Group #134648 — subgroups

Subgroups — #134648

Order 512.

280 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 92 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	#21722	C₂	1	1	yes	maximal, characteristic
2	#12884	C₂	1	1	yes	maximal, characteristic
3	#11935	C₂	1	2	yes	maximal
4	#8780	C₂	1	1	yes	maximal, characteristic
5	#7278	C₂	1	2	yes	maximal
6	C₂×#499	C₂²	1	1	yes	characteristic
7	#2330	—	2	1	no	—
8	#2150	C₂²	1	2	yes	—
9	#1992	—	2	1	no	—
10	#1943	C₄	1	2	yes	—
11	#1941	—	2	1	no	—
12	#1844	C₂²	1	1	yes	characteristic
13	#1776	C₄	1	2	yes	—
14	C₄×#106	—	2	1	no	—
15	#1726	C₂²	1	1	yes	characteristic
16	#1564	C₂²	1	2	yes	—
17	#1376	—	2	1	no	—
18	#1376	—	2	1	no	—
19	#1363	—	2	4	no	—
20	#1281	—	2	1	no	—
21	#1274	—	2	1	no	—
22	C₂×#125	—	2	1	no	—
23	C₂×M₃₂	—	2	2	no	—
24	#499	—	2	1	no	—
25	#499	—	2	1	no	—
26	#499	D₄	1	2	yes	—
27	#499	Q₈	1	2	yes	—
28	#431	—	2	2	no	—
29	C₂×#107	C₂×C₄	1	2	yes	—
30	C₂×#106	—	2	1	no	—
31	C₂×#106	—	2	1	no	—
32	C₂×#106	C₂×C₄	1	2	yes	—
33	C₂×#97	C₂×C₄	1	1	yes	characteristic
34	C₂×#97	C₂×C₄	1	1	yes	characteristic
35	C₂×#97	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#389	—	2	2	no	—
40	#389	—	4	2	no	—
41	#387	—	4	2	no	—
42	#384	—	2	1	no	—
43	C₄×C₄⋊C₄	—	2	2	no	—
44	#376	D₄	1	1	yes	characteristic
45	C₄⋊C₁₆	—	2	2	no	—
46	C₄⋊C₁₆	—	2	2	no	—
47	#340	—	2	2	no	—
48	#340	—	2	2	no	—
49	#335	—	2	2	no	—
50	#334	—	2	2	no	—
51	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
55	#125	—	2	2	no	—
56	#125	—	2	2	no	—
57	#125	—	4	2	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	C₂×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄⋊C₄	—	4	2	no	—
62	C₂×C₄⋊C₄	—	4	2	no	—
63	C₂×C₄²	—	4	2	no	—
64	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
67	M₃₂	—	4	2	no	—
68	M₃₂	—	4	2	no	—
69	C₂×C₁₆	—	4	2	no	—
70	#107	—	2	2	no	—
71	#107	C₄⋊C₄	1	4	yes	—
72	#106	—	2	2	no	—
73	#106	—	2	2	no	—
74	#106	—	4	2	no	—
75	#106	—	4	2	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₂²⋊C₈	—	4	2	no	—
78	#97	—	2	2	no	—
79	#97	—	2	2	no	—
80	#97	—	4	1	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	—	4	2	no	—
84	C₄×C₈	C₂×D₄	1	1	yes	characteristic
85	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
86	C₄×C₈	#41	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	#95	—	2	2	no	—
89	#95	—	4	2	no	—
90	#95	—	4	2	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	4	2	no	—
95	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	8	2	no	—
106	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₈	#117	1	1	yes	characteristic
108	C₂×C₈	#121	1	1	yes	characteristic
109	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂×C₈	#135	1	2	yes	—
111	C₂×C₈	#137	1	2	yes	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	4	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	4	2	no	—
118	C₄⋊C₄	—	16	2	no	—
119	C₄⋊C₄	—	16	2	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	—	16	2	no	—
124	C₄²	#101	1	2	yes	—
125	C₄²	#121	1	1	yes	characteristic
126	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
127	C₄²	#135	1	2	yes	—
128	C₁₆	—	8	2	no	—
129	C₂³	#418	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	4	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	#380	1	1	yes	characteristic
142	C₂×C₄	C₂×#101	1	1	yes	characteristic
143	C₂×C₄	#413	1	1	yes	characteristic
144	C₂×C₄	#416	1	1	yes	characteristic
145	C₂×C₄	#416	1	1	yes	characteristic
146	C₂×C₄	#419	1	1	yes	characteristic
147	C₂×C₄	#463	1	2	yes	—
148	C₂×C₄	#478	1	2	yes	—
149	C₈	—	2	1	no	—
150	C₈	—	2	1	no	—
151	C₈	—	4	1	no	—
152	C₈	—	4	1	no	—
153	C₈	—	16	2	no	—
154	C₈	#464	1	2	yes	—
155	C₈	#479	1	2	yes	—
156	C₂²	—	2	2	no	—
157	C₂²	#1553	1	1	yes	characteristic
158	C₂²	#1556	1	1	yes	characteristic
159	C₂²	#1804	1	1	yes	characteristic
160	C₄	—	2	2	no	—
161	C₄	—	2	2	no	—
162	C₄	—	4	2	no	—
163	C₄	—	8	1	no	—
164	C₄	—	8	4	no	—
165	C₄	—	16	2	no	—
166	C₄	#1544	1	1	yes	characteristic
167	C₄	#1554	1	1	yes	characteristic
168	C₄	#1801	1	1	yes	characteristic
169	C₄	#1805	1	1	yes	characteristic
170	C₂	—	2	1	no	—
171	C₂	—	2	1	no	—
172	C₂	#8593	1	1	yes	characteristic
173	C₂	#12828	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21722

C₂

yes

maximal, characteristic

#12884

C₂

yes

maximal, characteristic

#11935

C₂

yes

maximal

#8780

C₂

yes

maximal, characteristic

#7278

C₂

yes

maximal

C₂×#499

C₂²

yes

characteristic

#2330

—

#2150

C₂²

yes

—

#1992

—

#1943

C₄

yes

—

#1941

—

#1844

C₂²

yes

characteristic

#1776

C₄

yes

—

C₄×#106

—

#1726

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

—

—

—

C₂×M₃₂

—

#499

—

#499

—

#499

D₄

yes

—

#499

Q₈

yes

—

#431

—

C₂×#107

C₂×C₄

yes

—

C₂×#106

—

C₂×#106

—

C₂×#106

C₂×C₄

yes

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₄×C₄⋊C₄

—

#376

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#107

—

#107

C₄⋊C₄

yes

—

—

—

—

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

107

C₂×C₈

#117

yes

characteristic

108

C₂×C₈

#121

yes

characteristic

109

C₂×C₈

C₂²⋊Q₈

yes

characteristic

110

C₂×C₈

#135

yes

—

111

C₂×C₈

#137

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

#101

yes

—

125

C₄²

#121

yes

characteristic

126

C₄²

C₂²⋊Q₈

yes

characteristic

127

C₄²

#135

yes

—

128

C₁₆

—

129

C₂³

#418

yes

characteristic

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

#380

yes

characteristic

142

C₂×C₄

C₂×#101

yes

characteristic

143

C₂×C₄

#413

yes

characteristic

144

C₂×C₄

#416

yes

characteristic

145

C₂×C₄

#416

yes

characteristic

146

C₂×C₄

#419

yes

characteristic

147

C₂×C₄

#463

yes

—

148

C₂×C₄

#478

yes

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

#464

yes

—

155

C₈

#479

yes

—

156

C₂²

—

157

C₂²

#1553

yes

characteristic

158

C₂²

#1556

yes

characteristic

159

C₂²

#1804

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#1544

yes

characteristic

167

C₄

#1554

yes

characteristic

168

C₄

#1801

yes

characteristic

169

C₄

#1805

yes

characteristic

170

C₂

—

171

C₂

—

172

C₂

#8593

yes

characteristic

173

C₂

#12828

yes

—

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1804

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#418

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ 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
#21722	2
#12884	2
#11935	2
#8780	2
#7278	2

subgroup

index

Subgroups — #134648

Conjugacy classes of subgroups

Maximal subgroups