Group #93403 — subgroups

Subgroups — #93403

Order 512.

232 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 90 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	#12653	C₂	1	1	yes	maximal, characteristic
2	#12591	C₂	1	1	yes	maximal, characteristic
3	#8261	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₁₆	—	2	1	no	—
5	C₂×#346	—	2	1	no	—
6	C₂×#346	—	2	1	no	—
7	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
8	C₄×C₄⋊C₈	C₄	1	1	yes	characteristic
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₂×#322	—	2	1	no	—
11	#1374	—	2	1	no	—
12	#1374	C₄	1	2	yes	—
13	#1374	C₄	1	2	yes	—
14	#1273	—	2	1	no	—
15	C₂²×C₁₆	—	2	2	no	—
16	C₂²×C₁₆	—	2	2	no	—
17	C₂×C₄⋊C₈	—	2	1	no	—
18	C₂×C₄⋊C₈	—	2	1	no	—
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	—	2	4	no	—
29	#346	—	2	2	no	—
30	#346	—	2	2	no	—
31	#346	—	4	1	no	—
32	#346	—	4	1	no	—
33	#336	C₈	1	2	yes	—
34	#336	C₈	1	2	yes	—
35	#322	—	4	1	no	—
36	#322	—	4	1	no	—
37	C₂²×C₈	—	2	2	no	—
38	C₂²×C₈	—	2	2	no	—
39	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
41	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄²	C₄²	1	1	yes	characteristic
44	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄²	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄²	M₁₆	1	1	yes	characteristic
49	C₂×C₄²	M₁₆	1	1	yes	characteristic
50	C₂×C₁₆	—	2	4	no	—
51	C₂×C₁₆	—	2	4	no	—
52	C₂×C₁₆	—	4	2	no	—
53	C₂×C₁₆	—	4	2	no	—
54	C₂×C₁₆	—	4	2	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₄⋊C₈	—	2	4	no	—
57	C₄⋊C₈	—	4	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	2	1	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	4	1	no	—
63	C₄×C₈	—	4	1	no	—
64	C₄×C₈	—	4	1	no	—
65	C₄×C₈	—	4	1	no	—
66	C₄×C₈	—	4	1	no	—
67	C₄×C₈	—	4	1	no	—
68	C₄×C₈	D₈	1	1	yes	characteristic
69	C₄×C₈	SD₁₆	1	1	yes	characteristic
70	C₄×C₈	SD₁₆	1	1	yes	characteristic
71	C₄×C₈	Q₁₆	1	1	yes	characteristic
72	C₂²×C₄	#95	1	1	yes	characteristic
73	C₂²×C₄	C₄×C₈	1	1	yes	characteristic
74	C₂²×C₄	#97	1	1	yes	characteristic
75	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
76	C₂²×C₄	C₂²⋊C₈	1	1	yes	characteristic
77	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
78	C₂²×C₄	C₄⋊C₈	1	1	yes	characteristic
79	C₂×C₈	—	2	1	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	4	no	—
88	C₂×C₈	—	4	2	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	—	4	2	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	—	4	2	no	—
93	C₂×C₈	#102	1	1	yes	characteristic
94	C₂×C₈	#103	1	1	yes	characteristic
95	C₂×C₈	#104	1	2	yes	—
96	C₂×C₈	#106	1	1	yes	characteristic
97	C₂×C₈	#107	1	1	yes	characteristic
98	C₂×C₈	#108	1	2	yes	—
99	C₄²	—	2	1	no	—
100	C₄²	—	2	1	no	—
101	C₄²	—	2	1	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	4	1	no	—
106	C₄²	—	4	1	no	—
107	C₄²	#102	1	1	yes	characteristic
108	C₄²	#103	1	1	yes	characteristic
109	C₄²	#104	1	1	yes	characteristic
110	C₄²	#104	1	1	yes	characteristic
111	C₁₆	—	4	4	no	—
112	C₁₆	—	4	4	no	—
113	C₂³	#336	1	1	yes	characteristic
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	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
129	C₂×C₄	D₄⋊C₈	1	1	yes	characteristic
130	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
131	C₂×C₄	Q₈⋊C₈	1	1	yes	characteristic
132	C₂×C₄	#339	1	1	yes	characteristic
133	C₂×C₄	#339	1	1	yes	characteristic
134	C₂×C₄	#340	1	1	yes	characteristic
135	C₂×C₄	#341	1	1	yes	characteristic
136	C₂×C₄	#344	1	1	yes	characteristic
137	C₂×C₄	#344	1	1	yes	characteristic
138	C₂×C₄	#347	1	2	yes	—
139	C₂×C₄	#364	1	2	yes	—
140	C₈	—	2	2	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	1	no	—
145	C₈	—	2	1	no	—
146	C₈	—	4	4	no	—
147	C₈	—	8	2	no	—
148	C₂²	#1278	1	1	yes	characteristic
149	C₂²	#1279	1	1	yes	characteristic
150	C₂²	#1281	1	1	yes	characteristic
151	C₂²	#1375	1	1	yes	characteristic
152	C₂²	#1376	1	1	yes	characteristic
153	C₂²	#1378	1	1	yes	characteristic
154	C₂²	#1378	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	2	2	no	—
160	C₄	—	2	2	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	1	no	—
163	C₄	#1300	1	1	yes	characteristic
164	C₄	#1301	1	1	yes	characteristic
165	C₄	#1302	1	1	yes	characteristic
166	C₄	#1303	1	1	yes	characteristic
167	C₂	#7027	1	1	yes	characteristic
168	C₂	#7164	1	1	yes	characteristic
169	C₂	#7165	1	1	yes	characteristic
170	C₂	#7168	1	1	yes	characteristic
171	C₂	#7169	1	1	yes	characteristic
172	C₂	#7196	1	1	yes	characteristic
173	C₂	#7197	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12653

C₂

yes

maximal, characteristic

#12591

C₂

yes

maximal, characteristic

#8261

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

—

C₂×#346

—

C₂×#346

—

C₄×C₄⋊C₈

C₄

yes

characteristic

C₄×C₄⋊C₈

C₄

yes

characteristic

C₄²×C₈

C₂²

yes

—

—

C₄

yes

—

#1374

C₄

yes

—

#1273

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

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

—

—

—

—

—

C₈

yes

—

#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

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

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

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

#102

yes

characteristic

108

C₄²

#103

yes

characteristic

109

C₄²

#104

yes

characteristic

110

C₄²

#104

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₂³

#336

yes

characteristic

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

D₄⋊C₈

yes

characteristic

129

C₂×C₄

D₄⋊C₈

yes

characteristic

130

C₂×C₄

Q₈⋊C₈

yes

characteristic

131

C₂×C₄

Q₈⋊C₈

yes

characteristic

132

C₂×C₄

#339

yes

characteristic

133

C₂×C₄

#339

yes

characteristic

134

C₂×C₄

#340

yes

characteristic

135

C₂×C₄

#341

yes

characteristic

136

C₂×C₄

#344

yes

characteristic

137

C₂×C₄

#344

yes

characteristic

138

C₂×C₄

#347

yes

—

139

C₂×C₄

#364

yes

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₂²

#1278

yes

characteristic

149

C₂²

#1279

yes

characteristic

150

C₂²

#1281

yes

characteristic

151

C₂²

#1375

yes

characteristic

152

C₂²

#1376

yes

characteristic

153

C₂²

#1378

yes

characteristic

154

C₂²

#1378

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#1300

yes

characteristic

164

C₄

#1301

yes

characteristic

165

C₄

#1302

yes

characteristic

166

C₄

#1303

yes

characteristic

167

C₂

#7027

yes

characteristic

168

C₂

#7164

yes

characteristic

169

C₂

#7165

yes

characteristic

170

C₂

#7168

yes

characteristic

171

C₂

#7169

yes

characteristic

172

C₂

#7196

yes

characteristic

173

C₂

#7197

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
#12653	2
#12591	2
#8261	2

subgroup

index

#12653

#12591

#8261

Subgroups — #93403

Conjugacy classes of subgroups

Maximal subgroups