Group #146161 — subgroups

Subgroups — #146161

Order 512.

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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	#13187	C₂	1	2	yes	maximal
2	#13161	C₂	1	1	yes	maximal, characteristic
3	#12907	C₂	1	1	yes	maximal, characteristic
4	#7271	C₂	1	2	yes	maximal
5	#7160	C₂	1	1	yes	maximal, characteristic
6	C₂×#367	—	2	1	no	—
7	C₂×#366	—	2	1	no	—
8	C₂×#365	—	2	2	no	—
9	C₂×#364	C₂²	1	1	yes	characteristic
10	C₂×#364	C₂²	1	2	yes	—
11	#2151	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1846	C₂²	1	2	yes	—
14	#1772	C₂²	1	1	yes	characteristic
15	#1386	—	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	D₄	1	1	yes	characteristic
22	#427	D₄	1	1	yes	characteristic
23	C₂×#107	—	2	2	no	—
24	C₂×#106	—	2	2	no	—
25	#405	—	2	1	no	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	#382	—	2	2	no	—
28	#367	—	4	2	no	—
29	#366	—	4	2	no	—
30	#365	—	4	4	no	—
31	#364	—	2	2	no	—
32	#364	—	2	2	no	—
33	#364	Q₈	1	4	yes	—
34	C₄⋊C₁₆	—	4	2	no	—
35	C₄×C₁₆	—	2	2	no	—
36	#339	—	2	1	no	—
37	C₂×M₁₆	—	2	1	no	—
38	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
40	#117	—	2	1	no	—
41	C₂×C₄⋊C₄	—	4	2	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	SD₁₆	1	2	yes	—
54	C₂×C₁₆	SD₁₆	1	2	yes	—
55	#108	—	2	2	no	—
56	#107	—	4	2	no	—
57	#107	—	4	2	no	—
58	#107	Q₁₆	1	2	yes	—
59	#106	—	2	1	no	—
60	#106	—	4	2	no	—
61	#106	—	4	2	no	—
62	#97	—	4	1	no	—
63	C₄×C₈	—	4	1	no	—
64	C₄×C₈	C₂×Q₈	1	2	yes	—
65	C₄×C₈	#41	1	2	yes	—
66	#95	—	8	2	no	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	—	8	2	no	—
69	C₂²×C₄	#121	1	1	yes	characteristic
70	M₁₆	—	4	1	no	—
71	M₁₆	—	4	1	no	—
72	C₂×C₈	—	4	1	no	—
73	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
74	C₂×C₈	#123	1	1	yes	characteristic
75	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
76	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
79	C₂×C₈	#135	1	1	yes	characteristic
80	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
81	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
82	C₂×C₈	#137	1	1	yes	characteristic
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	8	2	no	—
85	C₄⋊C₄	—	8	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	#472	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	16	4	no	—
99	C₂×C₄	—	16	2	no	—
100	C₂×C₄	#398	1	1	yes	characteristic
101	C₂×C₄	#460	1	1	yes	characteristic
102	C₂×C₄	#461	1	1	yes	characteristic
103	C₂×C₄	#467	1	1	yes	characteristic
104	C₂×C₄	#469	1	1	yes	characteristic
105	C₂×C₄	#473	1	1	yes	characteristic
106	C₈	—	8	1	no	—
107	C₈	#477	1	2	yes	—
108	C₈	#478	1	2	yes	—
109	C₈	#481	1	1	yes	characteristic
110	C₈	#483	1	1	yes	characteristic
111	C₈	#498	1	1	yes	characteristic
112	C₈	#501	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	#1675	1	1	yes	characteristic
115	C₂²	#1688	1	1	yes	characteristic
116	C₂²	#2089	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	8	2	no	—
119	C₄	—	32	2	no	—
120	C₄	#1670	1	1	yes	characteristic
121	C₄	#1693	1	1	yes	characteristic
122	C₄	#2084	1	1	yes	characteristic
123	C₄	#2087	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	2	1	no	—
126	C₂	#10652	1	1	yes	characteristic
127	C₂	#12464	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13187

C₂

yes

maximal

#13161

C₂

yes

maximal, characteristic

#12907

C₂

yes

maximal, characteristic

#7271

C₂

yes

maximal

#7160

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

C₂×#364

C₂²

yes

—

#2151

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1846

C₂²

yes

—

#1772

C₂²

yes

characteristic

#1386

—

#1379

—

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₂×#108

D₄

yes

characteristic

#427

D₄

yes

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

Q₈

yes

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

SD₁₆

yes

—

—

—

—

Q₁₆

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

#125

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#472

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#398

yes

characteristic

101

C₂×C₄

#460

yes

characteristic

102

C₂×C₄

#461

yes

characteristic

103

C₂×C₄

#467

yes

characteristic

104

C₂×C₄

#469

yes

characteristic

105

C₂×C₄

#473

yes

characteristic

106

C₈

—

107

C₈

#477

yes

—

108

C₈

#478

yes

—

109

C₈

#481

yes

characteristic

110

C₈

#483

yes

characteristic

111

C₈

#498

yes

characteristic

112

C₈

#501

yes

characteristic

113

C₂²

—

114

C₂²

#1675

yes

characteristic

115

C₂²

#1688

yes

characteristic

116

C₂²

#2089

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#1670

yes

characteristic

121

C₄

#1693

yes

characteristic

122

C₄

#2084

yes

characteristic

123

C₄

#2087

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

#10652

yes

characteristic

127

C₂

#12464

yes

—

	subgroup	G/H
center, socle	C₂²	#1675
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1675

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#472

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ 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
#13187	2
#13161	2
#12907	2
#7271	2
#7160	2

subgroup

index

Subgroups — #146161

Conjugacy classes of subgroups

Maximal subgroups