Group #140351 — subgroups

Subgroups — #140351

Order 512.

260 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 122 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	#13159	C₂	1	2	yes	maximal
2	C₂×#1367	C₂	1	1	yes	maximal, characteristic
3	C₂×#1312	C₂	1	1	yes	maximal, characteristic
4	C₈×#107	C₂	1	1	yes	maximal, characteristic
5	#7167	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₁₆	—	2	2	no	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×#107	C₄	1	1	yes	characteristic
11	C₈×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×#335	C₄	1	1	yes	characteristic
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1374	—	2	2	no	—
15	#1367	—	2	2	no	—
16	#1312	—	2	2	no	—
17	#1279	C₄	1	2	yes	—
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	—	2	2	no	—
20	C₂²×C₁₆	D₄	1	2	yes	—
21	C₂×#107	C₈	1	2	yes	—
22	C₂×C₄⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₈	—	2	2	no	—
24	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄×C₄⋊C₄	C₂×C₄	1	2	yes	—
28	C₄⋊C₁₆	—	2	2	no	—
29	C₄⋊C₁₆	—	2	2	no	—
30	C₄⋊C₁₆	—	4	4	no	—
31	C₄×C₁₆	—	2	1	no	—
32	C₄×C₁₆	—	2	1	no	—
33	C₄×C₁₆	—	2	1	no	—
34	C₄×C₁₆	Q₈	1	2	yes	—
35	#340	C₈	1	1	yes	characteristic
36	#340	C₈	1	1	yes	characteristic
37	#336	—	2	2	no	—
38	#335	—	2	2	no	—
39	C₈²	D₄	1	2	yes	—
40	C₈²	Q₈	1	2	yes	—
41	C₂²×C₈	—	2	1	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	#41	1	1	yes	characteristic
47	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
48	C₂×C₄⋊C₄	C₂×C₈	1	2	yes	—
49	C₂×C₄²	—	2	2	no	—
50	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	2	2	no	—
54	C₂×C₁₆	—	2	2	no	—
55	C₂×C₁₆	—	4	1	no	—
56	C₂×C₁₆	—	4	1	no	—
57	C₂×C₁₆	—	4	2	no	—
58	C₂×C₁₆	—	4	2	no	—
59	C₂×C₁₆	—	4	2	no	—
60	#107	—	2	4	no	—
61	C₄⋊C₈	—	2	2	no	—
62	C₄⋊C₈	—	2	2	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	4	2	no	—
65	C₄×C₈	—	4	2	no	—
66	C₄×C₈	C₄⋊C₄	1	2	yes	—
67	C₄×C₈	C₄⋊C₄	1	2	yes	—
68	C₄×C₈	M₁₆	1	2	yes	—
69	C₄×C₈	C₂×D₄	1	1	yes	characteristic
70	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₄×C₈	#41	1	1	yes	characteristic
73	#95	—	2	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂²×C₈	1	1	yes	characteristic
78	C₂²×C₄	#131	1	1	yes	characteristic
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	4	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	C₄⋊C₈	1	2	yes	—
89	C₂×C₈	C₄⋊C₈	1	2	yes	—
90	C₂×C₈	#121	1	1	yes	characteristic
91	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
92	C₂×C₈	#125	1	1	yes	characteristic
93	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
94	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
95	C₂×C₈	#137	1	2	yes	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	M₃₂	1	4	yes	—
100	C₄²	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₄²	C₄×Q₈	1	1	yes	characteristic
104	C₄²	C₂×M₁₆	1	1	yes	characteristic
105	C₄²	#131	1	1	yes	characteristic
106	C₁₆	—	4	2	no	—
107	C₁₆	—	4	2	no	—
108	C₁₆	—	8	2	no	—
109	C₂³	C₈×D₄	1	1	yes	characteristic
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	#389	1	1	yes	characteristic
118	C₂×C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
119	C₂×C₄	#424	1	1	yes	characteristic
120	C₂×C₄	#436	1	1	yes	characteristic
121	C₂×C₄	#441	1	1	yes	characteristic
122	C₂×C₄	#442	1	1	yes	characteristic
123	C₂×C₄	#444	1	1	yes	characteristic
124	C₂×C₄	#444	1	1	yes	characteristic
125	C₂×C₄	C₈×Q₈	1	1	yes	characteristic
126	C₂×C₄	#446	1	1	yes	characteristic
127	C₂×C₄	C₂×M₃₂	1	2	yes	—
128	C₂×C₄	#504	1	2	yes	—
129	C₈	—	2	4	no	—
130	C₈	—	2	2	no	—
131	C₈	—	2	2	no	—
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	8	2	no	—
135	C₈	#468	1	2	yes	—
136	C₈	#501	1	2	yes	—
137	C₂²	#1579	1	1	yes	characteristic
138	C₂²	#1580	1	1	yes	characteristic
139	C₂²	#1939	1	1	yes	characteristic
140	C₂²	#2168	1	2	yes	—
141	C₂²	#2170	1	2	yes	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	—	8	2	no	—
146	C₄	#1593	1	1	yes	characteristic
147	C₄	#1594	1	1	yes	characteristic
148	C₄	#1943	1	1	yes	characteristic
149	C₄	#1946	1	1	yes	characteristic
150	C₄	#2150	1	1	yes	characteristic
151	C₄	#2151	1	1	yes	characteristic
152	C₄	#2183	1	2	yes	—
153	C₂	#9595	1	1	yes	characteristic
154	C₂	#12060	1	2	yes	—
155	C₂	#12063	1	2	yes	—
156	C₂	#13160	1	1	yes	characteristic
157	C₂	#13161	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13159

C₂

yes

maximal

C₂×#1367

C₂

yes

maximal, characteristic

C₂×#1312

C₂

yes

maximal, characteristic

C₈×#107

C₂

yes

maximal, characteristic

#7167

C₂

yes

maximal

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₄×#107

C₄

yes

characteristic

C₈×C₄⋊C₄

C₂²

yes

—

C₂×#335

C₄

yes

characteristic

C₂×C₈²

C₂²

yes

—

—

—

C₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

C₂×#107

C₈

yes

—

C₂×C₄⋊C₈

C₂×C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

C₂×C₄

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

Q₈

yes

—

#340

C₈

yes

characteristic

#340

C₈

yes

characteristic

#336

—

#335

—

C₈²

D₄

yes

—

C₈²

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×C₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄⋊C₄

C₂×C₈

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

C₂²×C₄

C₂²×C₈

yes

characteristic

C₂²×C₄

#131

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

C₄⋊C₈

yes

—

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

M₃₂

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂×C₄⋊C₄

yes

characteristic

103

C₄²

C₄×Q₈

yes

characteristic

104

C₄²

C₂×M₁₆

yes

characteristic

105

C₄²

#131

yes

characteristic

106

C₁₆

—

107

C₁₆

—

108

C₁₆

—

109

C₂³

C₈×D₄

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#389

yes

characteristic

118

C₂×C₄

C₂×C₄⋊C₈

yes

characteristic

119

C₂×C₄

#424

yes

characteristic

120

C₂×C₄

#436

yes

characteristic

121

C₂×C₄

#441

yes

characteristic

122

C₂×C₄

#442

yes

characteristic

123

C₂×C₄

#444

yes

characteristic

124

C₂×C₄

#444

yes

characteristic

125

C₂×C₄

C₈×Q₈

yes

characteristic

126

C₂×C₄

#446

yes

characteristic

127

C₂×C₄

C₂×M₃₂

yes

—

128

C₂×C₄

#504

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

#468

yes

—

136

C₈

#501

yes

—

137

C₂²

#1579

yes

characteristic

138

C₂²

#1580

yes

characteristic

139

C₂²

#1939

yes

characteristic

140

C₂²

#2168

yes

—

141

C₂²

#2170

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

#1593

yes

characteristic

147

C₄

#1594

yes

characteristic

148

C₄

#1943

yes

characteristic

149

C₄

#1946

yes

characteristic

150

C₄

#2150

yes

characteristic

151

C₄

#2151

yes

characteristic

152

C₄

#2183

yes

—

153

C₂

#9595

yes

characteristic

154

C₂

#12060

yes

—

155

C₂

#12063

yes

—

156

C₂

#13160

yes

characteristic

157

C₂

#13161

yes

characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₈

C₂×D₄

commutator

C₂²×C₄

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₈×D₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆ ◁ 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
#13159	2
C₂×#1367	2
C₂×#1312	2
C₈×#107	2
#7167	2

subgroup

index

Subgroups — #140351

Conjugacy classes of subgroups

Maximal subgroups