Group #140447 — subgroups

Subgroups — #140447

Order 512.

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

normal subgroups only

This group has 235 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	#13185	C₂	1	2	yes	maximal
2	#12004	C₂	1	1	yes	maximal, characteristic
3	#8823	C₂	1	1	yes	maximal, characteristic
4	C₈×#106	C₂	1	1	yes	maximal, characteristic
5	#7280	C₂	1	2	yes	maximal
6	C₂×#365	—	2	2	no	—
7	#2150	C₂²	1	2	yes	—
8	#1848	C₂²	1	2	yes	—
9	C₄×#106	C₂²	1	1	yes	characteristic
10	C₈×C₄⋊C₄	—	2	1	no	—
11	#1567	—	2	1	no	—
12	#1567	—	2	1	no	—
13	C₂×#334	C₂²	1	1	yes	characteristic
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1383	—	2	2	no	—
16	#1370	C₄	1	2	yes	complemented
17	#1370	C₄	1	2	yes	complemented
18	#1279	—	2	1	no	—
19	C₂×M₃₂	—	2	2	no	—
20	C₂×#108	—	2	2	no	—
21	C₂×#108	—	2	2	no	—
22	C₂×#106	D₄	1	1	yes	characteristic
23	C₂×#106	D₄	1	1	yes	characteristic
24	#423	—	2	2	no	—
25	C₂×C₄⋊C₈	—	2	1	no	—
26	C₂×C₄×C₈	—	4	1	no	—
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	C₄×C₄⋊C₄	—	2	1	no	—
31	#365	—	4	4	no	—
32	C₄⋊C₁₆	—	2	2	no	—
33	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
34	#340	—	2	1	no	—
35	#336	—	4	1	no	—
36	#335	—	4	2	no	—
37	#335	—	4	1	no	—
38	#335	—	4	1	no	—
39	#334	—	2	1	no	—
40	#334	Q₈	1	2	yes	—
41	C₈²	C₂×C₄	1	1	yes	characteristic
42	C₈²	C₂×C₄	1	1	yes	characteristic
43	C₈²	Q₈	1	1	yes	characteristic
44	C₈²	Q₈	1	1	yes	characteristic
45	C₂×M₁₆	—	4	2	no	—
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	—	4	1	no	—
48	C₂²×C₈	SD₁₆	1	1	yes	characteristic
49	C₂²×C₈	SD₁₆	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₈	#41	1	1	yes	characteristic
52	C₂×C₄⋊C₄	—	2	1	no	—
53	C₂×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄²	—	4	1	no	—
55	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
56	M₃₂	—	4	2	no	—
57	M₃₂	—	4	2	no	—
58	C₂×C₁₆	—	2	4	no	—
59	#108	—	4	2	no	—
60	#108	—	4	2	no	—
61	#108	—	8	2	no	—
62	#106	—	2	1	no	—
63	#106	—	2	1	no	—
64	#106	D₈	1	2	yes	—
65	#106	Q₁₆	1	2	yes	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	4	2	no	—
68	C₄⋊C₈	—	4	1	no	—
69	C₄⋊C₈	—	4	2	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	8	2	no	—
72	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
73	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₈	SD₁₆	1	1	yes	characteristic
77	C₄×C₈	SD₁₆	1	1	yes	characteristic
78	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
80	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₄×C₈	#41	1	1	yes	characteristic
82	#95	—	4	1	no	—
83	C₂²×C₄	—	4	1	no	—
84	C₂²×C₄	—	4	1	no	—
85	C₂²×C₄	#121	1	1	yes	characteristic
86	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
87	C₂²×C₄	#137	1	1	yes	characteristic
88	M₁₆	—	8	2	no	—
89	M₁₆	—	8	2	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	—	8	1	no	—
99	C₂×C₈	—	8	1	no	—
100	C₂×C₈	—	8	2	no	—
101	C₂×C₈	#106	1	1	yes	characteristic
102	C₂×C₈	#106	1	1	yes	characteristic
103	C₂×C₈	#106	1	1	yes	characteristic
104	C₂×C₈	#106	1	1	yes	characteristic
105	C₂×C₈	C₄×D₄	1	1	yes	characteristic
106	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
107	C₂×C₈	#125	1	1	yes	characteristic
108	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
109	C₂×C₈	C₂×D₈	1	1	yes	characteristic
110	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₈	#135	1	1	yes	characteristic
112	C₂×C₈	#135	1	1	yes	characteristic
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	4	1	no	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄²	—	8	1	no	—
118	C₄²	—	8	1	no	—
119	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
121	C₄²	C₂×SD₁₆	1	1	yes	characteristic
122	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₁₆	—	4	4	no	—
124	C₂³	#461	1	1	yes	characteristic
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	8	1	no	—
131	C₂×C₄	—	8	1	no	—
132	C₂×C₄	—	8	1	no	—
133	C₂×C₄	—	8	1	no	—
134	C₂×C₄	#389	1	1	yes	characteristic
135	C₂×C₄	C₂×#106	1	1	yes	characteristic
136	C₂×C₄	#428	1	1	yes	characteristic
137	C₂×C₄	#460	1	1	yes	characteristic
138	C₂×C₄	#466	1	1	yes	characteristic
139	C₂×C₄	#467	1	1	yes	characteristic
140	C₂×C₄	#472	1	1	yes	characteristic
141	C₂×C₄	#473	1	1	yes	characteristic
142	C₂×C₄	#475	1	1	yes	characteristic
143	C₂×C₄	#476	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	2	1	no	—
146	C₈	—	2	1	no	—
147	C₈	—	2	1	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	1	no	—
151	C₈	—	8	2	no	—
152	C₈	—	16	2	no	—
153	C₈	#365	1	2	yes	complemented
154	C₈	C₄×D₈	1	1	yes	characteristic
155	C₈	C₄×Q₁₆	1	1	yes	characteristic
156	C₈	#499	1	1	yes	characteristic
157	C₈	#500	1	1	yes	characteristic
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#1673	1	1	yes	characteristic
161	C₂²	#1675	1	1	yes	characteristic
162	C₂²	#1948	1	1	yes	characteristic
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	8	2	no	—
167	C₄	—	8	2	no	—
168	C₄	#1668	1	1	yes	characteristic
169	C₄	#1670	1	1	yes	characteristic
170	C₄	#1942	1	1	yes	characteristic
171	C₄	#1952	1	1	yes	characteristic
172	C₄	C₂×#365	1	1	yes	characteristic
173	C₄	#2155	1	1	yes	characteristic
174	C₂	—	2	1	no	—
175	C₂	—	2	1	no	—
176	C₂	#9615	1	1	yes	characteristic
177	C₂	#13185	1	1	yes	characteristic
178	C₂	#13188	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13185

C₂

yes

maximal

#12004

C₂

yes

maximal, characteristic

#8823

C₂

yes

maximal, characteristic

C₈×#106

C₂

yes

maximal, characteristic

#7280

C₂

yes

maximal

C₂×#365

—

#2150

C₂²

yes

—

#1848

C₂²

yes

—

C₄×#106

C₂²

yes

characteristic

C₈×C₄⋊C₄

—

#1567

—

#1567

—

C₂×#334

C₂²

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1383

—

#1370

C₄

yes

complemented

#1370

C₄

yes

complemented

#1279

—

C₂×M₃₂

—

C₂×#108

—

C₂×#108

—

C₂×#106

D₄

yes

characteristic

C₂×#106

D₄

yes

characteristic

#423

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

—

#365

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

—

—

—

—

—

—

Q₈

yes

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

SD₁₆

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

D₈

yes

—

#106

Q₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

C₄⋊C₄

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

#137

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

#106

yes

characteristic

102

C₂×C₈

#106

yes

characteristic

103

C₂×C₈

#106

yes

characteristic

104

C₂×C₈

#106

yes

characteristic

105

C₂×C₈

C₄×D₄

yes

characteristic

106

C₂×C₈

C₄×Q₈

yes

characteristic

107

C₂×C₈

#125

yes

characteristic

108

C₂×C₈

C₄⋊Q₈

yes

characteristic

109

C₂×C₈

C₂×D₈

yes

characteristic

110

C₂×C₈

C₂×Q₁₆

yes

characteristic

111

C₂×C₈

#135

yes

characteristic

112

C₂×C₈

#135

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

C₂×C₄⋊C₄

yes

characteristic

120

C₄²

C₂²⋊Q₈

yes

characteristic

121

C₄²

C₂×SD₁₆

yes

characteristic

122

C₄²

AGL(1,ℤ/8)

yes

characteristic

123

C₁₆

—

124

C₂³

#461

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#389

yes

characteristic

135

C₂×C₄

C₂×#106

yes

characteristic

136

C₂×C₄

#428

yes

characteristic

137

C₂×C₄

#460

yes

characteristic

138

C₂×C₄

#466

yes

characteristic

139

C₂×C₄

#467

yes

characteristic

140

C₂×C₄

#472

yes

characteristic

141

C₂×C₄

#473

yes

characteristic

142

C₂×C₄

#475

yes

characteristic

143

C₂×C₄

#476

yes

characteristic

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

#365

yes

complemented

154

C₈

C₄×D₈

yes

characteristic

155

C₈

C₄×Q₁₆

yes

characteristic

156

C₈

#499

yes

characteristic

157

C₈

#500

yes

characteristic

158

C₂²

—

159

C₂²

—

160

C₂²

#1673

yes

characteristic

161

C₂²

#1675

yes

characteristic

162

C₂²

#1948

yes

characteristic

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

#1668

yes

characteristic

169

C₄

#1670

yes

characteristic

170

C₄

#1942

yes

characteristic

171

C₄

#1952

yes

characteristic

172

C₄

C₂×#365

yes

characteristic

173

C₄

#2155

yes

characteristic

174

C₂

—

175

C₂

—

176

C₂

#9615

yes

characteristic

177

C₂

#13185

yes

characteristic

178

C₂

#13188

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1948
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#461
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1948

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#461

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. #1370	C₄	semidirect
17. #1370	C₄	semidirect
153. C₈	#365	semidirect

H ≅ G/N

type

16. #1370

C₄

semidirect

17. #1370

C₄

semidirect

153. C₈

#365

semidirect

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

subgroup	index
#13185	2
#12004	2
#8823	2
C₈×#106	2
#7280	2

subgroup

index

Subgroups — #140447

Conjugacy classes of subgroups

Complements

Maximal subgroups