Group #93455 — subgroups

Subgroups — #93455

Order 512.

204 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 98 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	#12900	C₂	1	1	yes	maximal, characteristic
2	C₄×#364	C₂	1	1	yes	maximal, characteristic
3	#12653	C₂	1	1	yes	maximal, characteristic
4	C₂×#364	—	2	1	no	—
5	C₂×#364	—	2	1	no	—
6	C₂×#364	C₄	1	2	yes	—
7	#2150	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄⋊C₁₆	C₄	1	1	yes	characteristic
10	C₄×M₃₂	—	2	1	no	—
11	C₂×#346	—	2	1	no	—
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	#1376	C₄	1	2	yes	—
14	#1374	—	2	1	no	—
15	C₂×M₃₂	—	2	2	no	—
16	C₂×M₃₂	—	2	2	no	—
17	C₂²×C₁₆	—	2	1	no	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
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	C₂×C₄×C₈	D₄	1	1	yes	characteristic
25	C₄³	Q₈	1	1	yes	characteristic
26	#364	—	2	2	no	—
27	#364	—	2	2	no	—
28	#364	—	4	2	no	—
29	#364	—	4	1	no	—
30	#364	—	4	1	no	—
31	C₄⋊C₁₆	—	2	1	no	—
32	C₄⋊C₁₆	—	2	1	no	—
33	C₄⋊C₁₆	—	4	1	no	—
34	C₄⋊C₁₆	—	4	1	no	—
35	C₄⋊C₁₆	C₈	1	2	yes	—
36	C₄⋊C₁₆	C₈	1	2	yes	—
37	#346	—	4	1	no	—
38	#346	—	4	1	no	—
39	#346	—	4	1	no	—
40	C₄×C₁₆	—	4	1	no	—
41	C₂²×C₈	C₄²	1	1	yes	characteristic
42	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄²	—	2	1	no	—
46	C₂×C₄²	—	2	1	no	—
47	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
50	M₃₂	—	2	4	no	—
51	M₃₂	—	4	2	no	—
52	M₃₂	—	4	2	no	—
53	M₃₂	—	4	2	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	2	2	no	—
56	C₂×C₁₆	—	4	2	no	—
57	C₂×C₁₆	—	4	1	no	—
58	C₂×C₁₆	—	4	1	no	—
59	C₂×C₁₆	—	4	1	no	—
60	C₂×C₁₆	—	4	1	no	—
61	C₂×C₁₆	—	4	2	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	4	1	no	—
67	C₄×C₈	C₂×C₈	1	1	yes	characteristic
68	C₄×C₈	C₂×C₈	1	1	yes	characteristic
69	C₄×C₈	M₁₆	1	1	yes	characteristic
70	C₄×C₈	M₁₆	1	1	yes	characteristic
71	C₄×C₈	D₈	1	1	yes	characteristic
72	C₄×C₈	D₈	1	1	yes	characteristic
73	C₄×C₈	SD₁₆	1	1	yes	characteristic
74	C₄×C₈	SD₁₆	1	1	yes	characteristic
75	C₄×C₈	SD₁₆	1	1	yes	characteristic
76	C₄×C₈	SD₁₆	1	1	yes	characteristic
77	C₄×C₈	Q₁₆	1	1	yes	characteristic
78	C₄×C₈	Q₁₆	1	1	yes	characteristic
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	#95	1	1	yes	characteristic
82	C₂²×C₄	#108	1	1	yes	characteristic
83	C₂²×C₄	#108	1	1	yes	characteristic
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	C₄×C₈	1	1	yes	characteristic
89	C₂×C₈	#97	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
92	C₂×C₈	#100	1	1	yes	characteristic
93	C₂×C₈	#101	1	1	yes	characteristic
94	C₂×C₈	#102	1	1	yes	characteristic
95	C₂×C₈	#102	1	1	yes	characteristic
96	C₂×C₈	#102	1	1	yes	characteristic
97	C₂×C₈	#102	1	1	yes	characteristic
98	C₂×C₈	#103	1	1	yes	characteristic
99	C₂×C₈	#103	1	1	yes	characteristic
100	C₂×C₈	#103	1	1	yes	characteristic
101	C₂×C₈	#103	1	1	yes	characteristic
102	C₂×C₈	#104	1	1	yes	characteristic
103	C₂×C₈	#104	1	1	yes	characteristic
104	C₄²	—	2	1	no	—
105	C₄²	—	2	1	no	—
106	C₄²	—	2	1	no	—
107	C₄²	—	4	2	no	—
108	C₄²	—	4	2	no	—
109	C₄²	C₄⋊C₈	1	1	yes	characteristic
110	C₄²	C₄⋊C₈	1	1	yes	characteristic
111	C₄²	#106	1	1	yes	characteristic
112	C₄²	#106	1	1	yes	characteristic
113	C₄²	#107	1	1	yes	characteristic
114	C₄²	#107	1	1	yes	characteristic
115	C₁₆	—	4	4	no	—
116	C₁₆	—	4	2	no	—
117	C₁₆	—	4	2	no	—
118	C₁₆	—	8	2	no	—
119	C₂³	#341	1	1	yes	characteristic
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	#334	1	1	yes	characteristic
130	C₂×C₄	#334	1	1	yes	characteristic
131	C₂×C₄	#335	1	1	yes	characteristic
132	C₂×C₄	#335	1	1	yes	characteristic
133	C₂×C₄	#336	1	1	yes	characteristic
134	C₂×C₄	#339	1	1	yes	characteristic
135	C₂×C₄	#340	1	1	yes	characteristic
136	C₂×C₄	#340	1	1	yes	characteristic
137	C₂×C₄	#343	1	1	yes	characteristic
138	C₂×C₄	#344	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	D₄⋊C₈	1	1	yes	characteristic
144	C₈	D₄⋊C₈	1	1	yes	characteristic
145	C₈	Q₈⋊C₈	1	1	yes	characteristic
146	C₈	Q₈⋊C₈	1	1	yes	characteristic
147	C₈	#331	1	1	yes	characteristic
148	C₈	#332	1	1	yes	characteristic
149	C₈	#332	1	1	yes	characteristic
150	C₈	#333	1	1	yes	characteristic
151	C₂²	—	2	1	no	—
152	C₂²	—	2	1	no	—
153	C₂²	#1277	1	1	yes	characteristic
154	C₂²	#1299	1	1	yes	characteristic
155	C₂²	#1301	1	1	yes	characteristic
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	4	1	no	—
161	C₄	—	4	1	no	—
162	C₄	#1279	1	1	yes	characteristic
163	C₄	#1283	1	1	yes	characteristic
164	C₄	#1296	1	1	yes	characteristic
165	C₄	#1303	1	1	yes	characteristic
166	C₄	#1369	1	2	yes	—
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#7031	1	1	yes	characteristic
170	C₂	#7161	1	1	yes	characteristic
171	C₂	#7162	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12900

C₂

yes

maximal, characteristic

C₄×#364

C₂

yes

maximal, characteristic

#12653

C₂

yes

maximal, characteristic

C₂×#364

—

C₂×#364

—

C₂×#364

C₄

yes

—

#2150

—

C₂×C₄⋊C₁₆

C₄

yes

characteristic

C₂×C₄⋊C₁₆

C₄

yes

characteristic

C₄×M₃₂

—

C₂×#346

—

C₄²×C₈

C₂²

yes

characteristic

#1376

C₄

yes

—

#1374

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

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

C₄³

Q₈

yes

—

—

—

—

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₈

yes

—

C₄⋊C₁₆

C₈

yes

—

#346

—

#346

—

#346

—

C₄×C₁₆

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

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

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

M₁₆

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

#97

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

100

C₂×C₈

#103

yes

characteristic

101

C₂×C₈

#103

yes

characteristic

102

C₂×C₈

#104

yes

characteristic

103

C₂×C₈

#104

yes

characteristic

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

C₄⋊C₈

yes

characteristic

110

C₄²

C₄⋊C₈

yes

characteristic

111

C₄²

#106

yes

characteristic

112

C₄²

#106

yes

characteristic

113

C₄²

#107

yes

characteristic

114

C₄²

#107

yes

characteristic

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

118

C₁₆

—

119

C₂³

#341

yes

characteristic

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

—

129

C₂×C₄

#334

yes

characteristic

130

C₂×C₄

#334

yes

characteristic

131

C₂×C₄

#335

yes

characteristic

132

C₂×C₄

#335

yes

characteristic

133

C₂×C₄

#336

yes

characteristic

134

C₂×C₄

#339

yes

characteristic

135

C₂×C₄

#340

yes

characteristic

136

C₂×C₄

#340

yes

characteristic

137

C₂×C₄

#343

yes

characteristic

138

C₂×C₄

#344

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

D₄⋊C₈

yes

characteristic

144

C₈

D₄⋊C₈

yes

characteristic

145

C₈

Q₈⋊C₈

yes

characteristic

146

C₈

Q₈⋊C₈

yes

characteristic

147

C₈

#331

yes

characteristic

148

C₈

#332

yes

characteristic

149

C₈

#332

yes

characteristic

150

C₈

#333

yes

characteristic

151

C₂²

—

152

C₂²

—

153

C₂²

#1277

yes

characteristic

154

C₂²

#1299

yes

characteristic

155

C₂²

#1301

yes

characteristic

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

#1279

yes

characteristic

163

C₄

#1283

yes

characteristic

164

C₄

#1296

yes

characteristic

165

C₄

#1303

yes

characteristic

166

C₄

#1369

yes

—

167

C₂

—

168

C₂

—

169

C₂

#7031

yes

characteristic

170

C₂

#7161

yes

characteristic

171

C₂

#7162

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#340
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1277
prime radical	C₂³	#341
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#340

commutator

C₂×C₈

C₄×C₈

Frattini, cosocle

C₄²×C₈

C₂²

C₂²

C₂³

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ 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
#12900	2
C₄×#364	2
#12653	2

subgroup

index

#12900

C₄×#364

#12653

Subgroups — #93455

Conjugacy classes of subgroups

Maximal subgroups