Group #146065 — subgroups

Subgroups — #146065

Order 512.

206 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 70 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	#13264	C₂	1	1	yes	maximal, characteristic
2	#12892	C₂	1	1	yes	maximal, characteristic
3	#8659	C₂	1	2	yes	maximal
4	#7277	C₂	1	2	yes	maximal
5	#7157	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#346	C₂²	1	1	yes	characteristic
9	#1848	C₂²	1	2	yes	—
10	#1564	C₂²	1	2	yes	—
11	#1487	—	2	2	no	—
12	#1381	—	2	2	no	—
13	#1374	—	2	2	no	—
14	#1301	—	2	2	no	—
15	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
16	C₂²×C₁₆	—	2	1	no	—
17	C₂²×C₁₆	D₄	1	2	yes	—
18	C₈×Q₈	—	2	2	no	—
19	C₈×Q₈	—	2	2	no	—
20	#431	—	2	2	no	—
21	C₂×#108	—	2	2	no	—
22	C₂×#108	—	2	2	no	—
23	#423	—	2	2	no	—
24	C₂×C₄⋊C₈	—	2	1	no	—
25	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	C₄⋊C₁₆	—	2	2	no	—
29	#346	—	2	2	no	—
30	#335	—	2	2	no	—
31	#334	—	2	2	no	—
32	Q₈⋊C₈	—	4	4	no	—
33	C₂²×Q₈	Q₁₆	1	1	yes	characteristic
34	C₂×M₁₆	—	4	2	no	—
35	C₂²×C₈	—	2	1	no	—
36	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₈	#41	1	1	yes	characteristic
38	C₄×Q₈	—	2	2	no	—
39	C₄×Q₈	—	2	2	no	—
40	C₂×C₄⋊C₄	—	2	1	no	—
41	C₂×C₄⋊C₄	Q₁₆	1	1	yes	characteristic
42	C₂×C₄²	—	2	1	no	—
43	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
44	C₂×C₁₆	—	2	2	no	—
45	C₂×C₁₆	—	2	2	no	—
46	C₂×C₁₆	—	2	2	no	—
47	C₂×C₁₆	—	4	1	no	—
48	C₂×C₁₆	—	4	2	no	—
49	#108	—	4	2	no	—
50	#108	—	4	2	no	—
51	#108	—	8	2	no	—
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	—	4	2	no	—
54	C₄⋊C₈	—	4	2	no	—
55	C₄⋊C₈	—	4	2	no	—
56	C₄⋊C₈	—	4	2	no	—
57	C₂²⋊C₈	—	4	2	no	—
58	C₄×C₈	—	4	1	no	—
59	C₄×C₈	—	4	1	no	—
60	C₄×C₈	—	4	2	no	—
61	C₄×C₈	#41	1	2	yes	—
62	C₄×C₈	#41	1	2	yes	—
63	C₂×Q₈	—	2	2	no	—
64	C₂×Q₈	—	4	2	no	—
65	C₂²×C₄	—	2	1	no	—
66	C₂²×C₄	—	2	1	no	—
67	C₂²×C₄	#121	1	1	yes	characteristic
68	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
69	C₂²×C₄	#135	1	1	yes	characteristic
70	M₁₆	—	8	2	no	—
71	M₁₆	—	8	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	4	2	no	—
75	C₂×C₈	—	8	2	no	—
76	C₂×C₈	—	8	2	no	—
77	C₂×C₈	#123	1	1	yes	characteristic
78	C₂×C₈	#124	1	1	yes	characteristic
79	C₂×C₈	#126	1	2	yes	—
80	C₂×C₈	#135	1	2	yes	—
81	C₂×C₈	#137	1	2	yes	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	4	2	no	—
84	C₄²	—	4	1	no	—
85	C₄²	—	4	1	no	—
86	C₄²	#123	1	2	yes	—
87	C₄²	AGL(1,ℤ/8)	1	2	yes	—
88	C₁₆	—	4	2	no	—
89	C₁₆	—	8	1	no	—
90	C₂³	#467	1	1	yes	characteristic
91	Q₈	—	2	4	no	—
92	Q₈	—	4	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	4	1	no	—
96	C₂×C₄	—	4	1	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	#397	1	1	yes	characteristic
101	C₂×C₄	#464	1	2	yes	—
102	C₂×C₄	#469	1	1	yes	characteristic
103	C₂×C₄	#471	1	2	yes	—
104	C₂×C₄	#482	1	2	yes	—
105	C₂×C₄	#484	1	2	yes	—
106	C₈	—	2	2	no	—
107	C₈	—	8	1	no	—
108	C₈	—	16	2	no	—
109	C₈	—	16	2	no	—
110	C₈	#485	1	2	yes	—
111	C₈	#488	1	1	yes	characteristic
112	C₈	#491	1	1	yes	characteristic
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	#1684	1	2	yes	—
116	C₂²	#2076	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	4	2	no	—
120	C₄	—	8	1	no	—
121	C₄	#1702	1	2	yes	—
122	C₄	#2074	1	1	yes	characteristic
123	C₄	#2079	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#10624	1	1	yes	characteristic
126	C₂	#13498	1	1	yes	characteristic
127	C₂	#13499	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13264

C₂

yes

maximal, characteristic

#12892

C₂

yes

maximal, characteristic

#8659

C₂

yes

maximal

#7277

C₂

yes

maximal

#7157

C₂

yes

maximal, characteristic

C₂×C₈×Q₈

C₂²

yes

characteristic

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1564

C₂²

yes

—

—

—

—

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

C₈×Q₈

—

C₈×Q₈

—

—

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

#346

—

#335

—

#334

—

Q₈⋊C₈

—

C₂²×Q₈

Q₁₆

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Q₁₆

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂²×C₄

#135

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

#135

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

#123

yes

—

C₄²

AGL(1,ℤ/8)

yes

—

C₁₆

—

C₁₆

—

C₂³

#467

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#397

yes

characteristic

101

C₂×C₄

#464

yes

—

102

C₂×C₄

#469

yes

characteristic

103

C₂×C₄

#471

yes

—

104

C₂×C₄

#482

yes

—

105

C₂×C₄

#484

yes

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

#485

yes

—

111

C₈

#488

yes

characteristic

112

C₈

#491

yes

characteristic

113

C₂²

—

114

C₂²

—

115

C₂²

#1684

yes

—

116

C₂²

#2076

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#1702

yes

—

122

C₄

#2074

yes

characteristic

123

C₄

#2079

yes

characteristic

124

C₂

—

125

C₂

#10624

yes

characteristic

126

C₂

#13498

yes

characteristic

127

C₂

#13499

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2076
prime radical	C₂³	#467
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#469

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

subgroup	index
#13264	2
#12892	2
#8659	2
#7277	2
#7157	2

subgroup

index

Subgroups — #146065

Conjugacy classes of subgroups

Maximal subgroups