Group #141081 — subgroups

Subgroups — #141081

Order 512.

384 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 126 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	#12879	C₂	1	1	yes	maximal, characteristic
3	C₂×#1364	C₂	1	4	yes	maximal
4	#8274	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	—	2	1	no	—
6	C₂×#500	C₂²	1	2	yes	—
7	#2868	—	2	1	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	#1845	—	2	2	no	—
10	#1845	C₄	1	4	yes	—
11	#1767	—	2	1	no	—
12	C₄²×C₈	C₂²	1	1	yes	characteristic
13	C₂×#335	C₂²	1	2	yes	—
14	#1374	—	2	2	no	—
15	#1364	—	2	8	no	—
16	#1277	—	2	1	no	—
17	C₂×C₄⋊Q₈	—	2	2	no	—
18	C₂×C₄⋊Q₈	—	4	1	no	—
19	C₂×C₄⋊Q₈	—	4	1	no	—
20	C₂²×C₁₆	—	2	2	no	—
21	#500	—	2	4	no	—
22	#500	—	2	4	no	—
23	#500	—	4	4	no	—
24	C₂×#107	—	2	4	no	—
25	C₂×#107	—	2	2	no	—
26	C₂×#107	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₈	—	2	2	no	—
28	C₂×#97	—	4	1	no	—
29	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄×C₈	D₄	1	2	yes	—
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	#384	—	2	4	no	—
34	#384	—	4	2	no	—
35	C₄³	D₄	1	1	yes	characteristic
36	C₄⋊C₁₆	—	2	4	no	—
37	#336	—	4	1	no	—
38	#335	—	2	2	no	—
39	#335	D₄	1	4	yes	—
40	C₂²×Q₈	—	4	2	no	—
41	C₂²×C₈	—	4	2	no	—
42	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Q₈	—	4	4	no	—
46	C₄⋊Q₈	—	4	4	no	—
47	C₄⋊Q₈	—	8	4	no	—
48	C₄⋊Q₈	—	8	2	no	—
49	C₄⋊Q₈	—	8	2	no	—
50	C₂×C₄⋊C₄	—	2	4	no	—
51	C₂×C₄⋊C₄	—	4	4	no	—
52	C₂×C₄⋊C₄	—	4	4	no	—
53	C₂×C₄⋊C₄	—	4	2	no	—
54	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄²	D₈	1	1	yes	characteristic
56	C₂×C₄²	D₈	1	1	yes	characteristic
57	C₂×C₄²	SD₁₆	1	2	yes	—
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	C₂×C₁₆	—	4	2	no	—
60	C₂×C₁₆	—	4	4	no	—
61	#107	—	2	8	no	—
62	#107	—	4	8	no	—
63	#107	—	4	2	no	—
64	#107	—	4	2	no	—
65	C₄⋊C₈	—	2	4	no	—
66	C₄⋊C₈	—	4	2	no	—
67	#97	—	8	2	no	—
68	C₄×C₈	—	2	4	no	—
69	C₄×C₈	—	2	4	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	C₂×D₄	1	2	yes	—
72	C₄×C₈	#41	1	2	yes	—
73	C₂×Q₈	—	8	4	no	—
74	C₂×Q₈	—	8	2	no	—
75	C₂×Q₈	—	8	2	no	—
76	C₂×Q₈	—	8	4	no	—
77	C₂²×C₄	—	4	4	no	—
78	C₂²×C₄	#102	1	2	yes	—
79	C₂²×C₄	#102	1	2	yes	—
80	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
82	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	4	4	no	—
87	C₂×C₈	—	8	2	no	—
88	C₂×C₈	—	8	2	no	—
89	C₂×C₈	#101	1	4	yes	—
90	C₂×C₈	C₄×D₄	1	2	yes	—
91	C₂×C₈	#124	1	1	yes	characteristic
92	C₂×C₈	#127	1	1	yes	characteristic
93	C₂×C₈	#137	1	4	yes	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	—	4	4	no	—
96	C₄⋊C₄	—	8	8	no	—
97	C₄⋊C₄	—	8	8	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄²	—	2	4	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	D₁₆	1	2	yes	—
104	C₄²	SD₃₂	1	2	yes	—
105	C₄²	#121	1	2	yes	—
106	C₄²	AGL(1,ℤ/8)	1	2	yes	—
107	C₁₆	—	8	2	no	—
108	C₂³	C₂×#102	1	1	yes	characteristic
109	Q₈	—	16	4	no	—
110	Q₈	—	16	4	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	8	4	no	—
117	C₂×C₄	—	8	4	no	—
118	C₂×C₄	—	8	4	no	—
119	C₂×C₄	#357	1	4	yes	—
120	C₂×C₄	#362	1	4	yes	—
121	C₂×C₄	#390	1	1	yes	characteristic
122	C₂×C₄	C₂×#101	1	2	yes	—
123	C₂×C₄	#418	1	1	yes	characteristic
124	C₂×C₄	#419	1	2	yes	—
125	C₂×C₄	#459	1	2	yes	—
126	C₂×C₄	#460	1	2	yes	—
127	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
128	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
129	C₂×C₄	#509	1	2	yes	—
130	C₈	—	2	4	no	—
131	C₈	—	2	2	no	—
132	C₈	—	8	4	no	—
133	C₈	#491	1	2	yes	—
134	C₈	#497	1	2	yes	—
135	C₂²	#1546	1	2	yes	—
136	C₂²	#1982	1	1	yes	characteristic
137	C₂²	C₂×#357	1	1	yes	characteristic
138	C₂²	#2140	1	1	yes	characteristic
139	C₂²	C₂×#362	1	2	yes	—
140	C₄	—	2	4	no	—
141	C₄	—	2	4	no	—
142	C₄	—	2	2	no	—
143	C₄	—	16	4	no	—
144	C₄	#1556	1	2	yes	—
145	C₄	#1960	1	1	yes	characteristic
146	C₄	#1966	1	1	yes	characteristic
147	C₄	#2206	1	2	yes	—
148	C₄	#2207	1	2	yes	—
149	C₂	#9705	1	1	yes	characteristic
150	C₂	#11971	1	2	yes	—
151	C₂	#11974	1	2	yes	—
152	C₂	#13252	1	1	yes	characteristic
153	C₂	#13256	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33312

C₂

yes

maximal, characteristic

#12879

C₂

yes

maximal, characteristic

C₂×#1364

C₂

yes

maximal

#8274

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

C₂²

yes

—

#2868

—

C₂×C₄⋊C₁₆

C₂²

yes

—

#1845

—

#1845

C₄

yes

—

#1767

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

#1374

—

#1364

—

#1277

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

—

—

—

—

—

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

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

—

#336

—

#335

—

#335

D₄

yes

—

C₂²×Q₈

—

C₂²×C₈

—

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×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

SD₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#102

yes

—

C₂²×C₄

#102

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#101

yes

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

D₁₆

yes

—

104

C₄²

SD₃₂

yes

—

105

C₄²

#121

yes

—

106

C₄²

AGL(1,ℤ/8)

yes

—

107

C₁₆

—

108

C₂³

C₂×#102

yes

characteristic

109

Q₈

—

110

Q₈

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#357

yes

—

120

C₂×C₄

#362

yes

—

121

C₂×C₄

#390

yes

characteristic

122

C₂×C₄

C₂×#101

yes

—

123

C₂×C₄

#418

yes

characteristic

124

C₂×C₄

#419

yes

—

125

C₂×C₄

#459

yes

—

126

C₂×C₄

#460

yes

—

127

C₂×C₄

C₂×D₁₆

yes

characteristic

128

C₂×C₄

C₂×SD₃₂

yes

characteristic

129

C₂×C₄

#509

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

#491

yes

—

134

C₈

#497

yes

—

135

C₂²

#1546

yes

—

136

C₂²

#1982

yes

characteristic

137

C₂²

C₂×#357

yes

characteristic

138

C₂²

#2140

yes

characteristic

139

C₂²

C₂×#362

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#1556

yes

—

145

C₄

#1960

yes

characteristic

146

C₄

#1966

yes

characteristic

147

C₄

#2206

yes

—

148

C₄

#2207

yes

—

149

C₂

#9705

yes

characteristic

150

C₂

#11971

yes

—

151

C₂

#11974

yes

—

152

C₂

#13252

yes

characteristic

153

C₂

#13256

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×#102

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ 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
#12879	2
C₂×#1364	2
#8274	2

subgroup

index

Subgroups — #141081

Conjugacy classes of subgroups

Maximal subgroups