Group #141086 — subgroups

Subgroups — #141086

Order 512.

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

normal subgroups only

This group has 366 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	#33312	C₂	1	1	yes	maximal, characteristic
2	#12883	C₂	1	1	yes	maximal, characteristic
3	#11941	C₂	1	2	yes	maximal
4	#11940	C₂	1	2	yes	maximal
5	#8267	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	—	2	1	no	—
7	C₂×#500	C₂²	1	2	yes	—
8	#2868	—	2	1	no	—
9	#2150	C₂²	1	2	yes	—
10	#1845	—	2	2	no	—
11	#1845	C₄	1	4	yes	—
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	#1564	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	1	yes	characteristic
15	#1452	—	2	1	no	—
16	#1376	—	2	2	no	—
17	#1365	—	2	4	no	—
18	#1364	—	2	4	no	—
19	#1276	—	2	1	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	4	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	#500	—	2	2	no	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	—	4	2	no	—
28	#500	—	4	2	no	—
29	#431	—	2	1	no	—
30	#431	—	2	1	no	—
31	C₂×#107	—	2	4	no	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	C₂×C₄	1	4	yes	—
34	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
35	C₂×C₄×C₈	D₄	1	2	yes	—
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	#384	—	2	4	no	—
39	#384	—	4	2	no	—
40	C₄³	D₄	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄⋊C₁₆	—	2	2	no	—
43	#335	D₄	1	2	yes	—
44	#335	D₄	1	2	yes	—
45	#334	—	2	1	no	—
46	#334	—	2	1	no	—
47	#322	—	4	2	no	—
48	C₂²×Q₈	—	4	2	no	—
49	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₄⋊Q₈	—	4	2	no	—
53	C₄⋊Q₈	—	4	2	no	—
54	C₄⋊Q₈	—	4	2	no	—
55	C₄⋊Q₈	—	4	2	no	—
56	C₄⋊Q₈	—	8	8	no	—
57	C₂×C₄⋊C₄	—	2	4	no	—
58	C₂×C₄⋊C₄	—	4	8	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄²	—	2	1	no	—
61	C₂×C₄²	—	2	1	no	—
62	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	M₃₂	—	4	4	no	—
65	C₂×C₁₆	—	4	2	no	—
66	#107	—	2	4	no	—
67	#107	—	2	4	no	—
68	#107	—	4	8	no	—
69	#107	—	4	2	no	—
70	#107	—	4	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₄⋊C₈	—	2	2	no	—
73	C₂²⋊C₈	—	4	1	no	—
74	C₂²⋊C₈	—	4	1	no	—
75	C₄×C₈	—	2	4	no	—
76	C₄×C₈	—	2	4	no	—
77	C₄×C₈	—	4	1	no	—
78	C₄×C₈	—	4	1	no	—
79	C₄×C₈	D₈	1	2	yes	—
80	C₄×C₈	SD₁₆	1	2	yes	—
81	C₄×C₈	C₂×D₄	1	1	yes	characteristic
82	C₄×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	C₄×C₈	#41	1	1	yes	characteristic
85	C₂×Q₈	—	8	8	no	—
86	C₂×Q₈	—	8	2	no	—
87	C₂×Q₈	—	8	2	no	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	4	4	no	—
90	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
92	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	#102	1	4	yes	—
99	C₂×C₈	C₄×D₄	1	1	yes	characteristic
100	C₂×C₈	C₄×D₄	1	1	yes	characteristic
101	C₂×C₈	#124	1	1	yes	characteristic
102	C₂×C₈	#127	1	1	yes	characteristic
103	C₂×C₈	#137	1	4	yes	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	8	16	no	—
107	C₄⋊C₄	—	8	2	no	—
108	C₄⋊C₄	—	8	2	no	—
109	C₄²	—	2	4	no	—
110	C₄²	—	2	4	no	—
111	C₄²	—	2	2	no	—
112	C₄²	#101	1	4	yes	—
113	C₄²	#121	1	1	yes	characteristic
114	C₄²	#121	1	1	yes	characteristic
115	C₄²	C₂×D₈	1	1	yes	characteristic
116	C₄²	C₂×SD₁₆	1	1	yes	characteristic
117	C₁₆	—	8	2	no	—
118	C₂³	#418	1	1	yes	characteristic
119	Q₈	—	16	4	no	—
120	Q₈	—	16	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	8	8	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	#390	1	1	yes	characteristic
129	C₂×C₄	C₂×#101	1	2	yes	—
130	C₂×C₄	C₂×#102	1	1	yes	characteristic
131	C₂×C₄	#419	1	2	yes	—
132	C₂×C₄	#459	1	1	yes	characteristic
133	C₂×C₄	#459	1	1	yes	characteristic
134	C₂×C₄	#460	1	1	yes	characteristic
135	C₂×C₄	#460	1	1	yes	characteristic
136	C₈	—	2	4	no	—
137	C₈	—	2	1	no	—
138	C₈	—	2	1	no	—
139	C₈	—	8	2	no	—
140	C₈	—	8	2	no	—
141	C₈	#491	1	2	yes	—
142	C₈	#497	1	2	yes	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#1556	1	2	yes	—
146	C₂²	#1982	1	1	yes	characteristic
147	C₄	—	2	8	no	—
148	C₄	—	2	1	no	—
149	C₄	—	2	1	no	—
150	C₄	—	16	4	no	—
151	C₄	#1546	1	2	yes	—
152	C₄	#1960	1	1	yes	characteristic
153	C₄	#1966	1	1	yes	characteristic
154	C₄	#2213	1	2	yes	—
155	C₄	#2214	1	2	yes	—
156	C₂	—	2	2	no	—
157	C₂	#9705	1	1	yes	characteristic
158	C₂	#13261	1	1	yes	characteristic
159	C₂	#13262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33312

C₂

yes

maximal, characteristic

#12883

C₂

yes

maximal, characteristic

#11941

C₂

yes

maximal

#11940

C₂

yes

maximal

#8267

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

C₂²

yes

—

#2868

—

#2150

C₂²

yes

—

#1845

—

#1845

C₄

yes

—

C₄²×C₈

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

—

—

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#384

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#335

D₄

yes

—

#335

D₄

yes

—

#334

—

#334

—

#322

—

C₂²×Q₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

C₄×D₄

yes

characteristic

100

C₂×C₈

C₄×D₄

yes

characteristic

101

C₂×C₈

#124

yes

characteristic

102

C₂×C₈

#127

yes

characteristic

103

C₂×C₈

#137

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

#101

yes

—

113

C₄²

#121

yes

characteristic

114

C₄²

#121

yes

characteristic

115

C₄²

C₂×D₈

yes

characteristic

116

C₄²

C₂×SD₁₆

yes

characteristic

117

C₁₆

—

118

C₂³

#418

yes

characteristic

119

Q₈

—

120

Q₈

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#390

yes

characteristic

129

C₂×C₄

C₂×#101

yes

—

130

C₂×C₄

C₂×#102

yes

characteristic

131

C₂×C₄

#419

yes

—

132

C₂×C₄

#459

yes

characteristic

133

C₂×C₄

#459

yes

characteristic

134

C₂×C₄

#460

yes

characteristic

135

C₂×C₄

#460

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

#491

yes

—

142

C₈

#497

yes

—

143

C₂²

—

144

C₂²

—

145

C₂²

#1556

yes

—

146

C₂²

#1982

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

#1546

yes

—

152

C₄

#1960

yes

characteristic

153

C₄

#1966

yes

characteristic

154

C₄

#2213

yes

—

155

C₄

#2214

yes

—

156

C₂

—

157

C₂

#9705

yes

characteristic

158

C₂

#13261

yes

characteristic

159

C₂

#13262

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1982
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 ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1982

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 ◁ 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
#33312	2
#12883	2
#11941	2
#11940	2
#8267	2

subgroup

index

Subgroups — #141086

Conjugacy classes of subgroups

Maximal subgroups