Group #140450 — subgroups

Subgroups — #140450

Order 512.

203 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 96 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	#13188	C₂	1	2	yes	maximal
2	#12909	C₂	1	1	yes	maximal, characteristic
3	#12664	C₂	1	1	yes	maximal, characteristic
4	C₂×#1373	C₂	1	1	yes	maximal, characteristic
5	#7273	C₂	1	2	yes	maximal
6	C₂×#368	—	2	1	no	—
7	C₂×#368	—	2	1	no	—
8	#2155	—	2	2	no	—
9	C₂×#364	C₂²	1	2	yes	—
10	C₂×#364	C₂²	1	1	yes	characteristic
11	#2108	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1848	C₂²	1	2	yes	—
14	#1772	C₂²	1	1	yes	characteristic
15	#1388	—	2	2	no	—
16	#1379	—	2	1	no	—
17	#1373	C₄	1	4	yes	complemented
18	C₂×M₃₂	—	2	1	no	—
19	C₂×M₃₂	—	2	2	no	—
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂×#108	—	2	2	no	—
23	C₂×#108	—	2	2	no	—
24	C₂×#108	D₄	1	1	yes	characteristic
25	#427	D₄	1	1	yes	characteristic
26	#423	—	2	2	no	—
27	#405	—	2	1	no	—
28	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
29	#368	—	4	2	no	—
30	#368	—	4	2	no	—
31	#365	—	4	4	no	—
32	#364	—	2	2	no	—
33	#364	—	2	1	no	—
34	#364	C₂×C₄	1	4	yes	—
35	#364	Q₈	1	2	yes	—
36	#346	—	4	1	no	—
37	C₄×C₁₆	—	4	1	no	—
38	C₄×C₁₆	C₂×C₄	1	2	yes	—
39	C₄×C₁₆	Q₈	1	2	yes	—
40	#339	—	2	1	no	—
41	C₂×M₁₆	—	2	1	no	—
42	C₂×M₁₆	—	4	2	no	—
43	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
45	#117	—	2	1	no	—
46	C₂×C₄²	#41	1	1	yes	characteristic
47	M₃₂	—	2	4	no	—
48	M₃₂	—	2	2	no	—
49	M₃₂	—	4	2	no	—
50	M₃₂	—	4	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	4	1	no	—
54	C₂×C₁₆	—	4	2	no	—
55	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
56	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
57	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
58	C₂×C₁₆	SD₁₆	1	1	yes	characteristic
59	C₂×C₁₆	SD₁₆	1	2	yes	—
60	#108	—	2	1	no	—
61	#108	—	4	2	no	—
62	#108	—	4	2	no	—
63	#108	—	4	2	no	—
64	#108	—	4	2	no	—
65	#108	Q₁₆	1	2	yes	—
66	#107	—	2	1	no	—
67	#106	D₈	1	2	yes	—
68	C₄⋊C₈	—	8	2	no	—
69	#97	—	4	1	no	—
70	C₄×C₈	—	4	1	no	—
71	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
73	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
74	C₄×C₈	#41	1	1	yes	characteristic
75	C₂²×C₄	—	2	1	no	—
76	C₂²×C₄	#121	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	4	1	no	—
79	M₁₆	—	8	2	no	—
80	M₁₆	—	8	2	no	—
81	C₂×C₈	—	4	1	no	—
82	C₂×C₈	—	8	2	no	—
83	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₈	C₄×D₄	1	1	yes	characteristic
85	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₈	C₂×D₈	1	1	yes	characteristic
88	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₈	#135	1	1	yes	characteristic
92	C₂×C₈	#135	1	1	yes	characteristic
93	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×C₈	#137	1	1	yes	characteristic
95	C₄⋊C₄	—	2	2	no	—
96	C₂²⋊C₄	—	4	1	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	4	1	no	—
99	C₄²	C₄×Q₈	1	1	yes	characteristic
100	C₄²	#125	1	1	yes	characteristic
101	C₁₆	—	2	2	no	—
102	C₁₆	—	4	4	no	—
103	C₁₆	—	4	2	no	—
104	C₁₆	#106	1	2	yes	—
105	C₁₆	#106	1	2	yes	—
106	C₂³	#473	1	1	yes	characteristic
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	#389	1	1	yes	characteristic
112	C₂×C₄	#460	1	1	yes	characteristic
113	C₂×C₄	#461	1	1	yes	characteristic
114	C₂×C₄	#466	1	1	yes	characteristic
115	C₂×C₄	#467	1	1	yes	characteristic
116	C₂×C₄	#472	1	1	yes	characteristic
117	C₈	—	4	2	no	—
118	C₈	—	16	2	no	—
119	C₈	C₂×#106	1	1	yes	characteristic
120	C₈	#428	1	1	yes	characteristic
121	C₈	C₄×D₈	1	1	yes	characteristic
122	C₈	C₄×Q₁₆	1	1	yes	characteristic
123	C₈	#475	1	1	yes	characteristic
124	C₈	#476	1	1	yes	characteristic
125	C₈	#499	1	1	yes	characteristic
126	C₈	#500	1	1	yes	characteristic
127	C₂²	—	4	1	no	—
128	C₂²	#1670	1	1	yes	characteristic
129	C₂²	#1673	1	1	yes	characteristic
130	C₂²	#1952	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	4	1	no	—
133	C₄	—	8	2	no	—
134	C₄	#1668	1	1	yes	characteristic
135	C₄	#1675	1	1	yes	characteristic
136	C₄	#1942	1	1	yes	characteristic
137	C₄	#1948	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	2	1	no	—
140	C₂	—	2	1	no	—
141	C₂	#9615	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13188

C₂

yes

maximal

#12909

C₂

yes

maximal, characteristic

#12664

C₂

yes

maximal, characteristic

C₂×#1373

C₂

yes

maximal, characteristic

#7273

C₂

yes

—

—

—

C₂²

yes

—

C₂×#364

C₂²

yes

characteristic

#2108

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1772

C₂²

yes

—

—

C₄

yes

complemented

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

—

—

—

—

—

C₂×C₄

yes

—

#364

Q₈

yes

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

Q₈

yes

—

#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

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

SD₁₆

yes

characteristic

C₂×C₁₆

SD₁₆

yes

characteristic

C₂×C₁₆

SD₁₆

yes

—

—

—

—

—

—

Q₁₆

yes

—

#107

—

#106

D₈

yes

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×D₈

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

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄×Q₈

yes

characteristic

100

C₄²

#125

yes

characteristic

101

C₁₆

—

102

C₁₆

—

103

C₁₆

—

104

C₁₆

#106

yes

—

105

C₁₆

#106

yes

—

106

C₂³

#473

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#389

yes

characteristic

112

C₂×C₄

#460

yes

characteristic

113

C₂×C₄

#461

yes

characteristic

114

C₂×C₄

#466

yes

characteristic

115

C₂×C₄

#467

yes

characteristic

116

C₂×C₄

#472

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

C₂×#106

yes

characteristic

120

C₈

#428

yes

characteristic

121

C₈

C₄×D₈

yes

characteristic

122

C₈

C₄×Q₁₆

yes

characteristic

123

C₈

#475

yes

characteristic

124

C₈

#476

yes

characteristic

125

C₈

#499

yes

characteristic

126

C₈

#500

yes

characteristic

127

C₂²

—

128

C₂²

#1670

yes

characteristic

129

C₂²

#1673

yes

characteristic

130

C₂²

#1952

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#1668

yes

characteristic

135

C₄

#1675

yes

characteristic

136

C₄

#1942

yes

characteristic

137

C₄

#1948

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

—

141

C₂

#9615

yes

characteristic

	subgroup	G/H
center	C₄	#1675
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9615
prime radical	C₂³	#473
LCS	G ▷ 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

#1675

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₂³

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
17. #1373	C₄	semidirect

H ≅ G/N

type

17. #1373

C₄

semidirect

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

subgroup	index
#13188	2
#12909	2
#12664	2
C₂×#1373	2
#7273	2

subgroup

index

Subgroups — #140450

Conjugacy classes of subgroups

Complements

Maximal subgroups