Group #141084 — subgroups

Subgroups — #141084

Order 512.

322 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 98 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	#33311	C₂	1	1	yes	maximal, characteristic
2	#12883	C₂	1	1	yes	maximal, characteristic
3	#11935	C₂	1	4	yes	maximal
4	#8267	C₂	1	1	yes	maximal, characteristic
5	C₂×#499	—	2	1	no	—
6	C₂×#499	C₂²	1	2	yes	—
7	#2859	—	2	1	no	—
8	#2150	C₂²	1	2	yes	—
9	#1846	—	2	1	no	—
10	#1846	C₄	1	2	yes	—
11	#1845	C₄	1	2	yes	—
12	#1844	—	2	1	no	—
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	#1564	C₂²	1	2	yes	—
15	#1452	—	2	1	no	—
16	#1376	—	2	1	no	—
17	#1376	—	2	1	no	—
18	#1363	—	2	8	no	—
19	#1276	—	2	1	no	—
20	C₂×#125	—	2	2	no	—
21	C₂×M₃₂	—	2	2	no	—
22	#499	—	2	4	no	—
23	#499	—	2	4	no	—
24	#499	—	4	2	no	—
25	#499	—	4	2	no	—
26	#431	—	2	2	no	—
27	C₂×#107	—	2	2	no	—
28	C₂×#107	C₂×C₄	1	4	yes	—
29	C₂×#106	—	2	2	no	—
30	C₂×#106	—	2	2	no	—
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
37	#389	—	4	4	no	—
38	#384	—	2	2	no	—
39	#382	—	2	2	no	—
40	C₄³	D₄	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	4	no	—
42	#335	—	2	2	no	—
43	#334	D₄	1	4	yes	—
44	#322	—	4	2	no	—
45	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	#125	—	4	4	no	—
50	#125	—	4	2	no	—
51	#125	—	4	2	no	—
52	C₂×C₄⋊C₄	—	2	4	no	—
53	C₂×C₄⋊C₄	—	4	4	no	—
54	C₂×C₄⋊C₄	—	4	4	no	—
55	C₂×C₄²	—	2	2	no	—
56	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	M₃₂	—	4	2	no	—
60	M₃₂	—	4	2	no	—
61	C₂×C₁₆	—	4	2	no	—
62	#107	—	2	8	no	—
63	#107	—	4	4	no	—
64	#106	—	4	4	no	—
65	#106	—	4	2	no	—
66	#106	—	4	2	no	—
67	C₄⋊C₈	—	2	4	no	—
68	C₂²⋊C₈	—	4	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	4	2	no	—
74	C₄×C₈	D₈	1	2	yes	—
75	C₄×C₈	SD₁₆	1	2	yes	—
76	C₄×C₈	C₂×D₄	1	2	yes	—
77	C₄×C₈	#41	1	2	yes	—
78	#95	—	4	4	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	2	1	no	—
81	C₂²×C₄	—	4	4	no	—
82	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	4	4	no	—
91	C₂×C₈	#102	1	2	yes	—
92	C₂×C₈	#102	1	2	yes	—
93	C₂×C₈	C₄×D₄	1	2	yes	—
94	C₂×C₈	#124	1	1	yes	characteristic
95	C₂×C₈	#127	1	1	yes	characteristic
96	C₂×C₈	#137	1	2	yes	—
97	C₂×C₈	#137	1	2	yes	—
98	C₄⋊C₄	—	4	4	no	—
99	C₄⋊C₄	—	4	4	no	—
100	C₄⋊C₄	—	8	8	no	—
101	C₄⋊C₄	—	8	4	no	—
102	C₄⋊C₄	—	8	4	no	—
103	C₄²	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	4	4	no	—
106	C₄²	#101	1	2	yes	—
107	C₄²	#101	1	2	yes	—
108	C₄²	#121	1	2	yes	—
109	C₄²	C₂×D₈	1	1	yes	characteristic
110	C₄²	C₂×SD₁₆	1	1	yes	characteristic
111	C₁₆	—	8	2	no	—
112	C₂³	#418	1	1	yes	characteristic
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	2	no	—
118	C₂×C₄	—	8	8	no	—
119	C₂×C₄	—	8	4	no	—
120	C₂×C₄	#390	1	1	yes	characteristic
121	C₂×C₄	C₂×#101	1	1	yes	characteristic
122	C₂×C₄	C₂×#101	1	1	yes	characteristic
123	C₂×C₄	C₂×#102	1	1	yes	characteristic
124	C₂×C₄	#419	1	1	yes	characteristic
125	C₂×C₄	#419	1	1	yes	characteristic
126	C₂×C₄	#459	1	2	yes	—
127	C₂×C₄	#460	1	2	yes	—
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	2	2	no	—
131	C₈	—	8	4	no	—
132	C₈	#491	1	2	yes	—
133	C₈	#497	1	2	yes	—
134	C₂²	—	2	2	no	—
135	C₂²	#1556	1	1	yes	characteristic
136	C₂²	#1556	1	1	yes	characteristic
137	C₂²	#1982	1	1	yes	characteristic
138	C₄	—	2	4	no	—
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	—	4	2	no	—
142	C₄	—	16	4	no	—
143	C₄	#1546	1	1	yes	characteristic
144	C₄	#1546	1	1	yes	characteristic
145	C₄	#1960	1	1	yes	characteristic
146	C₄	#1966	1	1	yes	characteristic
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#9705	1	1	yes	characteristic
150	C₂	#13259	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33311

C₂

yes

maximal, characteristic

#12883

C₂

yes

maximal, characteristic

#11935

C₂

yes

maximal

#8267

C₂

yes

maximal, characteristic

C₂×#499

—

C₂×#499

C₂²

yes

—

#2859

—

#2150

C₂²

yes

—

#1846

—

#1846

C₄

yes

—

#1845

C₄

yes

—

#1844

—

C₄²×C₈

C₂²

yes

characteristic

#1564

C₂²

yes

—

—

—

—

—

—

—

C₂×M₃₂

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×#106

—

C₂×#106

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄³

D₄

yes

characteristic

C₄⋊C₁₆

—

#335

—

#334

D₄

yes

—

#322

—

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×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

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

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

#101

yes

—

107

C₄²

#101

yes

—

108

C₄²

#121

yes

—

109

C₄²

C₂×D₈

yes

characteristic

110

C₄²

C₂×SD₁₆

yes

characteristic

111

C₁₆

—

112

C₂³

#418

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

#390

yes

characteristic

121

C₂×C₄

C₂×#101

yes

characteristic

122

C₂×C₄

C₂×#101

yes

characteristic

123

C₂×C₄

C₂×#102

yes

characteristic

124

C₂×C₄

#419

yes

characteristic

125

C₂×C₄

#419

yes

characteristic

126

C₂×C₄

#459

yes

—

127

C₂×C₄

#460

yes

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

#491

yes

—

133

C₈

#497

yes

—

134

C₂²

—

135

C₂²

#1556

yes

characteristic

136

C₂²

#1556

yes

characteristic

137

C₂²

#1982

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#1546

yes

characteristic

144

C₄

#1546

yes

characteristic

145

C₄

#1960

yes

characteristic

146

C₄

#1966

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

#9705

yes

characteristic

150

C₂

#13259

yes

—

	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
#33311	2
#12883	2
#11935	2
#8267	2

subgroup

index

Subgroups — #141084

Conjugacy classes of subgroups

Maximal subgroups