Group #93407 — subgroups

Subgroups — #93407

Order 512.

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

normal subgroups only

This group has 236 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	#12590	C₂	1	1	yes	maximal, characteristic
2	#8274	C₂	1	1	yes	maximal, characteristic
3	C₄×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₁₆	—	2	1	no	—
5	#1767	—	2	1	no	—
6	C₄×C₄⋊C₈	—	2	1	no	—
7	C₄²×C₈	C₂²	1	1	yes	characteristic
8	C₂×#335	C₄	1	2	yes	—
9	C₂×#334	—	2	1	no	—
10	#1374	—	2	1	no	—
11	#1374	C₄	1	2	yes	—
12	#1277	—	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×#97	—	4	1	no	—
18	C₂×C₄×C₈	—	4	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	Q₈	1	1	yes	characteristic
25	C₄³	D₄	1	1	yes	characteristic
26	C₄×C₁₆	—	2	4	no	—
27	#336	—	4	1	no	—
28	#336	—	4	1	no	—
29	#335	—	2	2	no	—
30	#335	C₈	1	4	yes	complemented
31	#334	—	2	4	no	—
32	C₂²×C₈	—	4	2	no	—
33	C₂²×C₈	—	4	2	no	—
34	C₂²×C₈	C₄²	1	1	yes	characteristic
35	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
36	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
37	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
40	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄²	D₈	1	1	yes	characteristic
42	C₂×C₄²	SD₁₆	1	1	yes	characteristic
43	C₂×C₄²	SD₁₆	1	1	yes	characteristic
44	C₂×C₄²	Q₁₆	1	1	yes	characteristic
45	C₂×C₁₆	—	2	4	no	—
46	C₂×C₁₆	—	4	2	no	—
47	C₂×C₁₆	—	4	2	no	—
48	C₄⋊C₈	—	2	4	no	—
49	C₄⋊C₈	—	2	4	no	—
50	C₄⋊C₈	—	4	2	no	—
51	C₄⋊C₈	—	4	2	no	—
52	#97	—	8	2	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	—	2	2	no	—
55	C₄×C₈	—	2	2	no	—
56	C₄×C₈	—	4	1	no	—
57	C₄×C₈	—	4	1	no	—
58	C₄×C₈	—	8	2	no	—
59	C₄×C₈	C₂×C₈	1	2	yes	—
60	C₄×C₈	M₁₆	1	2	yes	—
61	C₂²×C₄	#95	1	1	yes	characteristic
62	C₂²×C₄	#102	1	1	yes	characteristic
63	C₂²×C₄	#102	1	1	yes	characteristic
64	C₂²×C₄	#103	1	1	yes	characteristic
65	C₂²×C₄	#103	1	1	yes	characteristic
66	C₂²×C₄	#106	1	1	yes	characteristic
67	C₂²×C₄	#107	1	1	yes	characteristic
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	2	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	2	1	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	—	4	4	no	—
78	C₂×C₈	—	8	2	no	—
79	C₂×C₈	—	8	2	no	—
80	C₂×C₈	—	8	2	no	—
81	C₂×C₈	—	8	2	no	—
82	C₂×C₈	C₄×C₈	1	1	yes	characteristic
83	C₂×C₈	#97	1	1	yes	characteristic
84	C₂×C₈	#104	1	2	yes	—
85	C₂×C₈	C₄⋊C₈	1	2	yes	—
86	C₂×C₈	#108	1	2	yes	—
87	C₄²	—	2	1	no	—
88	C₄²	—	2	1	no	—
89	C₄²	—	2	1	no	—
90	C₄²	—	2	1	no	—
91	C₄²	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	2	1	no	—
94	C₄²	—	2	1	no	—
95	C₄²	—	2	2	no	—
96	C₄²	C₂²⋊C₈	1	2	yes	—
97	C₄²	#104	1	2	yes	—
98	C₁₆	—	4	4	no	—
99	C₂³	#340	1	1	yes	characteristic
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
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	D₄⋊C₈	1	2	yes	—
113	C₂×C₄	Q₈⋊C₈	1	2	yes	—
114	C₂×C₄	#336	1	1	yes	characteristic
115	C₂×C₄	#339	1	1	yes	characteristic
116	C₂×C₄	#339	1	1	yes	characteristic
117	C₂×C₄	#341	1	1	yes	characteristic
118	C₂×C₄	#344	1	1	yes	characteristic
119	C₂×C₄	#344	1	1	yes	characteristic
120	C₂×C₄	#359	1	2	yes	—
121	C₂×C₄	#360	1	2	yes	—
122	C₂×C₄	#361	1	1	yes	characteristic
123	C₂×C₄	#362	1	1	yes	characteristic
124	C₂×C₄	#368	1	2	yes	—
125	C₈	—	2	1	no	—
126	C₈	—	2	1	no	—
127	C₈	—	2	2	no	—
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	8	4	no	—
131	C₈	—	8	4	no	—
132	C₂²	#1279	1	1	yes	characteristic
133	C₂²	#1301	1	1	yes	characteristic
134	C₂²	#1303	1	1	yes	characteristic
135	C₂²	#1381	1	1	yes	characteristic
136	C₂²	#1382	1	1	yes	characteristic
137	C₂²	#1386	1	1	yes	characteristic
138	C₂²	#1387	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	4	1	no	—
144	C₄	#1278	1	1	yes	characteristic
145	C₄	#1281	1	1	yes	characteristic
146	C₄	#1300	1	1	yes	characteristic
147	C₄	#1302	1	1	yes	characteristic
148	C₄	#1333	1	2	yes	—
149	C₄	#1334	1	2	yes	—
150	C₂	#7027	1	1	yes	characteristic
151	C₂	#7172	1	1	yes	characteristic
152	C₂	#7178	1	1	yes	characteristic
153	C₂	#7290	1	1	yes	characteristic
154	C₂	#7291	1	1	yes	characteristic
155	C₂	#7298	1	1	yes	characteristic
156	C₂	#7299	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12590

C₂

yes

maximal, characteristic

#8274

C₂

yes

maximal, characteristic

C₄×#334

C₂

yes

maximal, characteristic

C₂×C₄×C₁₆

—

#1767

—

C₄×C₄⋊C₈

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₄

yes

—

C₂×#334

—

#1374

—

#1374

C₄

yes

—

#1277

—

#1277

C₄

yes

—

C₂²×C₁₆

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

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

—

—

—

—

C₈

yes

complemented

#334

—

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

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

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

—

C₂×C₈

C₄×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#108

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²⋊C₈

yes

—

C₄²

#104

yes

—

C₁₆

—

C₂³

#340

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

D₄⋊C₈

yes

—

113

C₂×C₄

Q₈⋊C₈

yes

—

114

C₂×C₄

#336

yes

characteristic

115

C₂×C₄

#339

yes

characteristic

116

C₂×C₄

#339

yes

characteristic

117

C₂×C₄

#341

yes

characteristic

118

C₂×C₄

#344

yes

characteristic

119

C₂×C₄

#344

yes

characteristic

120

C₂×C₄

#359

yes

—

121

C₂×C₄

#360

yes

—

122

C₂×C₄

#361

yes

characteristic

123

C₂×C₄

#362

yes

characteristic

124

C₂×C₄

#368

yes

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

#1279

yes

characteristic

133

C₂²

#1301

yes

characteristic

134

C₂²

#1303

yes

characteristic

135

C₂²

#1381

yes

characteristic

136

C₂²

#1382

yes

characteristic

137

C₂²

#1386

yes

characteristic

138

C₂²

#1387

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#1278

yes

characteristic

145

C₄

#1281

yes

characteristic

146

C₄

#1300

yes

characteristic

147

C₄

#1302

yes

characteristic

148

C₄

#1333

yes

—

149

C₄

#1334

yes

—

150

C₂

#7027

yes

characteristic

151

C₂

#7172

yes

characteristic

152

C₂

#7178

yes

characteristic

153

C₂

#7290

yes

characteristic

154

C₂

#7291

yes

characteristic

155

C₂

#7298

yes

characteristic

156

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
30. #335	C₈	semidirect

H ≅ G/N

type

30. #335

C₈

semidirect

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

subgroup	index
#12590	2
#8274	2
C₄×#334	2

subgroup

index

#12590

#8274

C₄×#334

Subgroups — #93407

Conjugacy classes of subgroups

Complements

Maximal subgroups