Group #140402 — subgroups

Subgroups — #140402

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	#12009	C₂	1	2	yes	maximal
3	#8806	C₂	1	1	yes	maximal, characteristic
4	#8267	C₂	1	1	yes	maximal, characteristic
5	#7159	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×#335	—	2	1	no	—
13	C₂×#335	C₂²	1	2	yes	—
14	#1452	—	2	1	no	—
15	#1376	C₄	1	1	yes	characteristic
16	#1376	C₄	1	1	yes	characteristic
17	#1372	—	2	2	no	—
18	#1371	—	2	2	no	—
19	#1276	—	2	1	no	—
20	C₂×M₃₂	C₂×C₄	1	2	yes	—
21	C₂×M₃₂	C₂×C₄	1	2	yes	—
22	#431	—	2	1	no	—
23	#431	—	2	1	no	—
24	C₂×C₄⋊C₈	—	2	2	no	—
25	C₂×C₄⋊C₈	—	4	2	no	—
26	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
30	C₂×C₄×C₈	Q₈	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
32	C₄³	D₄	1	1	yes	characteristic
33	#364	—	2	4	no	—
34	#346	—	2	2	no	—
35	C₄×C₁₆	—	2	2	no	—
36	#335	—	2	2	no	—
37	#335	—	2	1	no	—
38	#335	—	2	2	no	—
39	#335	—	4	2	no	—
40	#335	D₄	1	1	yes	characteristic
41	#335	D₄	1	1	yes	characteristic
42	#334	—	2	1	no	—
43	#334	Q₈	1	2	yes	—
44	#322	—	4	2	no	—
45	C₂²×C₈	—	4	2	no	—
46	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₈	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
55	M₃₂	—	2	2	no	—
56	M₃₂	—	2	2	no	—
57	M₃₂	—	2	2	no	—
58	M₃₂	M₁₆	1	4	yes	—
59	C₂×C₁₆	—	2	2	no	—
60	C₂×C₁₆	—	2	2	no	—
61	C₄⋊C₈	—	2	1	no	—
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	4	2	no	—
65	C₄⋊C₈	—	4	2	no	—
66	C₄⋊C₈	—	8	4	no	—
67	C₂²⋊C₈	—	4	1	no	—
68	C₂²⋊C₈	—	4	1	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	4	1	no	—
73	C₄×C₈	—	4	1	no	—
74	C₄×C₈	—	4	1	no	—
75	C₄×C₈	D₈	1	2	yes	—
76	C₄×C₈	Q₁₆	1	2	yes	—
77	C₄×C₈	C₂×D₄	1	1	yes	characteristic
78	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
79	C₄×C₈	#41	1	1	yes	characteristic
80	C₄×C₈	#41	1	1	yes	characteristic
81	C₂²×C₄	—	2	1	no	—
82	C₂²×C₄	—	2	1	no	—
83	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
85	C₂²×C₄	#137	1	1	yes	characteristic
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	4	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	2	no	—
95	C₂×C₈	—	8	4	no	—
96	C₂×C₈	—	8	2	no	—
97	C₂×C₈	#107	1	2	yes	—
98	C₂×C₈	#107	1	2	yes	—
99	C₂×C₈	C₄×D₄	1	1	yes	characteristic
100	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₈	#125	1	1	yes	characteristic
102	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
103	C₂×C₈	C₂×M₁₆	1	2	yes	—
104	C₂×C₈	#131	1	2	yes	—
105	C₄²	—	2	2	no	—
106	C₄²	—	2	2	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	2	1	no	—
110	C₄²	—	4	2	no	—
111	C₄²	#108	1	2	yes	—
112	C₄²	#121	1	1	yes	characteristic
113	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
114	C₄²	C₂×D₈	1	1	yes	characteristic
115	C₄²	C₂×Q₁₆	1	1	yes	characteristic
116	C₁₆	—	2	4	no	—
117	C₁₆	—	4	2	no	—
118	C₂³	#428	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	#389	1	1	yes	characteristic
129	C₂×C₄	#423	1	1	yes	characteristic
130	C₂×C₄	#424	1	1	yes	characteristic
131	C₂×C₄	C₂×#107	1	1	yes	characteristic
132	C₂×C₄	C₂×#108	1	1	yes	characteristic
133	C₂×C₄	#430	1	1	yes	characteristic
134	C₂×C₄	#459	1	1	yes	characteristic
135	C₂×C₄	#462	1	1	yes	characteristic
136	C₂×C₄	#474	1	1	yes	characteristic
137	C₂×C₄	#477	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	8	2	no	—
143	C₈	—	8	1	no	—
144	C₈	—	8	1	no	—
145	C₈	—	16	2	no	—
146	C₈	#435	1	2	yes	—
147	C₈	#446	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#1569	1	1	yes	characteristic
151	C₂²	#1573	1	1	yes	characteristic
152	C₂²	#1947	1	1	yes	characteristic
153	C₄	—	2	4	no	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	4	2	no	—
158	C₄	#1567	1	1	yes	characteristic
159	C₄	#1571	1	1	yes	characteristic
160	C₄	#1940	1	1	yes	characteristic
161	C₄	#1950	1	1	yes	characteristic
162	C₄	#2213	1	1	yes	characteristic
163	C₄	#2214	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	2	1	no	—
166	C₂	#9604	1	1	yes	characteristic
167	C₂	#13181	1	1	yes	characteristic
168	C₂	#13182	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#364

C₂

yes

maximal, characteristic

#12009

C₂

yes

maximal

#8806

C₂

yes

maximal, characteristic

#8267

C₂

yes

maximal, characteristic

#7159

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×#335

—

C₂×#335

C₂²

yes

—

#1452

—

#1376

C₄

yes

characteristic

#1376

C₄

yes

—

—

—

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

—

—

—

—

—

D₄

yes

characteristic

#335

D₄

yes

characteristic

#334

—

#334

Q₈

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

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#107

yes

—

C₂×C₈

#107

yes

—

C₂×C₈

C₄×D₄

yes

characteristic

100

C₂×C₈

C₄×Q₈

yes

characteristic

101

C₂×C₈

#125

yes

characteristic

102

C₂×C₈

C₄⋊Q₈

yes

characteristic

103

C₂×C₈

C₂×M₁₆

yes

—

104

C₂×C₈

#131

yes

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

#108

yes

—

112

C₄²

#121

yes

characteristic

113

C₄²

C₂²⋊Q₈

yes

characteristic

114

C₄²

C₂×D₈

yes

characteristic

115

C₄²

C₂×Q₁₆

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₂³

#428

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#389

yes

characteristic

129

C₂×C₄

#423

yes

characteristic

130

C₂×C₄

#424

yes

characteristic

131

C₂×C₄

C₂×#107

yes

characteristic

132

C₂×C₄

C₂×#108

yes

characteristic

133

C₂×C₄

#430

yes

characteristic

134

C₂×C₄

#459

yes

characteristic

135

C₂×C₄

#462

yes

characteristic

136

C₂×C₄

#474

yes

characteristic

137

C₂×C₄

#477

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#435

yes

—

147

C₈

#446

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

#1569

yes

characteristic

151

C₂²

#1573

yes

characteristic

152

C₂²

#1947

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

#1567

yes

characteristic

159

C₄

#1571

yes

characteristic

160

C₄

#1940

yes

characteristic

161

C₄

#1950

yes

characteristic

162

C₄

#2213

yes

characteristic

163

C₄

#2214

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

#9604

yes

characteristic

167

C₂

#13181

yes

characteristic

168

C₂

#13182

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1947
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

#459

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
#12009	2
#8806	2
#8267	2
#7159	2

subgroup

index

Subgroups — #140402

Conjugacy classes of subgroups

Maximal subgroups