Group #140405 — subgroups

Subgroups — #140405

Order 512.

242 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 102 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×#364	C₂	1	1	yes	maximal, characteristic
2	#12000	C₂	1	2	yes	maximal
3	#8805	C₂	1	1	yes	maximal, characteristic
4	#8267	C₂	1	1	yes	maximal, characteristic
5	#7158	C₂	1	2	yes	maximal
6	C₂×#364	C₄	1	2	yes	—
7	C₄×M₃₂	C₂²	1	2	yes	—
8	#1842	—	2	1	no	—
9	C₄²×C₈	C₂²	1	1	yes	characteristic
10	#1564	C₂²	1	1	yes	characteristic
11	#1564	C₂²	1	1	yes	characteristic
12	C₂×#334	—	2	1	no	—
13	C₂×#334	C₂²	1	2	yes	—
14	#1452	—	2	1	no	—
15	#1376	C₄	1	1	yes	characteristic
16	#1376	C₄	1	1	yes	characteristic
17	#1368	—	2	4	no	—
18	#1276	—	2	1	no	—
19	C₂×M₃₂	C₂×C₄	1	2	yes	—
20	C₂×M₃₂	C₂×C₄	1	2	yes	—
21	#431	—	2	1	no	—
22	#431	—	2	1	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	—	4	2	no	—
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
29	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	C₄³	D₄	1	1	yes	characteristic
32	#364	—	2	4	no	—
33	#346	—	2	2	no	—
34	C₄×C₁₆	—	2	2	no	—
35	#335	—	2	1	no	—
36	#335	Q₈	1	2	yes	—
37	#334	—	2	2	no	—
38	#334	—	2	1	no	—
39	#334	—	2	2	no	—
40	#334	—	4	2	no	—
41	#334	D₄	1	2	yes	—
42	#322	—	4	2	no	—
43	C₂²×C₈	—	4	2	no	—
44	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄²	—	2	1	no	—
49	C₂×C₄²	—	2	1	no	—
50	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄²	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
53	M₃₂	—	2	2	no	—
54	M₃₂	—	2	2	no	—
55	M₃₂	—	2	2	no	—
56	M₃₂	M₁₆	1	4	yes	—
57	C₂×C₁₆	—	2	2	no	—
58	C₂×C₁₆	—	2	2	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₄⋊C₈	—	4	2	no	—
62	C₄⋊C₈	—	4	2	no	—
63	C₄⋊C₈	—	8	4	no	—
64	C₂²⋊C₈	—	4	1	no	—
65	C₂²⋊C₈	—	4	1	no	—
66	C₄×C₈	—	2	2	no	—
67	C₄×C₈	—	2	2	no	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	—	4	1	no	—
70	C₄×C₈	—	4	1	no	—
71	C₄×C₈	—	4	1	no	—
72	C₄×C₈	SD₁₆	1	2	yes	—
73	C₄×C₈	SD₁₆	1	2	yes	—
74	C₄×C₈	C₂×D₄	1	1	yes	characteristic
75	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
76	C₄×C₈	#41	1	1	yes	characteristic
77	C₄×C₈	#41	1	1	yes	characteristic
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂²×C₄	#137	1	1	yes	characteristic
83	C₂×C₈	—	2	1	no	—
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	—	2	1	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	—	4	2	no	—
91	C₂×C₈	—	8	4	no	—
92	C₂×C₈	—	8	2	no	—
93	C₂×C₈	#106	1	2	yes	—
94	C₂×C₈	#106	1	2	yes	—
95	C₂×C₈	C₄×D₄	1	1	yes	characteristic
96	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
97	C₂×C₈	#125	1	1	yes	characteristic
98	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×C₈	C₂×M₁₆	1	2	yes	—
100	C₂×C₈	#131	1	2	yes	—
101	C₄²	—	2	4	no	—
102	C₄²	—	2	1	no	—
103	C₄²	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	4	2	no	—
106	C₄²	#108	1	2	yes	—
107	C₄²	#121	1	1	yes	characteristic
108	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
109	C₄²	C₂×SD₁₆	1	1	yes	characteristic
110	C₄²	C₂×SD₁₆	1	1	yes	characteristic
111	C₁₆	—	2	4	no	—
112	C₁₆	—	4	2	no	—
113	C₂³	#428	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	#389	1	1	yes	characteristic
123	C₂×C₄	#423	1	1	yes	characteristic
124	C₂×C₄	#424	1	1	yes	characteristic
125	C₂×C₄	C₂×#106	1	1	yes	characteristic
126	C₂×C₄	C₂×#108	1	1	yes	characteristic
127	C₂×C₄	#430	1	1	yes	characteristic
128	C₂×C₄	#460	1	1	yes	characteristic
129	C₂×C₄	#461	1	1	yes	characteristic
130	C₂×C₄	#475	1	1	yes	characteristic
131	C₂×C₄	#476	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	—	2	2	no	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	8	2	no	—
137	C₈	—	8	2	no	—
138	C₈	—	16	2	no	—
139	C₈	#435	1	2	yes	—
140	C₈	#446	1	2	yes	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#1569	1	1	yes	characteristic
144	C₂²	#1573	1	1	yes	characteristic
145	C₂²	#1948	1	1	yes	characteristic
146	C₄	—	2	4	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	4	2	no	—
151	C₄	#1566	1	1	yes	characteristic
152	C₄	#1570	1	1	yes	characteristic
153	C₄	#1940	1	1	yes	characteristic
154	C₄	#1950	1	1	yes	characteristic
155	C₄	#2230	1	2	yes	—
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	#9605	1	1	yes	characteristic
159	C₂	#13186	1	1	yes	characteristic
160	C₂	#13188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#364

C₂

yes

maximal, characteristic

#12000

C₂

yes

maximal

#8805

C₂

yes

maximal, characteristic

#8267

C₂

yes

maximal, characteristic

#7158

C₂

yes

maximal

C₂×#364

C₄

yes

—

C₄×M₃₂

C₂²

yes

—

#1842

—

C₄²×C₈

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

C₂×#334

—

C₂×#334

C₂²

yes

—

#1452

—

#1376

C₄

yes

characteristic

#1376

C₄

yes

characteristic

#1368

—

#1276

—

C₂×M₃₂

C₂×C₄

yes

—

C₂×M₃₂

C₂×C₄

yes

—

#431

—

#431

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

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

D₄

yes

characteristic

#364

—

#346

—

C₄×C₁₆

—

#335

—

#335

Q₈

yes

—

—

—

—

—

D₄

yes

—

#322

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₄⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

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

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

M₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

100

C₂×C₈

#131

yes

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

#108

yes

—

107

C₄²

#121

yes

characteristic

108

C₄²

C₂²⋊Q₈

yes

characteristic

109

C₄²

C₂×SD₁₆

yes

characteristic

110

C₄²

C₂×SD₁₆

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₂³

#428

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#389

yes

characteristic

123

C₂×C₄

#423

yes

characteristic

124

C₂×C₄

#424

yes

characteristic

125

C₂×C₄

C₂×#106

yes

characteristic

126

C₂×C₄

C₂×#108

yes

characteristic

127

C₂×C₄

#430

yes

characteristic

128

C₂×C₄

#460

yes

characteristic

129

C₂×C₄

#461

yes

characteristic

130

C₂×C₄

#475

yes

characteristic

131

C₂×C₄

#476

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

#435

yes

—

140

C₈

#446

yes

—

141

C₂²

—

142

C₂²

—

143

C₂²

#1569

yes

characteristic

144

C₂²

#1573

yes

characteristic

145

C₂²

#1948

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#1566

yes

characteristic

152

C₄

#1570

yes

characteristic

153

C₄

#1940

yes

characteristic

154

C₄

#1950

yes

characteristic

155

C₄

#2230

yes

—

156

C₂

—

157

C₂

—

158

C₂

#9605

yes

characteristic

159

C₂

#13186

yes

characteristic

160

C₂

#13188

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#476
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1948
prime radical	C₂³	#428
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#476

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×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
C₄×#364	2
#12000	2
#8805	2
#8267	2
#7158	2

subgroup

index

Subgroups — #140405

Conjugacy classes of subgroups

Maximal subgroups