Group #140474 — subgroups

Subgroups — #140474

Order 512.

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

normal subgroups only

This group has 236 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	#11996	C₂	1	1	yes	maximal, characteristic
3	#8814	C₂	1	1	yes	maximal, characteristic
4	#8293	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	#1451	—	2	1	no	—
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1383	—	2	2	no	—
16	#1367	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	#322	—	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	2	no	—
55	#108	—	2	2	no	—
56	#107	—	8	4	no	—
57	#106	—	4	2	no	—
58	#106	—	4	2	no	—
59	#106	—	8	2	no	—
60	C₄⋊C₈	—	2	2	no	—
61	C₂²⋊C₈	—	4	1	no	—
62	#97	—	4	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	4	1	no	—
65	C₄×C₈	—	4	1	no	—
66	C₄×C₈	C₄⋊C₄	1	2	yes	—
67	C₄×C₈	C₄⋊C₄	1	2	yes	—
68	C₄×C₈	SD₁₆	1	1	yes	characteristic
69	C₄×C₈	SD₁₆	1	1	yes	characteristic
70	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
71	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
72	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₄×C₈	#41	1	1	yes	characteristic
74	C₂²×C₄	—	8	2	no	—
75	C₂²×C₄	#121	1	1	yes	characteristic
76	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
77	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
78	M₁₆	—	4	1	no	—
79	M₁₆	—	4	1	no	—
80	M₁₆	—	4	1	no	—
81	M₁₆	—	4	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	4	2	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	—	4	2	no	—
90	C₂×C₈	#106	1	2	yes	—
91	C₂×C₈	#106	1	2	yes	—
92	C₂×C₈	C₄×D₄	1	1	yes	characteristic
93	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₈	#125	1	1	yes	characteristic
95	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₈	#137	1	2	yes	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄⋊C₄	—	8	4	no	—
100	C₄⋊C₄	—	8	2	no	—
101	C₄⋊C₄	—	8	2	no	—
102	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
104	C₄²	C₂×SD₁₆	1	1	yes	characteristic
105	C₄²	#137	1	1	yes	characteristic
106	C₁₆	—	4	4	no	—
107	C₂³	#460	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	16	2	no	—
111	C₂×C₄	—	16	2	no	—
112	C₂×C₄	—	16	2	no	—
113	C₂×C₄	#389	1	1	yes	characteristic
114	C₂×C₄	C₂×#106	1	1	yes	characteristic
115	C₂×C₄	#428	1	1	yes	characteristic
116	C₂×C₄	#461	1	1	yes	characteristic
117	C₂×C₄	#469	1	1	yes	characteristic
118	C₂×C₄	#470	1	1	yes	characteristic
119	C₂×C₄	#472	1	1	yes	characteristic
120	C₂×C₄	#473	1	1	yes	characteristic
121	C₂×C₄	#475	1	1	yes	characteristic
122	C₂×C₄	#476	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	2	1	no	—
125	C₈	—	2	1	no	—
126	C₈	—	4	2	no	—
127	C₈	—	8	1	no	—
128	C₈	—	8	1	no	—
129	C₈	—	8	2	no	—
130	C₈	#365	1	4	yes	complemented
131	C₈	#440	1	2	yes	—
132	C₈	#501	1	2	yes	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#1688	1	1	yes	characteristic
136	C₂²	#1690	1	1	yes	characteristic
137	C₂²	#1948	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	32	2	no	—
141	C₄	#1693	1	1	yes	characteristic
142	C₄	#1695	1	1	yes	characteristic
143	C₄	#1944	1	1	yes	characteristic
144	C₄	#1952	1	1	yes	characteristic
145	C₄	C₂×#365	1	2	yes	—
146	C₄	#2231	1	2	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	#9621	1	1	yes	characteristic
150	C₂	#13183	1	1	yes	characteristic
151	C₂	#13187	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13183

C₂

yes

maximal

#11996

C₂

yes

maximal, characteristic

#8814

C₂

yes

maximal, characteristic

#8293

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

#1451

—

C₂×C₈²

C₂²

yes

characteristic

#1383

—

#1367

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

—

#322

—

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

—

—

—

—

—

—

—

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

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

C₂×C₄⋊C₄

yes

characteristic

103

C₄²

C₂²⋊Q₈

yes

characteristic

104

C₄²

C₂×SD₁₆

yes

characteristic

105

C₄²

#137

yes

characteristic

106

C₁₆

—

107

C₂³

#460

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#389

yes

characteristic

114

C₂×C₄

C₂×#106

yes

characteristic

115

C₂×C₄

#428

yes

characteristic

116

C₂×C₄

#461

yes

characteristic

117

C₂×C₄

#469

yes

characteristic

118

C₂×C₄

#470

yes

characteristic

119

C₂×C₄

#472

yes

characteristic

120

C₂×C₄

#473

yes

characteristic

121

C₂×C₄

#475

yes

characteristic

122

C₂×C₄

#476

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

#365

yes

complemented

131

C₈

#440

yes

—

132

C₈

#501

yes

—

133

C₂²

—

134

C₂²

—

135

C₂²

#1688

yes

characteristic

136

C₂²

#1690

yes

characteristic

137

C₂²

#1948

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#1693

yes

characteristic

142

C₄

#1695

yes

characteristic

143

C₄

#1944

yes

characteristic

144

C₄

#1952

yes

characteristic

145

C₄

C₂×#365

yes

—

146

C₄

#2231

yes

—

147

C₂

—

148

C₂

—

149

C₂

#9621

yes

characteristic

150

C₂

#13183

yes

characteristic

151

C₂

#13187

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. #1367	C₄	semidirect
130. C₈	#365	semidirect

H ≅ G/N

type

16. #1367

C₄

semidirect

130. C₈

#365

semidirect

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

subgroup	index
#13183	2
#11996	2
#8814	2
#8293	2
#7278	2

subgroup

index

Subgroups — #140474

Conjugacy classes of subgroups

Complements

Maximal subgroups