Group #146263 — subgroups

Subgroups — #146263

Order 512.

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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	#13288	C₂	1	1	yes	maximal, characteristic
2	#13285	C₂	1	1	yes	maximal, characteristic
3	#12907	C₂	1	1	yes	maximal, characteristic
4	#12887	C₂	1	1	yes	maximal, characteristic
5	#9755	C₂	1	1	yes	maximal, characteristic
6	#8793	C₂	1	1	yes	maximal, characteristic
7	#7141	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	1	yes	characteristic
9	C₂×#367	—	2	1	no	—
10	C₂×#366	—	2	1	no	—
11	C₄×M₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1848	C₂²	1	1	yes	characteristic
14	#1846	C₂²	1	1	yes	characteristic
15	#1846	C₂²	1	1	yes	characteristic
16	#1518	—	2	2	no	—
17	#1450	C₂²	1	1	yes	characteristic
18	#1386	—	2	2	no	—
19	#1383	—	2	2	no	—
20	#1382	—	2	1	no	—
21	#1382	—	2	1	no	—
22	#1286	—	2	2	no	—
23	#533	—	2	1	no	—
24	C₂×M₃₂	D₄	1	2	yes	—
25	C₂²×C₁₆	D₄	1	1	yes	characteristic
26	C₂²×C₁₆	D₄	1	1	yes	characteristic
27	#500	Q₈	1	2	yes	—
28	#499	—	2	1	no	—
29	#499	—	2	1	no	—
30	#498	—	2	1	no	—
31	#431	D₄	1	2	yes	—
32	C₂×#108	—	2	1	no	—
33	C₂×#108	—	2	1	no	—
34	#427	—	2	1	no	—
35	#427	—	2	1	no	—
36	C₂×#107	—	2	1	no	—
37	C₂×#107	—	2	1	no	—
38	C₂×#106	—	2	1	no	—
39	C₂×#106	—	2	1	no	—
40	#423	—	2	1	no	—
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	#382	—	2	1	no	—
43	#382	—	2	1	no	—
44	#367	—	4	2	no	—
45	#366	—	4	2	no	—
46	#346	—	2	1	no	—
47	C₄×C₁₆	—	2	1	no	—
48	C₄×C₁₆	—	2	1	no	—
49	C₄×C₁₆	—	2	1	no	—
50	#328	—	4	4	no	—
51	#322	—	2	1	no	—
52	C₈²	—	2	1	no	—
53	C₂×M₁₆	—	4	1	no	—
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₄⋊Q₈	—	2	2	no	—
57	#125	—	4	1	no	—
58	C₂²⋊Q₈	—	4	2	no	—
59	C₄×Q₈	—	4	2	no	—
60	#117	—	4	1	no	—
61	C₂×C₄⋊C₄	—	4	1	no	—
62	C₂×C₄⋊C₄	—	4	1	no	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	M₃₂	—	2	2	no	—
65	M₃₂	—	2	2	no	—
66	C₂×C₁₆	—	2	1	no	—
67	C₂×C₁₆	—	2	1	no	—
68	C₂×C₁₆	—	2	1	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	2	2	no	—
71	C₂×C₁₆	Q₁₆	1	2	yes	—
72	C₂×C₁₆	Q₁₆	1	2	yes	—
73	#108	—	4	1	no	—
74	#108	—	4	1	no	—
75	#108	—	4	2	no	—
76	#107	—	2	2	no	—
77	#107	—	2	2	no	—
78	#107	—	4	1	no	—
79	#107	—	4	1	no	—
80	#107	—	4	1	no	—
81	#107	—	4	1	no	—
82	#106	—	4	1	no	—
83	#106	—	4	1	no	—
84	#106	—	4	2	no	—
85	#106	—	4	2	no	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	4	2	no	—
88	C₂²⋊C₈	—	2	2	no	—
89	C₄×C₈	—	2	2	no	—
90	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	C₄×C₈	#41	1	1	yes	characteristic
93	C₄×C₈	#41	1	1	yes	characteristic
94	#95	—	4	2	no	—
95	#95	—	4	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂²×C₄	—	8	1	no	—
98	C₂²×C₄	—	8	1	no	—
99	C₂²×C₄	#127	1	1	yes	characteristic
100	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
101	M₁₆	—	8	1	no	—
102	M₁₆	—	8	1	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	8	1	no	—
105	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₈	#123	1	1	yes	characteristic
108	C₂×C₈	#123	1	1	yes	characteristic
109	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
110	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
111	C₂×C₈	#135	1	2	yes	—
112	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
113	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
114	C₂×C₈	#137	1	2	yes	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₄⋊C₄	—	8	1	no	—
119	C₄⋊C₄	—	8	1	no	—
120	C₄⋊C₄	—	8	1	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₂²⋊C₄	—	8	1	no	—
123	C₄²	—	8	1	no	—
124	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
125	C₄²	#123	1	1	yes	characteristic
126	C₄²	#135	1	2	yes	—
127	C₁₆	—	2	2	no	—
128	C₁₆	—	2	2	no	—
129	C₁₆	—	4	2	no	—
130	C₂³	#496	1	1	yes	characteristic
131	Q₈	—	16	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	—	16	2	no	—
135	C₂×C₄	—	16	2	no	—
136	C₂×C₄	—	16	1	no	—
137	C₂×C₄	—	16	1	no	—
138	C₂×C₄	#399	1	1	yes	characteristic
139	C₂×C₄	#478	1	2	yes	—
140	C₂×C₄	#482	1	2	yes	—
141	C₂×C₄	#494	1	1	yes	characteristic
142	C₂×C₄	#495	1	1	yes	characteristic
143	C₂×C₄	#495	1	1	yes	characteristic
144	C₂×C₄	#496	1	1	yes	characteristic
145	C₂×C₄	#497	1	1	yes	characteristic
146	C₈	—	8	2	no	—
147	C₈	—	16	1	no	—
148	C₈	#477	1	1	yes	characteristic
149	C₈	#477	1	1	yes	characteristic
150	C₈	#479	1	2	yes	—
151	C₈	#484	1	1	yes	characteristic
152	C₈	#484	1	1	yes	characteristic
153	C₈	#485	1	2	yes	—
154	C₂²	—	2	2	no	—
155	C₂²	#1717	1	1	yes	characteristic
156	C₂²	#1723	1	1	yes	characteristic
157	C₂²	#2092	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	16	2	no	—
161	C₄	—	32	1	no	—
162	C₄	—	32	1	no	—
163	C₄	#1710	1	1	yes	characteristic
164	C₄	#1720	1	1	yes	characteristic
165	C₄	#2095	1	1	yes	characteristic
166	C₄	#2097	1	1	yes	characteristic
167	C₂	—	2	1	no	—
168	C₂	—	2	1	no	—
169	C₂	#10679	1	1	yes	characteristic
170	C₂	#13525	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13288

C₂

yes

maximal, characteristic

#13285

C₂

yes

maximal, characteristic

#12907

C₂

yes

maximal, characteristic

#12887

C₂

yes

maximal, characteristic

#9755

C₂

yes

maximal, characteristic

#8793

C₂

yes

maximal, characteristic

#7141

C₂

yes

maximal, characteristic

#3160

C₂²

yes

characteristic

C₂×#367

—

C₂×#366

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1846

C₂²

yes

characteristic

#1846

C₂²

yes

characteristic

#1518

—

#1450

C₂²

yes

—

—

—

—

—

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

#500

Q₈

yes

—

—

—

—

D₄

yes

—

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#328

—

#322

—

C₈²

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

—

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

100

C₂²×C₄

AGL(1,ℤ/8)

yes

—

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²⋊Q₈

yes

characteristic

106

C₂×C₈

C₂²⋊Q₈

yes

characteristic

107

C₂×C₈

#123

yes

characteristic

108

C₂×C₈

#123

yes

characteristic

109

C₂×C₈

C₂×Q₁₆

yes

characteristic

110

C₂×C₈

C₂×Q₁₆

yes

characteristic

111

C₂×C₈

#135

yes

—

112

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

113

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

114

C₂×C₈

#137

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

C₂²⋊Q₈

yes

characteristic

125

C₄²

#123

yes

characteristic

126

C₄²

#135

yes

—

127

C₁₆

—

128

C₁₆

—

129

C₁₆

—

130

C₂³

#496

yes

characteristic

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#399

yes

characteristic

139

C₂×C₄

#478

yes

—

140

C₂×C₄

#482

yes

—

141

C₂×C₄

#494

yes

characteristic

142

C₂×C₄

#495

yes

characteristic

143

C₂×C₄

#495

yes

characteristic

144

C₂×C₄

#496

yes

characteristic

145

C₂×C₄

#497

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

#477

yes

characteristic

149

C₈

#477

yes

characteristic

150

C₈

#479

yes

—

151

C₈

#484

yes

characteristic

152

C₈

#484

yes

characteristic

153

C₈

#485

yes

—

154

C₂²

—

155

C₂²

#1717

yes

characteristic

156

C₂²

#1723

yes

characteristic

157

C₂²

#2092

yes

characteristic

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#1710

yes

characteristic

164

C₄

#1720

yes

characteristic

165

C₄

#2095

yes

characteristic

166

C₄

#2097

yes

characteristic

167

C₂

—

168

C₂

—

169

C₂

#10679

yes

characteristic

170

C₂

#13525

yes

—

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2092

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#496

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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
#13288	2
#13285	2
#12907	2
#12887	2
#9755	2
#8793	2
#7141	2

subgroup

index

Subgroups — #146263

Conjugacy classes of subgroups

Maximal subgroups