Group #93270 — subgroups

Subgroups — #93270

Order 512.

272 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 134 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×#346	C₂	1	1	yes	maximal, characteristic
2	C₄×#335	C₂	1	1	yes	maximal, characteristic
3	C₄×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×#346	—	2	1	no	—
5	C₂×#346	C₄	1	2	yes	—
6	C₂×C₄×C₁₆	—	2	1	no	—
7	C₄×C₄⋊C₈	—	2	1	no	—
8	C₄×C₄⋊C₈	—	2	1	no	—
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	C₂×#335	C₄	1	2	yes	—
11	C₂×#334	C₄	1	2	yes	—
12	#1277	—	2	1	no	—
13	#1277	—	2	1	no	—
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	—	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	#346	—	2	4	no	—
27	#346	—	2	2	no	—
28	#346	C₈	1	4	yes	complemented
29	C₄×C₁₆	—	2	4	no	—
30	#336	—	4	1	no	—
31	#336	—	4	1	no	—
32	#335	—	2	2	no	—
33	#335	C₈	1	4	yes	complemented
34	#334	—	2	2	no	—
35	#334	C₈	1	4	yes	complemented
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	—	2	4	no	—
51	C₂×C₁₆	—	2	4	no	—
52	C₄⋊C₈	—	2	4	no	—
53	C₄⋊C₈	—	2	4	no	—
54	C₄⋊C₈	—	4	2	no	—
55	C₄⋊C₈	—	4	2	no	—
56	C₄×C₈	—	2	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₄×C₈	—	8	2	no	—
60	C₄×C₈	—	8	2	no	—
61	C₄×C₈	C₂×C₈	1	2	yes	—
62	C₄×C₈	C₂×C₈	1	2	yes	—
63	C₄×C₈	C₂×C₈	1	2	yes	—
64	C₄×C₈	M₁₆	1	2	yes	—
65	C₄×C₈	M₁₆	1	2	yes	—
66	C₄×C₈	M₁₆	1	2	yes	—
67	C₂²×C₄	#95	1	1	yes	characteristic
68	C₂²×C₄	#102	1	1	yes	characteristic
69	C₂²×C₄	#102	1	1	yes	characteristic
70	C₂²×C₄	#103	1	1	yes	characteristic
71	C₂²×C₄	#103	1	1	yes	characteristic
72	C₂²×C₄	#106	1	1	yes	characteristic
73	C₂²×C₄	#107	1	1	yes	characteristic
74	C₂×C₈	—	2	1	no	—
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	—	4	4	no	—
81	C₂×C₈	—	4	4	no	—
82	C₂×C₈	—	8	2	no	—
83	C₂×C₈	—	8	2	no	—
84	C₂×C₈	—	8	2	no	—
85	C₂×C₈	—	8	2	no	—
86	C₂×C₈	C₄×C₈	1	1	yes	characteristic
87	C₂×C₈	C₄×C₈	1	1	yes	characteristic
88	C₂×C₈	C₄×C₈	1	1	yes	characteristic
89	C₂×C₈	#97	1	1	yes	characteristic
90	C₂×C₈	#97	1	1	yes	characteristic
91	C₂×C₈	#97	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊C₈	1	2	yes	—
93	C₂×C₈	C₄⋊C₈	1	2	yes	—
94	C₂×C₈	C₄⋊C₈	1	2	yes	—
95	C₄²	—	2	2	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	4	1	no	—
99	C₄²	—	4	1	no	—
100	C₄²	C₂²⋊C₈	1	2	yes	—
101	C₄²	C₂²⋊C₈	1	2	yes	—
102	C₄²	#104	1	2	yes	—
103	C₄²	#104	1	2	yes	—
104	C₄²	C₄⋊C₈	1	2	yes	—
105	C₄²	#108	1	2	yes	—
106	C₁₆	—	2	8	no	—
107	C₂³	#340	1	1	yes	characteristic
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	D₄⋊C₈	1	2	yes	—
119	C₂×C₄	D₄⋊C₈	1	2	yes	—
120	C₂×C₄	Q₈⋊C₈	1	2	yes	—
121	C₂×C₄	Q₈⋊C₈	1	2	yes	—
122	C₂×C₄	#334	1	2	yes	—
123	C₂×C₄	#335	1	2	yes	—
124	C₂×C₄	#336	1	1	yes	characteristic
125	C₂×C₄	#336	1	1	yes	characteristic
126	C₂×C₄	#336	1	1	yes	characteristic
127	C₂×C₄	#339	1	1	yes	characteristic
128	C₂×C₄	#339	1	1	yes	characteristic
129	C₂×C₄	#341	1	1	yes	characteristic
130	C₂×C₄	#360	1	2	yes	—
131	C₂×C₄	#361	1	1	yes	characteristic
132	C₂×C₄	#362	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	2	no	—
135	C₈	—	2	2	no	—
136	C₈	—	8	4	no	—
137	C₈	—	8	4	no	—
138	C₈	C₈²	1	1	yes	characteristic
139	C₈	#322	1	1	yes	characteristic
140	C₈	#322	1	1	yes	characteristic
141	C₈	#322	1	1	yes	characteristic
142	C₂²	#1277	1	1	yes	characteristic
143	C₂²	#1279	1	1	yes	characteristic
144	C₂²	#1279	1	1	yes	characteristic
145	C₂²	#1386	1	1	yes	characteristic
146	C₂²	#1387	1	1	yes	characteristic
147	C₂²	#1388	1	1	yes	characteristic
148	C₂²	#1389	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	#1273	1	1	yes	characteristic
155	C₄	#1276	1	1	yes	characteristic
156	C₄	#1278	1	1	yes	characteristic
157	C₄	#1278	1	1	yes	characteristic
158	C₄	#1333	1	2	yes	—
159	C₄	#1334	1	2	yes	—
160	C₂	#7015	1	1	yes	characteristic
161	C₂	#7158	1	1	yes	characteristic
162	C₂	#7159	1	1	yes	characteristic
163	C₂	#7175	1	1	yes	characteristic
164	C₂	#7176	1	1	yes	characteristic
165	C₂	#7177	1	1	yes	characteristic
166	C₂	#7178	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#346

C₂

yes

maximal, characteristic

C₄×#335

C₂

yes

maximal, characteristic

C₄×#334

C₂

yes

maximal, characteristic

C₂×#346

—

C₂×#346

C₄

yes

—

C₂×C₄×C₁₆

—

C₄×C₄⋊C₈

—

C₄×C₄⋊C₈

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₄

yes

—

C₂×#334

C₄

yes

—

#1277

—

#1277

—

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

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

—

—

C₈

yes

complemented

C₄×C₁₆

—

—

—

—

C₈

yes

complemented

#334

—

#334

C₈

yes

complemented

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

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

M₁₆

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

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

characteristic

C₂×C₈

C₄×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂²⋊C₈

yes

—

101

C₄²

C₂²⋊C₈

yes

—

102

C₄²

#104

yes

—

103

C₄²

#104

yes

—

104

C₄²

C₄⋊C₈

yes

—

105

C₄²

#108

yes

—

106

C₁₆

—

107

C₂³

#340

yes

characteristic

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

D₄⋊C₈

yes

—

119

C₂×C₄

D₄⋊C₈

yes

—

120

C₂×C₄

Q₈⋊C₈

yes

—

121

C₂×C₄

Q₈⋊C₈

yes

—

122

C₂×C₄

#334

yes

—

123

C₂×C₄

#335

yes

—

124

C₂×C₄

#336

yes

characteristic

125

C₂×C₄

#336

yes

characteristic

126

C₂×C₄

#336

yes

characteristic

127

C₂×C₄

#339

yes

characteristic

128

C₂×C₄

#339

yes

characteristic

129

C₂×C₄

#341

yes

characteristic

130

C₂×C₄

#360

yes

—

131

C₂×C₄

#361

yes

characteristic

132

C₂×C₄

#362

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

C₈²

yes

characteristic

139

C₈

#322

yes

characteristic

140

C₈

#322

yes

characteristic

141

C₈

#322

yes

characteristic

142

C₂²

#1277

yes

characteristic

143

C₂²

#1279

yes

characteristic

144

C₂²

#1279

yes

characteristic

145

C₂²

#1386

yes

characteristic

146

C₂²

#1387

yes

characteristic

147

C₂²

#1388

yes

characteristic

148

C₂²

#1389

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#1273

yes

characteristic

155

C₄

#1276

yes

characteristic

156

C₄

#1278

yes

characteristic

157

C₄

#1278

yes

characteristic

158

C₄

#1333

yes

—

159

C₄

#1334

yes

—

160

C₂

#7015

yes

characteristic

161

C₂

#7158

yes

characteristic

162

C₂

#7159

yes

characteristic

163

C₂

#7175

yes

characteristic

164

C₂

#7176

yes

characteristic

165

C₂

#7177

yes

characteristic

166

C₂

#7178

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#102
commutator	C₈	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle, prime radical	C₂³	#340
LCS	G ▷ C₈ ▷ 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

C₈

C₈²

Frattini, cosocle

C₄²×C₈

C₂²

socle, prime radical

C₂³

#340

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
28. #346	C₈	semidirect
33. #335	C₈	semidirect
35. #334	C₈	semidirect

H ≅ G/N

type

28. #346

C₈

semidirect

33. #335

C₈

semidirect

35. #334

C₈

semidirect

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

subgroup	index
C₄×#346	2
C₄×#335	2
C₄×#334	2

subgroup

index

C₄×#346

C₄×#335

C₄×#334

Subgroups — #93270

Conjugacy classes of subgroups

Complements

Maximal subgroups