Group #93451 — subgroups

Subgroups — #93451

Order 512.

238 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₄×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#8807	C₂	1	1	yes	maximal, characteristic
3	C₄×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₁₆	C₄	1	2	yes	—
5	C₂×C₄×C₁₆	—	2	1	no	—
6	#1831	—	2	1	no	—
7	C₄×C₄⋊C₈	—	2	1	no	—
8	C₄²×C₈	C₂²	1	1	yes	characteristic
9	C₂×#335	C₄	1	2	yes	—
10	C₂×#334	—	2	1	no	—
11	C₂×#334	—	2	1	no	—
12	#1374	—	2	1	no	—
13	#1277	C₄	1	2	yes	—
14	C₂²×C₁₆	—	2	2	no	—
15	C₂×C₄⋊C₈	—	2	2	no	—
16	C₂×C₄⋊C₈	—	2	2	no	—
17	C₂×C₄×C₈	—	4	1	no	—
18	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₈	D₄	1	1	yes	characteristic
22	C₂×C₄×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
24	C₄³	D₄	1	1	yes	characteristic
25	C₄⋊C₁₆	—	2	2	no	—
26	C₄⋊C₁₆	C₈	1	4	yes	complemented
27	C₄×C₁₆	—	4	1	no	—
28	C₄×C₁₆	—	4	1	no	—
29	#336	—	4	1	no	—
30	#336	—	4	1	no	—
31	#336	—	4	1	no	—
32	#335	—	4	2	no	—
33	#334	—	2	4	no	—
34	#334	—	4	1	no	—
35	#334	—	4	1	no	—
36	C₂²×C₈	—	4	2	no	—
37	C₂²×C₈	—	4	2	no	—
38	C₂²×C₈	C₄²	1	1	yes	characteristic
39	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄²	D₈	1	1	yes	characteristic
46	C₂×C₄²	SD₁₆	1	1	yes	characteristic
47	C₂×C₄²	SD₁₆	1	1	yes	characteristic
48	C₂×C₄²	Q₁₆	1	1	yes	characteristic
49	C₂×C₁₆	—	2	4	no	—
50	C₂×C₁₆	—	4	2	no	—
51	C₂×C₁₆	—	4	2	no	—
52	C₄⋊C₈	—	2	4	no	—
53	C₄⋊C₈	—	4	2	no	—
54	C₄⋊C₈	—	4	2	no	—
55	C₄⋊C₈	—	4	2	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	—	2	1	no	—
59	C₄×C₈	—	2	2	no	—
60	C₄×C₈	—	4	1	no	—
61	C₄×C₈	—	8	2	no	—
62	C₄×C₈	C₂×C₈	1	2	yes	—
63	C₄×C₈	M₁₆	1	2	yes	—
64	C₄×C₈	D₈	1	1	yes	characteristic
65	C₄×C₈	SD₁₆	1	1	yes	characteristic
66	C₄×C₈	SD₁₆	1	1	yes	characteristic
67	C₄×C₈	Q₁₆	1	1	yes	characteristic
68	C₂²×C₄	#95	1	1	yes	characteristic
69	C₂²×C₄	#102	1	1	yes	characteristic
70	C₂²×C₄	#102	1	1	yes	characteristic
71	C₂²×C₄	#103	1	1	yes	characteristic
72	C₂²×C₄	#103	1	1	yes	characteristic
73	C₂²×C₄	#106	1	1	yes	characteristic
74	C₂²×C₄	#107	1	1	yes	characteristic
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	4	4	no	—
83	C₂×C₈	—	8	2	no	—
84	C₂×C₈	—	8	2	no	—
85	C₂×C₈	—	8	2	no	—
86	C₂×C₈	—	8	2	no	—
87	C₂×C₈	—	8	2	no	—
88	C₂×C₈	C₄×C₈	1	1	yes	characteristic
89	C₂×C₈	#97	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊C₈	1	2	yes	—
91	C₂×C₈	#102	1	1	yes	characteristic
92	C₂×C₈	#103	1	1	yes	characteristic
93	C₂×C₈	#104	1	2	yes	—
94	C₂×C₈	#106	1	1	yes	characteristic
95	C₂×C₈	#107	1	1	yes	characteristic
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	C₄²	#100	1	1	yes	characteristic
103	C₄²	#101	1	1	yes	characteristic
104	C₄²	#102	1	1	yes	characteristic
105	C₄²	#103	1	1	yes	characteristic
106	C₄²	C₄⋊C₈	1	2	yes	—
107	C₄²	#108	1	2	yes	—
108	C₁₆	—	4	4	no	—
109	C₂³	#340	1	1	yes	characteristic
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	—	2	1	no	—
121	C₂×C₄	#331	1	1	yes	characteristic
122	C₂×C₄	#332	1	1	yes	characteristic
123	C₂×C₄	#332	1	1	yes	characteristic
124	C₂×C₄	#333	1	1	yes	characteristic
125	C₂×C₄	#334	1	2	yes	—
126	C₂×C₄	#335	1	2	yes	—
127	C₂×C₄	#336	1	1	yes	characteristic
128	C₂×C₄	#339	1	1	yes	characteristic
129	C₂×C₄	#340	1	1	yes	characteristic
130	C₂×C₄	#341	1	1	yes	characteristic
131	C₂×C₄	#343	1	1	yes	characteristic
132	C₂×C₄	#344	1	1	yes	characteristic
133	C₂×C₄	#359	1	2	yes	—
134	C₂×C₄	#359	1	2	yes	—
135	C₂×C₄	#360	1	2	yes	—
136	C₈	—	2	1	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	2	1	no	—
140	C₈	—	4	1	no	—
141	C₈	—	8	4	no	—
142	C₈	—	16	2	no	—
143	C₈	D₄⋊C₈	1	2	yes	—
144	C₈	Q₈⋊C₈	1	2	yes	—
145	C₂²	#1277	1	1	yes	characteristic
146	C₂²	#1296	1	1	yes	characteristic
147	C₂²	#1303	1	1	yes	characteristic
148	C₂²	#1382	1	1	yes	characteristic
149	C₂²	#1382	1	1	yes	characteristic
150	C₂²	#1386	1	1	yes	characteristic
151	C₂²	#1388	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	2	2	no	—
157	C₄	—	4	1	no	—
158	C₄	#1279	1	1	yes	characteristic
159	C₄	#1283	1	1	yes	characteristic
160	C₄	#1299	1	1	yes	characteristic
161	C₄	#1301	1	1	yes	characteristic
162	C₄	#1360	1	2	yes	—
163	C₄	#1363	1	2	yes	—
164	C₂	#7031	1	1	yes	characteristic
165	C₂	#7155	1	1	yes	characteristic
166	C₂	#7158	1	1	yes	characteristic
167	C₂	#7275	1	1	yes	characteristic
168	C₂	#7277	1	1	yes	characteristic
169	C₂	#7297	1	1	yes	characteristic
170	C₂	#7299	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#8807

C₂

yes

maximal, characteristic

C₄×#334

C₂

yes

maximal, characteristic

C₂×C₄⋊C₁₆

C₄

yes

—

C₂×C₄×C₁₆

—

#1831

—

C₄×C₄⋊C₈

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₄

yes

—

—

—

—

C₄

yes

—

C₂²×C₁₆

—

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

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

characteristic

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

C₈

yes

complemented

C₄×C₁₆

—

C₄×C₁₆

—

—

—

—

—

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₄²

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

SD₁₆

yes

characteristic

C₂×C₄²

Q₁₆

yes

characteristic

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

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

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

C₄×C₈

yes

characteristic

C₂×C₈

#97

yes

characteristic

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

#100

yes

characteristic

103

C₄²

#101

yes

characteristic

104

C₄²

#102

yes

characteristic

105

C₄²

#103

yes

characteristic

106

C₄²

C₄⋊C₈

yes

—

107

C₄²

#108

yes

—

108

C₁₆

—

109

C₂³

#340

yes

characteristic

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

—

121

C₂×C₄

#331

yes

characteristic

122

C₂×C₄

#332

yes

characteristic

123

C₂×C₄

#332

yes

characteristic

124

C₂×C₄

#333

yes

characteristic

125

C₂×C₄

#334

yes

—

126

C₂×C₄

#335

yes

—

127

C₂×C₄

#336

yes

characteristic

128

C₂×C₄

#339

yes

characteristic

129

C₂×C₄

#340

yes

characteristic

130

C₂×C₄

#341

yes

characteristic

131

C₂×C₄

#343

yes

characteristic

132

C₂×C₄

#344

yes

characteristic

133

C₂×C₄

#359

yes

—

134

C₂×C₄

#359

yes

—

135

C₂×C₄

#360

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

D₄⋊C₈

yes

—

144

C₈

Q₈⋊C₈

yes

—

145

C₂²

#1277

yes

characteristic

146

C₂²

#1296

yes

characteristic

147

C₂²

#1303

yes

characteristic

148

C₂²

#1382

yes

characteristic

149

C₂²

#1382

yes

characteristic

150

C₂²

#1386

yes

characteristic

151

C₂²

#1388

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#1279

yes

characteristic

159

C₄

#1283

yes

characteristic

160

C₄

#1299

yes

characteristic

161

C₄

#1301

yes

characteristic

162

C₄

#1360

yes

—

163

C₄

#1363

yes

—

164

C₂

#7031

yes

characteristic

165

C₂

#7155

yes

characteristic

166

C₂

#7158

yes

characteristic

167

C₂

#7275

yes

characteristic

168

C₂

#7277

yes

characteristic

169

C₂

#7297

yes

characteristic

170

C₂

#7299

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₂×C₈	C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#340
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#102

commutator

C₂×C₈

C₄×C₈

Frattini, cosocle

C₄²×C₈

C₂²

socle, prime radical

C₂³

#340

LCS

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
26. C₄⋊C₁₆	C₈	semidirect

H ≅ G/N

type

26. C₄⋊C₁₆

C₈

semidirect

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

subgroup	index
C₄×C₄⋊C₁₆	2
#8807	2
C₄×#334	2

subgroup

index

C₄×C₄⋊C₁₆

#8807

C₄×#334

Subgroups — #93451

Conjugacy classes of subgroups

Complements

Maximal subgroups