Group #146152 — subgroups

Subgroups — #146152

Order 512.

185 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 185 conjugacy classes of subgroups — too many to draw. Only its 76 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	#13182	C₂	1	2	yes	maximal
2	#13162	C₂	1	1	yes	maximal, characteristic
3	#12910	C₂	1	1	yes	maximal, characteristic
4	#7270	C₂	1	2	yes	maximal
5	#7161	C₂	1	1	yes	maximal, characteristic
6	#2161	—	2	2	no	—
7	C₂×#368	—	2	1	no	—
8	C₂×#368	—	2	1	no	—
9	C₂×#364	C₂²	1	1	yes	characteristic
10	C₂×#364	C₂²	1	2	yes	—
11	#2151	—	2	1	no	—
12	C₂×#346	C₂²	1	1	yes	characteristic
13	#1848	C₂²	1	2	yes	—
14	#1772	C₂²	1	1	yes	characteristic
15	#1389	—	2	2	no	—
16	#1379	—	2	1	no	—
17	C₂×M₃₂	—	2	2	no	—
18	C₂×M₃₂	—	2	1	no	—
19	C₂²×C₁₆	D₄	1	1	yes	characteristic
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂×#108	—	2	2	no	—
22	C₂×#108	—	2	2	no	—
23	C₂×#108	D₄	1	1	yes	characteristic
24	#427	D₄	1	1	yes	characteristic
25	#423	—	2	2	no	—
26	#405	—	2	1	no	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#368	—	4	4	no	—
29	#368	—	4	2	no	—
30	#368	—	4	2	no	—
31	#364	—	2	2	no	—
32	#364	—	2	2	no	—
33	#364	Q₈	1	4	yes	—
34	C₄⋊C₁₆	—	4	2	no	—
35	#346	—	2	2	no	—
36	#339	—	2	1	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂×M₁₆	—	4	2	no	—
39	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
41	#117	—	2	1	no	—
42	C₂×C₄²	#41	1	1	yes	characteristic
43	M₃₂	—	2	4	no	—
44	M₃₂	—	4	2	no	—
45	M₃₂	—	4	1	no	—
46	M₃₂	—	4	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	2	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	4	2	no	—
52	C₂×C₁₆	—	4	1	no	—
53	C₂×C₁₆	D₈	1	2	yes	—
54	C₂×C₁₆	Q₁₆	1	2	yes	—
55	#108	—	2	2	no	—
56	#108	—	4	2	no	—
57	#108	—	4	2	no	—
58	#108	—	4	2	no	—
59	#108	—	4	2	no	—
60	#107	—	2	1	no	—
61	#106	Q₁₆	1	2	yes	—
62	C₄⋊C₈	—	8	2	no	—
63	#97	—	4	1	no	—
64	C₄×C₈	—	4	1	no	—
65	C₄×C₈	C₂×Q₈	1	2	yes	—
66	C₄×C₈	#41	1	2	yes	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	#121	1	1	yes	characteristic
69	M₁₆	—	4	1	no	—
70	M₁₆	—	4	1	no	—
71	M₁₆	—	8	2	no	—
72	M₁₆	—	8	2	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	8	2	no	—
75	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
76	C₂×C₈	#123	1	1	yes	characteristic
77	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₈	C₂×D₈	1	1	yes	characteristic
79	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₈	#135	1	1	yes	characteristic
82	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
83	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
84	C₂×C₈	#137	1	1	yes	characteristic
85	C₄⋊C₄	—	2	2	no	—
86	C₂²⋊C₄	—	4	1	no	—
87	C₄²	—	2	1	no	—
88	C₄²	—	4	1	no	—
89	C₄²	#125	1	2	yes	—
90	C₁₆	—	2	2	no	—
91	C₁₆	—	2	2	no	—
92	C₁₆	—	4	4	no	—
93	C₁₆	—	8	1	no	—
94	C₂³	#473	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	8	1	no	—
98	C₂×C₄	#398	1	1	yes	characteristic
99	C₂×C₄	#459	1	1	yes	characteristic
100	C₂×C₄	#462	1	1	yes	characteristic
101	C₂×C₄	#467	1	1	yes	characteristic
102	C₂×C₄	#469	1	1	yes	characteristic
103	C₂×C₄	#472	1	1	yes	characteristic
104	C₈	—	8	1	no	—
105	C₈	—	16	2	no	—
106	C₈	#477	1	2	yes	—
107	C₈	#478	1	2	yes	—
108	C₈	#480	1	1	yes	characteristic
109	C₈	#484	1	1	yes	characteristic
110	C₈	#500	1	1	yes	characteristic
111	C₈	#501	1	1	yes	characteristic
112	C₂²	—	4	1	no	—
113	C₂²	#1669	1	1	yes	characteristic
114	C₂²	#1694	1	1	yes	characteristic
115	C₂²	#2089	1	1	yes	characteristic
116	C₄	—	4	2	no	—
117	C₄	—	8	2	no	—
118	C₄	#1676	1	1	yes	characteristic
119	C₄	#1687	1	1	yes	characteristic
120	C₄	#2083	1	1	yes	characteristic
121	C₄	#2087	1	1	yes	characteristic
122	C₂	—	2	1	no	—
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#10650	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13182

C₂

yes

maximal

#13162

C₂

yes

maximal, characteristic

#12910

C₂

yes

maximal, characteristic

#7270

C₂

yes

maximal

#7161

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

C₂×#364

C₂²

yes

—

#2151

—

C₂×#346

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1772

C₂²

yes

characteristic

#1389

—

#1379

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

D₄

yes

characteristic

#427

D₄

yes

characteristic

#423

—

#405

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

Q₈

yes

—

C₄⋊C₁₆

—

#346

—

#339

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

—

Q₁₆

yes

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

#125

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#473

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#462

yes

characteristic

101

C₂×C₄

#467

yes

characteristic

102

C₂×C₄

#469

yes

characteristic

103

C₂×C₄

#472

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

#477

yes

—

107

C₈

#478

yes

—

108

C₈

#480

yes

characteristic

109

C₈

#484

yes

characteristic

110

C₈

#500

yes

characteristic

111

C₈

#501

yes

characteristic

112

C₂²

—

113

C₂²

#1669

yes

characteristic

114

C₂²

#1694

yes

characteristic

115

C₂²

#2089

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

#1676

yes

characteristic

119

C₄

#1687

yes

characteristic

120

C₄

#2083

yes

characteristic

121

C₄

#2087

yes

characteristic

122

C₂

—

123

C₂

—

124

C₂

—

125

C₂

#10650

yes

characteristic

	subgroup	G/H
center	C₄	#1687
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#10650
prime radical	C₂³	#473
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#1687

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ 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
#13182	2
#13162	2
#12910	2
#7270	2
#7161	2

subgroup

index

Subgroups — #146152

Conjugacy classes of subgroups

Maximal subgroups