic

C₄×#101

C₄

yes

characteristic

#1461

C₄

yes

characteristic

C₈×M₁₆

—

—

—

—

C₄

yes

—

#1283

—

C₂×C₄×Q₈

C₂×C₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

—

C₈×Q₈

—

C₈×Q₈

—

#431

—

C₂×C₄⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄⋊C₈

Q₈

yes

characteristic

C₂×#101

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

C₄×C₁₆

—

#343

C₈

yes

—

#324

—

#324

C₈

yes

—

#322

—

C₈²

—

C₂²×Q₈

M₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

C₂×C₈

yes

characteristic

C₂×C₄⋊C₄

C₂×C₈

yes

characteristic

C₂×C₄⋊C₄

M₁₆

yes

characteristic

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#95

yes

characteristic

C₂²×C₄

C₄×C₈

yes

characteristic

C₂²×C₄

#97

yes

characteristic

C₂²×C₄

C₂²⋊C₈

yes

characteristic

C₂²×C₄

C₂²⋊C₈

yes

characteristic

C₂²×C₄

C₄⋊C₈

yes

characteristic

C₂²×C₄

C₄⋊C₈

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₁₆

—

109

C₁₆

—

110

C₁₆

—

111

C₂³

#336

yes

characteristic

112

Q₈

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

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

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#342

yes

characteristic

129

C₂×C₄

#344

yes

characteristic

130

C₂×C₄

#350

yes

—

131

C₂×C₄

#364

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

#355

yes

characteristic

139

C₈

#356

yes

characteristic

140

C₂²

—

141

C₂²

—

142

C₂²

#1282

yes

characteristic

143

C₂²

#1376

yes

characteristic

144

C₂²

#1379

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

#1394

yes

characteristic

153

C₄

#1395

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#7202

yes

characteristic

157

C₂

#7206

yes

characteristic

158

C₂

#7207

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂×C₈×Q₈	C₂²
socle	C₂²	#1282
prime radical	C₂³	#336
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₈

#99

commutator

C₂²×C₄

C₄×C₈

Frattini, cosocle

C₂×C₈×Q₈

C₂²

C₂²

C₂³

G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₈×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

subgroup	index
#13264	2
#11493	2
C₈×#101	2

subgroup

index

#13264

#11493

C₈×#101

Subgroups — #93010

Conjugacy classes of subgroups

Maximal subgroups