Group #134647 — subgroups

Subgroups — #134647

Order 512.

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

normal subgroups only

This group has 265 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	#21733	C₂	1	1	yes	maximal, characteristic
2	#12880	C₂	1	1	yes	maximal, characteristic
3	#11939	C₂	1	2	yes	maximal
4	#8780	C₂	1	1	yes	maximal, characteristic
5	#7276	C₂	1	2	yes	maximal
6	#3160	C₂²	1	1	yes	characteristic
7	#2961	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	#1995	—	2	1	no	—
10	#1946	C₄	1	2	yes	—
11	#1945	—	2	1	no	—
12	#1848	C₂²	1	1	yes	characteristic
13	#1773	C₄	1	2	yes	—
14	#1772	—	2	1	no	—
15	#1726	C₂²	1	1	yes	characteristic
16	#1564	C₂²	1	2	yes	—
17	#1374	—	2	1	no	—
18	#1374	—	2	1	no	—
19	#1365	—	2	2	no	—
20	#1364	—	2	2	no	—
21	#1281	—	2	1	no	—
22	#1274	—	2	1	no	—
23	#533	—	2	1	no	—
24	C₂²×C₁₆	—	2	2	no	—
25	#500	D₄	1	2	yes	—
26	#499	—	2	1	no	—
27	#499	Q₈	1	2	yes	—
28	#498	—	2	1	no	—
29	#446	—	4	4	no	—
30	#431	—	2	2	no	—
31	C₂×#108	—	2	1	no	—
32	C₂×#108	C₂×C₄	1	2	yes	—
33	#427	—	2	1	no	—
34	#427	C₂×C₄	1	2	yes	—
35	#424	—	2	2	no	—
36	#423	—	2	1	no	—
37	#405	—	2	2	no	—
38	C₂×#97	C₂×C₄	1	1	yes	characteristic
39	C₂×#97	C₂×C₄	1	1	yes	characteristic
40	C₂×#97	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#376	D₄	1	1	yes	characteristic
45	C₄⋊C₁₆	—	2	2	no	—
46	C₄⋊C₁₆	—	2	2	no	—
47	#339	—	2	2	no	—
48	#339	—	2	2	no	—
49	#335	—	2	2	no	—
50	#334	—	2	2	no	—
51	C₂×M₁₆	—	2	2	no	—
52	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
56	C₄⋊Q₈	—	2	2	no	—
57	#125	—	2	2	no	—
58	C₂²⋊Q₈	—	4	2	no	—
59	C₄×Q₈	—	4	2	no	—
60	#117	—	2	2	no	—
61	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₁₆	—	4	2	no	—
65	C₂×C₁₆	—	4	2	no	—
66	C₂×C₁₆	—	4	2	no	—
67	#108	—	4	2	no	—
68	#108	—	4	2	no	—
69	#107	—	2	2	no	—
70	#107	C₄⋊C₄	1	4	yes	—
71	#106	—	2	2	no	—
72	#106	—	2	2	no	—
73	C₄⋊C₈	—	4	4	no	—
74	C₄⋊C₈	—	4	2	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₂²⋊C₈	—	4	2	no	—
77	#97	—	2	2	no	—
78	#97	—	2	2	no	—
79	#97	—	4	1	no	—
80	#97	—	4	2	no	—
81	C₄×C₈	—	2	2	no	—
82	C₄×C₈	—	2	2	no	—
83	C₄×C₈	—	4	2	no	—
84	C₄×C₈	—	4	2	no	—
85	C₄×C₈	C₂×D₄	1	1	yes	characteristic
86	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
87	C₄×C₈	#41	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	#95	—	2	2	no	—
90	C₂×Q₈	—	4	2	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₄	#135	1	2	yes	—
95	M₁₆	—	4	2	no	—
96	M₁₆	—	4	2	no	—
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	—	4	2	no	—
105	C₂×C₈	—	8	2	no	—
106	C₂×C₈	#101	1	2	yes	—
107	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₈	#117	1	1	yes	characteristic
109	C₂×C₈	#121	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₈	#135	1	2	yes	—
112	C₂×C₈	#137	1	2	yes	—
113	C₄⋊C₄	—	2	4	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₂²⋊C₄	—	4	2	no	—
117	C₄²	—	2	1	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	4	2	no	—
122	C₄²	#121	1	1	yes	characteristic
123	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
124	C₄²	AGL(1,ℤ/8)	1	2	yes	—
125	C₁₆	—	8	2	no	—
126	C₂³	#416	1	1	yes	characteristic
127	Q₈	—	16	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	#380	1	1	yes	characteristic
135	C₂×C₄	C₂×#101	1	1	yes	characteristic
136	C₂×C₄	#413	1	1	yes	characteristic
137	C₂×C₄	#416	1	1	yes	characteristic
138	C₂×C₄	#418	1	1	yes	characteristic
139	C₂×C₄	#419	1	1	yes	characteristic
140	C₂×C₄	#463	1	2	yes	—
141	C₂×C₄	#478	1	2	yes	—
142	C₈	—	2	1	no	—
143	C₈	—	2	1	no	—
144	C₈	—	2	2	no	—
145	C₈	—	4	1	no	—
146	C₈	—	8	2	no	—
147	C₈	—	16	2	no	—
148	C₈	#464	1	2	yes	—
149	C₈	#479	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	#1544	1	1	yes	characteristic
152	C₂²	#1554	1	1	yes	characteristic
153	C₂²	#1804	1	1	yes	characteristic
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	8	1	no	—
158	C₄	—	8	4	no	—
159	C₄	#1553	1	1	yes	characteristic
160	C₄	#1556	1	1	yes	characteristic
161	C₄	#1801	1	1	yes	characteristic
162	C₄	#1805	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#8593	1	1	yes	characteristic
166	C₂	#12829	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21733

C₂

yes

maximal, characteristic

#12880

C₂

yes

maximal, characteristic

#11939

C₂

yes

maximal

#8780

C₂

yes

maximal, characteristic

#7276

C₂

yes

maximal

#3160

C₂²

yes

characteristic

#2961

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1995

—

#1946

C₄

yes

—

#1945

—

#1848

C₂²

yes

characteristic

#1773

C₄

yes

—

#1772

—

#1726

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

—

—

—

—

C₂²×C₁₆

—

#500

D₄

yes

—

#499

—

#499

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

#427

—

#427

C₂×C₄

yes

—

#424

—

#423

—

#405

—

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

characteristic

#376

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

C₄⋊C₄

yes

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

#135

yes

—

M₁₆

—

M₁₆

—

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

#101

yes

—

107

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

108

C₂×C₈

#117

yes

characteristic

109

C₂×C₈

#121

yes

characteristic

110

C₂×C₈

C₂²⋊Q₈

yes

characteristic

111

C₂×C₈

#135

yes

—

112

C₂×C₈

#137

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

#121

yes

characteristic

123

C₄²

C₂²⋊Q₈

yes

characteristic

124

C₄²

AGL(1,ℤ/8)

yes

—

125

C₁₆

—

126

C₂³

#416

yes

characteristic

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#380

yes

characteristic

135

C₂×C₄

C₂×#101

yes

characteristic

136

C₂×C₄

#413

yes

characteristic

137

C₂×C₄

#416

yes

characteristic

138

C₂×C₄

#418

yes

characteristic

139

C₂×C₄

#419

yes

characteristic

140

C₂×C₄

#463

yes

—

141

C₂×C₄

#478

yes

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

#464

yes

—

149

C₈

#479

yes

—

150

C₂²

—

151

C₂²

#1544

yes

characteristic

152

C₂²

#1554

yes

characteristic

153

C₂²

#1804

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

#1553

yes

characteristic

160

C₄

#1556

yes

characteristic

161

C₄

#1801

yes

characteristic

162

C₄

#1805

yes

characteristic

163

C₂

—

164

C₂

—

165

C₂

#8593

yes

characteristic

166

C₂

#12829

yes

—

	subgroup	G/H
center	C₂×C₄	#418
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1804
prime radical	C₂³	#416
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#418

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ 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
#21733	2
#12880	2
#11939	2
#8780	2
#7276	2

subgroup

index

Subgroups — #134647

Conjugacy classes of subgroups

Maximal subgroups