Group #140446 — subgroups

Subgroups — #140446

Order 512.

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

normal subgroups only

This group has 242 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	#13183	C₂	1	2	yes	maximal
2	#12004	C₂	1	1	yes	maximal, characteristic
3	#8814	C₂	1	1	yes	maximal, characteristic
4	C₈×#108	C₂	1	1	yes	maximal, characteristic
5	#7278	C₂	1	2	yes	maximal
6	C₂×#365	—	2	2	no	—
7	#2150	C₂²	1	2	yes	—
8	#1845	—	2	1	no	—
9	#1844	C₂²	1	2	yes	—
10	#1834	—	2	1	no	—
11	C₄×#108	C₂²	1	1	yes	characteristic
12	#1564	C₂²	1	1	yes	characteristic
13	C₈×M₁₆	—	2	1	no	—
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1383	—	2	2	no	—
16	#1370	C₄	1	4	yes	complemented
17	#1278	—	2	1	no	—
18	C₂×M₃₂	—	2	2	no	—
19	#431	—	2	1	no	—
20	C₂×#108	D₄	1	1	yes	characteristic
21	C₂×#108	D₄	1	1	yes	characteristic
22	C₂×#107	—	4	2	no	—
23	C₂×#106	—	2	2	no	—
24	C₂×#106	—	2	2	no	—
25	C₄×M₁₆	—	2	1	no	—
26	C₂×C₄×C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	#384	—	2	2	no	—
30	#365	—	4	4	no	—
31	C₄⋊C₁₆	—	2	2	no	—
32	C₄⋊C₁₆	C₂×C₄	1	4	yes	—
33	#341	—	2	1	no	—
34	#340	—	4	2	no	—
35	#335	Q₈	1	2	yes	—
36	#334	—	2	1	no	—
37	#322	—	4	1	no	—
38	C₈²	—	4	1	no	—
39	C₈²	C₂×C₄	1	2	yes	—
40	C₈²	Q₈	1	2	yes	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	—	2	1	no	—
43	C₂²×C₈	—	2	1	no	—
44	C₂²×C₈	SD₁₆	1	2	yes	—
45	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₈	#41	1	1	yes	characteristic
47	C₂×C₄⋊C₄	—	4	2	no	—
48	C₂×C₄⋊C₄	—	4	2	no	—
49	C₂×C₄⋊C₄	—	4	2	no	—
50	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
51	M₃₂	—	4	2	no	—
52	M₃₂	—	4	2	no	—
53	C₂×C₁₆	—	2	4	no	—
54	#108	—	2	1	no	—
55	#108	—	2	1	no	—
56	#108	D₈	1	2	yes	—
57	#108	Q₁₆	1	2	yes	—
58	#107	—	8	4	no	—
59	#106	—	4	2	no	—
60	#106	—	4	2	no	—
61	#106	—	8	2	no	—
62	C₄⋊C₈	—	2	2	no	—
63	C₂²⋊C₈	—	4	1	no	—
64	#97	—	4	1	no	—
65	C₄×C₈	—	2	1	no	—
66	C₄×C₈	—	4	1	no	—
67	C₄×C₈	—	4	1	no	—
68	C₄×C₈	C₄⋊C₄	1	2	yes	—
69	C₄×C₈	C₄⋊C₄	1	2	yes	—
70	C₄×C₈	SD₁₆	1	1	yes	characteristic
71	C₄×C₈	SD₁₆	1	1	yes	characteristic
72	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
75	C₄×C₈	#41	1	1	yes	characteristic
76	C₂²×C₄	—	8	2	no	—
77	C₂²×C₄	#121	1	1	yes	characteristic
78	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
79	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
80	M₁₆	—	2	2	no	—
81	M₁₆	—	2	2	no	—
82	M₁₆	—	4	1	no	—
83	M₁₆	—	4	1	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	1	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	4	2	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	2	no	—
92	C₂×C₈	#106	1	2	yes	—
93	C₂×C₈	#106	1	2	yes	—
94	C₂×C₈	C₄×D₄	1	1	yes	characteristic
95	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
96	C₂×C₈	#125	1	1	yes	characteristic
97	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₈	C₂×D₈	1	1	yes	characteristic
99	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₈	#135	1	1	yes	characteristic
101	C₂×C₈	#135	1	1	yes	characteristic
102	C₄⋊C₄	—	8	4	no	—
103	C₄⋊C₄	—	8	4	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₄⋊C₄	—	8	2	no	—
106	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
108	C₄²	C₂×SD₁₆	1	1	yes	characteristic
109	C₄²	#137	1	1	yes	characteristic
110	C₁₆	—	4	4	no	—
111	C₂³	#460	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	16	2	no	—
115	C₂×C₄	—	16	2	no	—
116	C₂×C₄	—	16	2	no	—
117	C₂×C₄	#389	1	1	yes	characteristic
118	C₂×C₄	C₂×#106	1	1	yes	characteristic
119	C₂×C₄	#428	1	1	yes	characteristic
120	C₂×C₄	#461	1	1	yes	characteristic
121	C₂×C₄	#466	1	1	yes	characteristic
122	C₂×C₄	#467	1	1	yes	characteristic
123	C₂×C₄	#472	1	1	yes	characteristic
124	C₂×C₄	#473	1	1	yes	characteristic
125	C₂×C₄	#475	1	1	yes	characteristic
126	C₂×C₄	#476	1	1	yes	characteristic
127	C₈	—	2	4	no	—
128	C₈	—	2	2	no	—
129	C₈	—	2	2	no	—
130	C₈	—	2	1	no	—
131	C₈	—	2	1	no	—
132	C₈	—	4	2	no	—
133	C₈	—	4	2	no	—
134	C₈	—	8	2	no	—
135	C₈	C₄×D₈	1	1	yes	characteristic
136	C₈	C₄×Q₁₆	1	1	yes	characteristic
137	C₈	#499	1	1	yes	characteristic
138	C₈	#500	1	1	yes	characteristic
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#1668	1	1	yes	characteristic
142	C₂²	#1670	1	1	yes	characteristic
143	C₂²	#1948	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	32	2	no	—
147	C₄	#1673	1	1	yes	characteristic
148	C₄	#1675	1	1	yes	characteristic
149	C₄	#1942	1	1	yes	characteristic
150	C₄	#1952	1	1	yes	characteristic
151	C₄	#2155	1	2	yes	—
152	C₄	#2230	1	2	yes	—
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	#9615	1	1	yes	characteristic
156	C₂	#13184	1	1	yes	characteristic
157	C₂	#13186	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13183

C₂

yes

maximal

#12004

C₂

yes

maximal, characteristic

#8814

C₂

yes

maximal, characteristic

C₈×#108

C₂

yes

maximal, characteristic

#7278

C₂

yes

maximal

C₂×#365

—

#2150

C₂²

yes

—

#1845

—

#1844

C₂²

yes

—

#1834

—

C₄×#108

C₂²

yes

characteristic

#1564

C₂²

yes

characteristic

C₈×M₁₆

—

C₂×C₈²

C₂²

yes

characteristic

#1383

—

#1370

C₄

yes

complemented

#1278

—

C₂×M₃₂

—

#431

—

C₂×#108

D₄

yes

characteristic

C₂×#108

D₄

yes

—

—

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#365

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

#341

—

#340

—

#335

Q₈

yes

—

#334

—

#322

—

C₈²

—

C₈²

C₂×C₄

yes

—

C₈²

Q₈

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

#108

—

#108

—

#108

D₈

yes

—

#108

Q₁₆

yes

—

—

—

—

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

C₄⋊C₄

yes

—

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#106

yes

—

C₂×C₈

#106

yes

—

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

100

C₂×C₈

#135

yes

characteristic

101

C₂×C₈

#135

yes

characteristic

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂×C₄⋊C₄

yes

characteristic

107

C₄²

C₂²⋊Q₈

yes

characteristic

108

C₄²

C₂×SD₁₆

yes

characteristic

109

C₄²

#137

yes

characteristic

110

C₁₆

—

111

C₂³

#460

yes

characteristic

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×#106

yes

characteristic

119

C₂×C₄

#428

yes

characteristic

120

C₂×C₄

#461

yes

characteristic

121

C₂×C₄

#466

yes

characteristic

122

C₂×C₄

#467

yes

characteristic

123

C₂×C₄

#472

yes

characteristic

124

C₂×C₄

#473

yes

characteristic

125

C₂×C₄

#475

yes

characteristic

126

C₂×C₄

#476

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

C₄×D₈

yes

characteristic

136

C₈

C₄×Q₁₆

yes

characteristic

137

C₈

#499

yes

characteristic

138

C₈

#500

yes

characteristic

139

C₂²

—

140

C₂²

—

141

C₂²

#1668

yes

characteristic

142

C₂²

#1670

yes

characteristic

143

C₂²

#1948

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#1673

yes

characteristic

148

C₄

#1675

yes

characteristic

149

C₄

#1942

yes

characteristic

150

C₄

#1952

yes

characteristic

151

C₄

#2155

yes

—

152

C₄

#2230

yes

—

153

C₂

—

154

C₂

—

155

C₂

#9615

yes

characteristic

156

C₂

#13184

yes

characteristic

157

C₂

#13186

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1948
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#460
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1948

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#460

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. #1370	C₄	semidirect

H ≅ G/N

type

16. #1370

C₄

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13183	2
#12004	2
#8814	2
C₈×#108	2
#7278	2

subgroup

index

Subgroups — #140446

Conjugacy classes of subgroups

Complements

Maximal subgroups