Group #129164 — subgroups

Subgroups — #129164

Order 512.

234 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 130 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	#12671	C₂	1	1	yes	maximal, characteristic
2	#12665	C₂	1	2	yes	maximal
3	C₂×#1369	C₂	1	1	yes	maximal, characteristic
4	C₂×#1313	C₂	1	1	yes	maximal, characteristic
5	#7182	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	#2108	—	2	2	no	—
10	C₂×#346	C₂²	1	1	yes	characteristic
11	C₂×#346	C₂²	1	1	yes	characteristic
12	#1772	C₂²	1	2	yes	—
13	C₂×#322	C₂²	1	1	yes	characteristic
14	#1382	C₄	1	2	yes	—
15	#1379	—	2	2	no	—
16	#1369	C₄	1	4	yes	complemented
17	#1313	C₄	1	4	yes	complemented
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	2	yes	—
22	C₂×#108	C₂×C₄	1	2	yes	—
23	#427	C₂×C₄	1	2	yes	—
24	#405	—	2	2	no	—
25	C₂×C₄×C₈	—	2	1	no	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	#368	C₈	1	4	yes	complemented
28	#367	C₈	1	2	yes	complemented
29	#366	C₈	1	2	yes	complemented
30	#364	—	2	2	no	—
31	#364	C₂×C₄	1	4	yes	—
32	#346	—	4	2	no	—
33	#346	C₂×C₄	1	2	yes	—
34	#346	C₂×C₄	1	2	yes	—
35	#346	C₂×C₄	1	2	yes	—
36	#346	Q₈	1	2	yes	—
37	C₄×C₁₆	—	4	2	no	—
38	#339	—	2	2	no	—
39	#322	—	2	1	no	—
40	#322	C₂×C₄	1	2	yes	—
41	C₂×M₁₆	—	2	2	no	—
42	C₂²×C₈	—	4	1	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	—	2	4	no	—
48	M₃₂	—	4	2	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	—	2	2	no	—
53	C₂×C₁₆	—	4	2	no	—
54	C₂×C₁₆	C₄²	1	2	yes	—
55	C₂×C₁₆	C₄²	1	2	yes	—
56	C₂×C₁₆	C₄⋊C₄	1	4	yes	—
57	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
58	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
59	C₂×C₁₆	C₂×C₈	1	2	yes	—
60	#108	—	2	2	no	—
61	#108	C₂×C₈	1	4	yes	—
62	#107	C₂×C₈	1	4	yes	—
63	#106	—	2	2	no	—
64	#97	—	4	2	no	—
65	C₄×C₈	—	2	2	no	—
66	C₄×C₈	—	4	1	no	—
67	C₄×C₈	—	4	2	no	—
68	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
69	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₄×C₈	#41	1	1	yes	characteristic
72	C₂²×C₄	—	2	1	no	—
73	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
74	M₁₆	—	2	4	no	—
75	M₁₆	—	4	2	no	—
76	C₂×C₈	—	4	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	8	2	no	—
79	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
80	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₈	C₄×D₄	1	1	yes	characteristic
82	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
83	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
85	C₂×C₈	#131	1	1	yes	characteristic
86	C₂×C₈	#131	1	1	yes	characteristic
87	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₈	#137	1	2	yes	—
89	C₄⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	4	2	no	—
91	C₄²	—	2	1	no	—
92	C₄²	—	4	2	no	—
93	C₄²	#117	1	1	yes	characteristic
94	C₄²	C₄×Q₈	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₁₆	—	2	4	no	—
97	C₁₆	—	4	4	no	—
98	C₁₆	C₄×C₈	1	2	yes	—
99	C₁₆	C₄×C₈	1	2	yes	—
100	C₂³	#444	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	4	no	—
105	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄	C₈×D₄	1	1	yes	characteristic
107	C₂×C₄	C₈×D₄	1	1	yes	characteristic
108	C₂×C₄	#441	1	1	yes	characteristic
109	C₂×C₄	#442	1	1	yes	characteristic
110	C₂×C₄	#444	1	1	yes	characteristic
111	C₈	—	4	4	no	—
112	C₈	—	8	2	no	—
113	C₈	C₂×C₄×C₈	1	1	yes	characteristic
114	C₈	#405	1	1	yes	characteristic
115	C₈	#428	1	2	yes	—
116	C₈	#440	1	2	yes	—
117	C₈	C₈×Q₈	1	1	yes	characteristic
118	C₈	C₈×Q₈	1	1	yes	characteristic
119	C₂²	—	2	2	no	—
120	C₂²	#1579	1	1	yes	characteristic
121	C₂²	#1580	1	1	yes	characteristic
122	C₂²	#1773	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	—	4	1	no	—
125	C₄	—	8	4	no	—
126	C₄	#1579	1	1	yes	characteristic
127	C₄	#1580	1	1	yes	characteristic
128	C₄	C₈×C₄⋊C₄	1	1	yes	characteristic
129	C₄	#1776	1	1	yes	characteristic
130	C₂	—	2	1	no	—
131	C₂	—	2	1	no	—
132	C₂	#8284	1	1	yes	characteristic
133	C₂	#12068	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12671

C₂

yes

maximal, characteristic

#12665

C₂

yes

maximal

C₂×#1369

C₂

yes

maximal, characteristic

C₂×#1313

C₂

yes

maximal, characteristic

#7182

C₂

yes

maximal

C₂×#368

C₄

yes

characteristic

#2158

C₄

yes

characteristic

C₂×#364

C₂²

yes

—

#2108

—

C₂×#346

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1772

C₂²

yes

—

C₂×#322

C₂²

yes

characteristic

#1382

C₄

yes

—

#1379

—

#1369

C₄

yes

complemented

#1313

C₄

yes

complemented

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

C₂×#108

C₂×C₄

yes

—

#427

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

#368

C₈

yes

complemented

#367

C₈

yes

complemented

#366

C₈

yes

complemented

#364

—

#364

C₂×C₄

yes

—

#346

—

#346

C₂×C₄

yes

—

#346

C₂×C₄

yes

—

#346

C₂×C₄

yes

—

#346

Q₈

yes

—

C₄×C₁₆

—

#339

—

#322

—

#322

C₂×C₄

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

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₂×C₈

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

characteristic

C₂×C₁₆

C₂×C₈

yes

—

#108

—

#108

C₂×C₈

yes

—

#107

C₂×C₈

yes

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂²×C₄

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×C₄²

yes

characteristic

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂²×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

#117

yes

characteristic

C₄²

C₄×Q₈

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

C₄×C₈

yes

—

C₁₆

C₄×C₈

yes

—

100

C₂³

#444

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

106

C₂×C₄

C₈×D₄

yes

characteristic

107

C₂×C₄

C₈×D₄

yes

characteristic

108

C₂×C₄

#441

yes

characteristic

109

C₂×C₄

#442

yes

characteristic

110

C₂×C₄

#444

yes

characteristic

111

C₈

—

112

C₈

—

113

C₈

C₂×C₄×C₈

yes

characteristic

114

C₈

#405

yes

characteristic

115

C₈

#428

yes

—

116

C₈

#440

yes

—

117

C₈

C₈×Q₈

yes

characteristic

118

C₈

C₈×Q₈

yes

characteristic

119

C₂²

—

120

C₂²

#1579

yes

characteristic

121

C₂²

#1580

yes

characteristic

122

C₂²

#1773

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#1579

yes

characteristic

127

C₄

#1580

yes

characteristic

128

C₄

C₈×C₄⋊C₄

yes

characteristic

129

C₄

#1776

yes

characteristic

130

C₂

—

131

C₂

—

132

C₂

#8284

yes

characteristic

133

C₂

#12068

yes

—

	subgroup	G/H
center	C₂×C₄	#442
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1580
prime radical	C₂³	#444
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#442

commutator

C₈

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#346 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
16. #1369	C₄	semidirect
17. #1313	C₄	semidirect
27. #368	C₈	semidirect
28. #367	C₈	semidirect
29. #366	C₈	semidirect

H ≅ G/N

type

16. #1369

C₄

semidirect

17. #1313

C₄

semidirect

27. #368

C₈

semidirect

28. #367

C₈

semidirect

29. #366

C₈

semidirect

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

subgroup	index
#12671	2
#12665	2
C₂×#1369	2
C₂×#1313	2
#7182	2

subgroup

index

Subgroups — #129164

Conjugacy classes of subgroups

Complements

Maximal subgroups