Group #140347 — subgroups

Subgroups — #140347

Order 512.

234 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 106 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	#13161	C₂	1	2	yes	maximal
2	#12670	C₂	1	1	yes	maximal, characteristic
3	C₂×#1373	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#7179	C₂	1	2	yes	maximal
6	C₂×#368	C₄	1	1	yes	characteristic
7	#2158	C₄	1	1	yes	characteristic
8	C₂×#364	C₂²	1	2	yes	—
9	#2151	—	2	2	no	—
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1772	C₂²	1	2	yes	—
13	C₂×C₈²	C₂²	1	1	yes	characteristic
14	#1382	C₄	1	2	yes	—
15	#1379	—	2	2	no	—
16	#1373	—	2	2	no	—
17	C₈×C₁₆	—	2	2	no	—
18	C₂×M₃₂	—	2	2	no	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	C₂×#108	C₂×C₄	1	2	yes	—
24	#427	C₂×C₄	1	2	yes	—
25	#405	—	2	2	no	—
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#368	—	2	2	no	—
29	#367	C₈	1	2	yes	complemented
30	#366	C₈	1	2	yes	complemented
31	#364	—	2	2	no	—
32	#364	—	2	2	no	—
33	C₄⋊C₁₆	—	4	4	no	—
34	C₄×C₁₆	—	2	1	no	—
35	C₄×C₁₆	—	2	1	no	—
36	C₄×C₁₆	D₄	1	2	yes	—
37	C₄×C₁₆	Q₈	1	2	yes	—
38	#339	—	2	2	no	—
39	C₈²	—	2	1	no	—
40	C₈²	Q₈	1	2	yes	—
41	C₂×M₁₆	—	2	2	no	—
42	C₂²×C₈	—	2	2	no	—
43	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
45	#117	—	2	2	no	—
46	C₂×C₄²	#41	1	1	yes	characteristic
47	M₃₂	—	4	2	no	—
48	M₃₂	—	4	2	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	4	2	no	—
56	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
57	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
58	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
59	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
60	C₂×C₁₆	M₁₆	1	2	yes	—
61	C₂×C₁₆	D₈	1	2	yes	—
62	C₂×C₁₆	Q₁₆	1	2	yes	—
63	#108	—	2	2	no	—
64	#108	—	2	2	no	—
65	#107	C₂×C₈	1	4	yes	—
66	#106	—	2	2	no	—
67	#97	—	4	2	no	—
68	C₄×C₈	—	2	2	no	—
69	C₄×C₈	—	2	2	no	—
70	C₄×C₈	—	4	2	no	—
71	C₄×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
77	M₁₆	—	4	2	no	—
78	M₁₆	—	4	2	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	8	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₈	#121	1	1	yes	characteristic
85	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
86	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₈	#131	1	1	yes	characteristic
89	C₂×C₈	#131	1	1	yes	characteristic
90	C₂×C₈	C₂×D₈	1	1	yes	characteristic
91	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×C₈	#135	1	1	yes	characteristic
93	C₂×C₈	#135	1	1	yes	characteristic
94	C₄⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	4	2	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	4	2	no	—
98	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
99	C₄²	#125	1	1	yes	characteristic
100	C₁₆	—	2	2	no	—
101	C₁₆	—	2	2	no	—
102	C₁₆	—	2	2	no	—
103	C₁₆	—	8	2	no	—
104	C₁₆	C₄⋊C₈	1	2	yes	—
105	C₁₆	C₄⋊C₈	1	2	yes	—
106	C₂³	#443	1	1	yes	characteristic
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	#389	1	1	yes	characteristic
112	C₂×C₄	C₈×D₄	1	1	yes	characteristic
113	C₂×C₄	#436	1	1	yes	characteristic
114	C₂×C₄	C₄×D₈	1	1	yes	characteristic
115	C₂×C₄	C₄×Q₁₆	1	1	yes	characteristic
116	C₂×C₄	#443	1	1	yes	characteristic
117	C₈	—	2	8	no	—
118	C₈	—	8	2	no	—
119	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
120	C₈	#424	1	1	yes	characteristic
121	C₈	C₈×Q₈	1	1	yes	characteristic
122	C₈	#446	1	1	yes	characteristic
123	C₈	#466	1	1	yes	characteristic
124	C₈	#467	1	1	yes	characteristic
125	C₈	#499	1	1	yes	characteristic
126	C₈	#500	1	1	yes	characteristic
127	C₂²	—	2	2	no	—
128	C₂²	C₈×D₈	1	1	yes	characteristic
129	C₂²	#1591	1	1	yes	characteristic
130	C₂²	#1945	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	8	4	no	—
134	C₄	C₈×Q₁₆	1	1	yes	characteristic
135	C₄	#1589	1	1	yes	characteristic
136	C₄	#1939	1	1	yes	characteristic
137	C₄	#1942	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	#9594	1	1	yes	characteristic
141	C₂	#12046	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13161

C₂

yes

maximal

#12670

C₂

yes

maximal, characteristic

C₂×#1373

C₂

yes

maximal, characteristic

C₂×C₈×C₁₆

C₂

yes

maximal, characteristic

#7179

C₂

yes

maximal

C₂×#368

C₄

yes

characteristic

#2158

C₄

yes

characteristic

C₂×#364

C₂²

yes

—

#2151

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1772

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1382

C₄

yes

—

#1379

—

#1373

—

C₈×C₁₆

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂×#108

C₂×C₄

yes

—

#427

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

#368

—

#367

C₈

yes

complemented

#366

C₈

yes

complemented

#364

—

#364

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

C₄×C₁₆

Q₈

yes

—

#339

—

C₈²

—

C₈²

Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₂×C₈

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

characteristic

C₂×C₁₆

M₁₆

yes

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

Q₁₆

yes

—

#108

—

#108

—

#107

C₂×C₈

yes

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

#125

yes

characteristic

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

103

C₁₆

—

104

C₁₆

C₄⋊C₈

yes

—

105

C₁₆

C₄⋊C₈

yes

—

106

C₂³

#443

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#389

yes

characteristic

112

C₂×C₄

C₈×D₄

yes

characteristic

113

C₂×C₄

#436

yes

characteristic

114

C₂×C₄

C₄×D₈

yes

characteristic

115

C₂×C₄

C₄×Q₁₆

yes

characteristic

116

C₂×C₄

#443

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

C₂×C₄⋊C₈

yes

characteristic

120

C₈

#424

yes

characteristic

121

C₈

C₈×Q₈

yes

characteristic

122

C₈

#446

yes

characteristic

123

C₈

#466

yes

characteristic

124

C₈

#467

yes

characteristic

125

C₈

#499

yes

characteristic

126

C₈

#500

yes

characteristic

127

C₂²

—

128

C₂²

C₈×D₈

yes

characteristic

129

C₂²

#1591

yes

characteristic

130

C₂²

#1945

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

C₈×Q₁₆

yes

characteristic

135

C₄

#1589

yes

characteristic

136

C₄

#1939

yes

characteristic

137

C₄

#1942

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

#9594

yes

characteristic

141

C₂

#12046

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₈×D₈
prime radical	C₂³	#443
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

C₂×D₈

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle

C₂²

C₈×D₈

prime radical

C₂³

#443

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
29. #367	C₈	semidirect
30. #366	C₈	semidirect

H ≅ G/N

type

29. #367

C₈

semidirect

30. #366

C₈

semidirect

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

subgroup	index
#13161	2
#12670	2
C₂×#1373	2
C₂×C₈×C₁₆	2
#7179	2

subgroup

index

C₂×C₈×C₁₆

Subgroups — #140347

Conjugacy classes of subgroups

Complements

Maximal subgroups