Group #140375 — subgroups

Subgroups — #140375

Order 512.

242 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 118 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×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#12005	C₂	1	2	yes	maximal
4	#8266	C₂	1	1	yes	maximal, characteristic
5	#7153	C₂	1	2	yes	maximal
6	C₂×#364	—	2	1	no	—
7	C₂×#364	—	2	1	no	—
8	#2150	C₂²	1	2	yes	—
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	—	2	1	no	—
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	#1564	C₄	1	1	yes	characteristic
14	#1564	C₄	1	1	yes	characteristic
15	#1450	C₂²	1	2	yes	—
16	#1374	—	2	1	no	—
17	#1370	—	2	4	no	—
18	#1276	C₄	1	2	yes	—
19	C₂×M₃₂	—	2	2	no	—
20	C₂²×C₁₆	—	2	1	no	—
21	C₂²×C₁₆	—	2	1	no	—
22	#431	C₂×C₄	1	2	yes	—
23	#431	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
28	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	C₄³	C₂×C₄	1	1	yes	characteristic
31	#364	—	4	2	no	—
32	#364	—	4	2	no	—
33	C₄⋊C₁₆	—	2	2	no	—
34	C₄⋊C₁₆	—	2	1	no	—
35	C₄⋊C₁₆	—	2	1	no	—
36	C₄⋊C₁₆	—	2	2	no	—
37	C₄⋊C₁₆	D₄	1	2	yes	—
38	C₄⋊C₁₆	Q₈	1	2	yes	—
39	C₄×C₁₆	—	4	2	no	—
40	#335	—	2	1	no	—
41	#335	C₈	1	2	yes	—
42	#334	—	2	1	no	—
43	#334	C₈	1	2	yes	—
44	#322	—	2	2	no	—
45	C₈²	—	2	2	no	—
46	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄²	—	2	1	no	—
51	C₂×C₄²	—	2	1	no	—
52	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
53	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
55	M₃₂	—	4	2	no	—
56	M₃₂	—	4	2	no	—
57	C₂×C₁₆	—	2	2	no	—
58	C₂×C₁₆	—	2	2	no	—
59	C₂×C₁₆	—	4	1	no	—
60	C₂×C₁₆	—	4	1	no	—
61	C₂×C₁₆	—	4	1	no	—
62	C₂×C₁₆	—	4	1	no	—
63	C₂×C₁₆	—	4	2	no	—
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	C₂×C₈	1	4	yes	—
66	C₂²⋊C₈	—	2	2	no	—
67	C₂²⋊C₈	—	2	2	no	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	4	1	no	—
74	C₄×C₈	C₂×C₈	1	1	yes	characteristic
75	C₄×C₈	C₂×C₈	1	1	yes	characteristic
76	C₄×C₈	M₁₆	1	1	yes	characteristic
77	C₄×C₈	M₁₆	1	1	yes	characteristic
78	C₄×C₈	D₈	1	2	yes	—
79	C₄×C₈	Q₁₆	1	2	yes	—
80	C₄×C₈	C₂×D₄	1	1	yes	characteristic
81	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
82	C₄×C₈	#41	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄	#131	1	1	yes	characteristic
88	C₂²×C₄	#131	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
96	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
97	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
98	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
99	C₂×C₈	#107	1	2	yes	—
100	C₂×C₈	#108	1	2	yes	—
101	C₂×C₈	#121	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₈	#125	1	1	yes	characteristic
104	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
105	C₂×C₈	C₂×D₈	1	1	yes	characteristic
106	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₈	#137	1	1	yes	characteristic
109	C₄²	—	2	4	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	4	2	no	—
114	C₄²	#107	1	2	yes	—
115	C₄²	C₄×D₄	1	1	yes	characteristic
116	C₄²	C₄×Q₈	1	1	yes	characteristic
117	C₄²	C₂²×C₈	1	1	yes	characteristic
118	C₄²	C₂×M₁₆	1	1	yes	characteristic
119	C₁₆	—	4	2	no	—
120	C₁₆	—	4	2	no	—
121	C₁₆	—	8	2	no	—
122	C₂³	#424	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	#389	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
133	C₂×C₄	C₂×#107	1	1	yes	characteristic
134	C₂×C₄	#428	1	1	yes	characteristic
135	C₂×C₄	C₂×#108	1	1	yes	characteristic
136	C₂×C₄	#430	1	1	yes	characteristic
137	C₂×C₄	C₈×D₄	1	1	yes	characteristic
138	C₂×C₄	#436	1	1	yes	characteristic
139	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
140	C₂×C₄	#446	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	4	4	no	—
146	C₈	—	8	2	no	—
147	C₈	#335	1	2	yes	—
148	C₈	#335	1	2	yes	—
149	C₈	#459	1	1	yes	characteristic
150	C₈	#462	1	1	yes	characteristic
151	C₈	#474	1	1	yes	characteristic
152	C₈	#477	1	1	yes	characteristic
153	C₂²	—	2	1	no	—
154	C₂²	—	2	1	no	—
155	C₂²	#1571	1	1	yes	characteristic
156	C₂²	#1574	1	1	yes	characteristic
157	C₂²	#1939	1	1	yes	characteristic
158	C₄	—	2	4	no	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	4	2	no	—
163	C₄	C₂×#335	1	1	yes	characteristic
164	C₄	#1565	1	1	yes	characteristic
165	C₄	#1947	1	1	yes	characteristic
166	C₄	#1950	1	1	yes	characteristic
167	C₄	#2170	1	2	yes	—
168	C₂	—	2	1	no	—
169	C₂	—	2	1	no	—
170	C₂	#9601	1	1	yes	characteristic
171	C₂	#13161	1	1	yes	characteristic
172	C₂	#13164	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#12900

C₂

yes

maximal, characteristic

C₄×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#12005

C₂

yes

maximal

#8266

C₂

yes

maximal, characteristic

#7153

C₂

yes

—

—

C₂²

yes

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

—

C₄²×C₈

C₂²

yes

characteristic

#1564

C₄

yes

characteristic

#1564

C₄

yes

characteristic

#1450

C₂²

yes

—

#1374

—

#1370

—

#1276

C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

—

#431

C₂×C₄

yes

—

#431

C₂×C₄

yes

—

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

C₂×C₄×C₈

Q₈

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

#364

—

#364

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₄

yes

—

C₄⋊C₁₆

Q₈

yes

—

C₄×C₁₆

—

#335

—

#335

C₈

yes

—

#334

—

#334

C₈

yes

—

#322

—

C₈²

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₈

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

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

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄⋊C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

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

characteristic

C₂×C₈

C₄⋊C₈

yes

characteristic

C₂×C₈

#107

yes

—

100

C₂×C₈

#108

yes

—

101

C₂×C₈

#121

yes

characteristic

102

C₂×C₈

C₂²⋊Q₈

yes

characteristic

103

C₂×C₈

#125

yes

characteristic

104

C₂×C₈

C₄⋊Q₈

yes

characteristic

105

C₂×C₈

C₂×D₈

yes

characteristic

106

C₂×C₈

C₂×Q₁₆

yes

characteristic

107

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

108

C₂×C₈

#137

yes

characteristic

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

#107

yes

—

115

C₄²

C₄×D₄

yes

characteristic

116

C₄²

C₄×Q₈

yes

characteristic

117

C₄²

C₂²×C₈

yes

characteristic

118

C₄²

C₂×M₁₆

yes

characteristic

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

122

C₂³

#424

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#389

yes

characteristic

132

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

133

C₂×C₄

C₂×#107

yes

characteristic

134

C₂×C₄

#428

yes

characteristic

135

C₂×C₄

C₂×#108

yes

characteristic

136

C₂×C₄

#430

yes

characteristic

137

C₂×C₄

C₈×D₄

yes

characteristic

138

C₂×C₄

#436

yes

characteristic

139

C₂×C₄

C₈×Q₈

yes

characteristic

140

C₂×C₄

#446

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

#335

yes

—

148

C₈

#335

yes

—

149

C₈

#459

yes

characteristic

150

C₈

#462

yes

characteristic

151

C₈

#474

yes

characteristic

152

C₈

#477

yes

characteristic

153

C₂²

—

154

C₂²

—

155

C₂²

#1571

yes

characteristic

156

C₂²

#1574

yes

characteristic

157

C₂²

#1939

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

C₂×#335

yes

characteristic

164

C₄

#1565

yes

characteristic

165

C₄

#1947

yes

characteristic

166

C₄

#1950

yes

characteristic

167

C₄

#2170

yes

—

168

C₂

—

169

C₂

—

170

C₂

#9601

yes

characteristic

171

C₂

#13161

yes

characteristic

172

C₂

#13164

yes

characteristic

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1939
prime radical	C₂³	#424
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#121

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×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×C₄⋊C₁₆	2
#12005	2
#8266	2
#7153	2

subgroup

index

C₄×C₄⋊C₁₆

Subgroups — #140375

Conjugacy classes of subgroups

Maximal subgroups