Group #129113 — subgroups

Subgroups — #129113

Order 512.

292 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 158 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	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#1372	C₂	1	2	yes	maximal
3	#8274	C₂	1	2	yes	maximal
4	#7154	C₂	1	2	yes	maximal
5	C₂×C₄×C₁₆	C₄	1	2	yes	—
6	C₂×C₄×C₁₆	C₄	1	2	yes	—
7	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
9	#1767	—	2	2	no	—
10	C₄²×C₈	C₂²	1	1	yes	characteristic
11	C₂×#335	C₂²	1	4	yes	—
12	#1372	C₄	1	8	yes	complemented
13	#1277	—	2	2	no	—
14	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₁₆	C₂×C₄	1	2	yes	—
16	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄⋊C₈	—	2	4	no	—
18	C₂×#97	—	4	2	no	—
19	C₂×C₄×C₈	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₈	D₄	1	1	yes	characteristic
21	C₂×C₄×C₈	Q₈	1	2	yes	—
22	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
23	C₄³	D₄	1	1	yes	characteristic
24	C₄×C₁₆	—	2	4	no	—
25	C₄×C₁₆	—	2	4	no	—
26	C₄×C₁₆	—	2	2	no	—
27	C₄×C₁₆	C₂×C₄	1	4	yes	—
28	#336	—	4	2	no	—
29	#335	—	2	4	no	—
30	#335	C₂×C₄	1	8	yes	—
31	C₂²×C₈	—	4	4	no	—
32	C₂²×C₈	C₄⋊C₄	1	2	yes	—
33	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
34	C₂²×C₈	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄²	C₄⋊C₄	1	2	yes	—
36	C₂×C₄²	D₈	1	1	yes	characteristic
37	C₂×C₄²	D₈	1	1	yes	characteristic
38	C₂×C₄²	Q₁₆	1	2	yes	—
39	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
40	C₂×C₁₆	—	2	2	no	—
41	C₂×C₁₆	—	2	2	no	—
42	C₂×C₁₆	—	2	2	no	—
43	C₂×C₁₆	C₄²	1	2	yes	—
44	C₂×C₁₆	C₄²	1	2	yes	—
45	C₂×C₁₆	M₁₆	1	4	yes	—
46	C₂×C₁₆	M₁₆	1	4	yes	—
47	C₄⋊C₈	—	2	8	no	—
48	C₄⋊C₈	—	4	4	no	—
49	#97	—	8	4	no	—
50	C₄×C₈	—	2	4	no	—
51	C₄×C₈	—	2	2	no	—
52	C₄×C₈	—	2	2	no	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	C₂²×C₄	1	2	yes	—
55	C₄×C₈	#41	1	2	yes	—
56	C₂²×C₄	#107	1	2	yes	—
57	C₂²×C₄	#107	1	2	yes	—
58	C₂²×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
60	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	4	8	no	—
65	C₂×C₈	—	8	4	no	—
66	C₂×C₈	—	8	4	no	—
67	C₂×C₈	#108	1	4	yes	—
68	C₂×C₈	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₈	#117	1	1	yes	characteristic
70	C₂×C₈	C₄×Q₈	1	2	yes	—
71	C₂×C₈	C₂×M₁₆	1	2	yes	—
72	C₂×C₈	C₂×M₁₆	1	2	yes	—
73	C₄²	—	2	4	no	—
74	C₄²	—	2	2	no	—
75	C₄²	—	2	2	no	—
76	C₄²	—	2	2	no	—
77	C₄²	SD₃₂	1	4	yes	—
78	C₄²	C₄×D₄	1	2	yes	—
79	C₄²	#135	1	2	yes	—
80	C₁₆	—	2	2	no	—
81	C₁₆	—	2	4	no	—
82	C₁₆	#97	1	4	yes	complemented
83	C₂³	C₂×#107	1	1	yes	characteristic
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	#367	1	4	yes	—
90	C₂×C₄	#368	1	4	yes	—
91	C₂×C₄	C₄×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄	#423	1	2	yes	—
93	C₂×C₄	#427	1	1	yes	characteristic
94	C₂×C₄	C₂×#108	1	2	yes	—
95	C₂×C₄	C₄×D₈	1	2	yes	—
96	C₂×C₄	C₄×Q₁₆	1	2	yes	—
97	C₂×C₄	C₂×SD₃₂	1	2	yes	—
98	C₂×C₄	#508	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₈	—	2	2	no	—
101	C₈	—	8	8	no	—
102	C₈	C₂×#97	1	1	yes	characteristic
103	C₈	C₄×M₁₆	1	1	yes	characteristic
104	C₈	#432	1	2	yes	—
105	C₂²	#1567	1	2	yes	—
106	C₂²	C₄×#107	1	1	yes	characteristic
107	C₂²	C₂×#367	1	1	yes	characteristic
108	C₂²	#2158	1	1	yes	characteristic
109	C₂²	C₂×#368	1	2	yes	—
110	C₄	—	2	2	no	—
111	C₄	—	2	4	no	—
112	C₄	—	2	4	no	—
113	C₄	#1568	1	2	yes	—
114	C₄	#1767	1	1	yes	characteristic
115	C₄	C₄×#108	1	1	yes	characteristic
116	C₄	C₄×SD₃₂	1	4	yes	—
117	C₂	#8274	1	1	yes	characteristic
118	C₂	#12023	1	2	yes	—
119	C₂	#12024	1	2	yes	—
120	C₂	C₄×#367	1	1	yes	characteristic
121	C₂	C₄×#368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄²×C₁₆

C₂

yes

maximal, characteristic

C₂×#1372

C₂

yes

maximal

#8274

C₂

yes

maximal

#7154

C₂

yes

maximal

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₄

yes

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1767

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

#1372

C₄

yes

complemented

#1277

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

C₂×C₄

yes

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

Q₈

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

D₄

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

#336

—

#335

—

#335

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₄⋊C₄

yes

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×Q₈

yes

characteristic

C₂×C₄²

C₄⋊C₄

yes

—

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

D₈

yes

characteristic

C₂×C₄²

Q₁₆

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

C₄²

yes

—

C₂×C₁₆

M₁₆

yes

—

C₂×C₁₆

M₁₆

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

#107

yes

—

C₂²×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#108

yes

—

C₂×C₈

C₂×C₄²

yes

characteristic

C₂×C₈

#117

yes

characteristic

C₂×C₈

C₄×Q₈

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

C₂×M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

SD₃₂

yes

—

C₄²

C₄×D₄

yes

—

C₄²

#135

yes

—

C₁₆

—

C₁₆

—

C₁₆

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#367

yes

—

C₂×C₄

#368

yes

—

C₂×C₄

C₄×C₄⋊C₄

yes

characteristic

C₂×C₄

#423

yes

—

C₂×C₄

#427

yes

characteristic

C₂×C₄

C₂×#108

yes

—

C₂×C₄

C₄×D₈

yes

—

C₂×C₄

C₄×Q₁₆

yes

—

C₂×C₄

C₂×SD₃₂

yes

—

C₂×C₄

#508

yes

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

C₂×#97

yes

characteristic

103

C₈

C₄×M₁₆

yes

characteristic

104

C₈

#432

yes

—

105

C₂²

#1567

yes

—

106

C₂²

C₄×#107

yes

characteristic

107

C₂²

C₂×#367

yes

characteristic

108

C₂²

#2158

yes

characteristic

109

C₂²

C₂×#368

yes

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#1568

yes

—

114

C₄

#1767

yes

characteristic

115

C₄

C₄×#108

yes

characteristic

116

C₄

C₄×SD₃₂

yes

—

117

C₂

#8274

yes

characteristic

118

C₂

#12023

yes

—

119

C₂

#12024

yes

—

120

C₂

C₄×#367

yes

characteristic

121

C₂

C₄×#368

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₈

commutator

C₂×C₈

C₂×C₄²

Frattini, cosocle

C₂×C₄×C₈

C₂³

socle, prime radical

C₂³

C₂×#107

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
12. #1372	C₄	semidirect
82. C₁₆	#97	semidirect

H ≅ G/N

type

12. #1372

C₄

semidirect

82. C₁₆

#97

semidirect

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

subgroup	index
C₄²×C₁₆	2
C₂×#1372	2
#8274	2
#7154	2

subgroup

index

C₄²×C₁₆

C₂×#1372

#8274

#7154

Subgroups — #129113

Conjugacy classes of subgroups

Complements

Maximal subgroups