Group #93439 — subgroups

Subgroups — #93439

Order 512.

234 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 86 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	#12879	C₂	1	1	yes	maximal, characteristic
2	#12591	C₂	1	1	yes	maximal, characteristic
3	#7842	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
5	C₂×#346	—	2	1	no	—
6	#1744	C₄	1	1	yes	characteristic
7	C₄×#97	C₄	1	1	yes	characteristic
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#322	—	2	1	no	—
10	C₂×#322	—	2	1	no	—
11	#1374	—	2	1	no	—
12	#1374	—	2	1	no	—
13	#1374	—	2	1	no	—
14	#1374	C₄	1	2	yes	—
15	C₂²×C₁₆	—	2	2	no	—
16	C₂²×C₁₆	—	2	2	no	—
17	C₂×#97	—	2	1	no	—
18	C₂×#97	C₈	1	2	yes	—
19	C₂×C₄×C₈	—	2	1	no	—
20	C₂×C₄×C₈	—	2	1	no	—
21	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
27	C₄³	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₁₆	—	4	2	no	—
29	#346	—	2	4	no	—
30	#336	—	2	1	no	—
31	#336	C₈	1	2	yes	—
32	#322	—	2	4	no	—
33	#322	—	2	4	no	—
34	C₂²×C₈	—	2	2	no	—
35	C₂²×C₈	—	2	2	no	—
36	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄²	C₄²	1	1	yes	characteristic
41	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄²	M₁₆	1	1	yes	characteristic
46	C₂×C₄²	M₁₆	1	1	yes	characteristic
47	C₂×C₁₆	—	2	4	no	—
48	C₂×C₁₆	—	4	2	no	—
49	C₂×C₁₆	—	4	2	no	—
50	C₂×C₁₆	—	4	2	no	—
51	C₂×C₁₆	—	4	2	no	—
52	C₂×C₁₆	—	4	2	no	—
53	#97	—	4	2	no	—
54	#97	—	4	2	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	1	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	2	no	—
64	C₄×C₈	—	2	2	no	—
65	C₄×C₈	—	4	1	no	—
66	C₂²×C₄	#95	1	1	yes	characteristic
67	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
68	C₂²×C₄	#97	1	1	yes	characteristic
69	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
70	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
71	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
72	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	4	2	no	—
83	C₂×C₈	—	4	2	no	—
84	C₂×C₈	—	4	2	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	#100	1	1	yes	characteristic
87	C₂×C₈	#101	1	1	yes	characteristic
88	C₂×C₈	#104	1	2	yes	—
89	C₂×C₈	#104	1	2	yes	—
90	C₂×C₈	#104	1	2	yes	—
91	C₂×C₈	#108	1	2	yes	—
92	C₄²	—	2	1	no	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	1	no	—
95	C₄²	—	2	1	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	4	1	no	—
102	C₁₆	—	4	4	no	—
103	C₁₆	—	8	2	no	—
104	C₂³	#336	1	1	yes	characteristic
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	1	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	#339	1	1	yes	characteristic
121	C₂×C₄	#339	1	1	yes	characteristic
122	C₂×C₄	#339	1	1	yes	characteristic
123	C₂×C₄	#341	1	1	yes	characteristic
124	C₂×C₄	#343	1	1	yes	characteristic
125	C₂×C₄	#344	1	1	yes	characteristic
126	C₂×C₄	#347	1	2	yes	—
127	C₂×C₄	#349	1	2	yes	—
128	C₂×C₄	#350	1	2	yes	—
129	C₈	—	2	2	no	—
130	C₈	—	2	2	no	—
131	C₈	—	4	4	no	—
132	C₈	—	4	4	no	—
133	C₈	—	4	1	no	—
134	C₈	#329	1	2	yes	—
135	C₈	#330	1	2	yes	—
136	C₂²	#1276	1	1	yes	characteristic
137	C₂²	#1278	1	1	yes	characteristic
138	C₂²	#1283	1	1	yes	characteristic
139	C₂²	#1375	1	1	yes	characteristic
140	C₂²	#1377	1	1	yes	characteristic
141	C₂²	#1378	1	1	yes	characteristic
142	C₂²	#1379	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	4	1	no	—
147	C₄	—	4	1	no	—
148	C₄	—	4	1	no	—
149	C₄	#1295	1	1	yes	characteristic
150	C₄	#1299	1	1	yes	characteristic
151	C₄	#1300	1	1	yes	characteristic
152	C₄	#1302	1	1	yes	characteristic
153	C₂	#7030	1	1	yes	characteristic
154	C₂	#7149	1	1	yes	characteristic
155	C₂	#7150	1	1	yes	characteristic
156	C₂	#7164	1	1	yes	characteristic
157	C₂	#7166	1	1	yes	characteristic
158	C₂	#7210	1	1	yes	characteristic
159	C₂	#7211	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12879

C₂

yes

maximal, characteristic

#12591

C₂

yes

maximal, characteristic

#7842

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×#346

—

#1744

C₄

yes

characteristic

C₄×#97

C₄

yes

characteristic

C₄²×C₈

C₂²

yes

—

—

—

—

—

C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#97

—

C₂×#97

C₈

yes

—

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

Q₈

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

#346

—

#336

—

#336

C₈

yes

—

#322

—

#322

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄²

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

C₂×C₈

yes

characteristic

C₂×C₄²

M₁₆

yes

characteristic

C₂×C₄²

M₁₆

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

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

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

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#104

yes

—

C₂×C₈

#104

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₁₆

—

103

C₁₆

—

104

C₂³

#336

yes

characteristic

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#339

yes

characteristic

121

C₂×C₄

#339

yes

characteristic

122

C₂×C₄

#339

yes

characteristic

123

C₂×C₄

#341

yes

characteristic

124

C₂×C₄

#343

yes

characteristic

125

C₂×C₄

#344

yes

characteristic

126

C₂×C₄

#347

yes

—

127

C₂×C₄

#349

yes

—

128

C₂×C₄

#350

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

#329

yes

—

135

C₈

#330

yes

—

136

C₂²

#1276

yes

characteristic

137

C₂²

#1278

yes

characteristic

138

C₂²

#1283

yes

characteristic

139

C₂²

#1375

yes

characteristic

140

C₂²

#1377

yes

characteristic

141

C₂²

#1378

yes

characteristic

142

C₂²

#1379

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#1295

yes

characteristic

150

C₄

#1299

yes

characteristic

151

C₄

#1300

yes

characteristic

152

C₄

#1302

yes

characteristic

153

C₂

#7030

yes

characteristic

154

C₂

#7149

yes

characteristic

155

C₂

#7150

yes

characteristic

156

C₂

#7164

yes

characteristic

157

C₂

#7166

yes

characteristic

158

C₂

#7210

yes

characteristic

159

C₂

#7211

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#336
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#95

commutator

C₂²×C₄

C₄×C₈

Frattini, cosocle

C₄²×C₈

C₂²

socle, prime radical

C₂³

#336

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#12879	2
#12591	2
#7842	2

subgroup

index

#12879

#12591

#7842

Subgroups — #93439

Conjugacy classes of subgroups

Maximal subgroups