Group #140509 — subgroups

Subgroups — #140509

Order 512.

356 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 134 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	#21732	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	C₄²×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#11608	C₂	1	4	yes	maximal
5	#3160	C₄	1	1	yes	characteristic
6	#2960	C₂²	1	2	yes	—
7	C₂×#364	—	2	2	no	—
8	#2150	C₂²	1	2	yes	—
9	C₂×C₄×C₁₆	—	2	2	no	—
10	C₂×C₄×C₁₆	C₂²	1	2	yes	—
11	#1994	C₄	1	2	yes	—
12	#1945	—	2	2	no	—
13	#1848	C₄	1	1	yes	characteristic
14	#1772	—	2	2	no	—
15	C₄²×C₈	C₂²	1	1	yes	characteristic
16	Q₈⋊C₁₆	—	2	8	no	—
17	#533	C₂×C₄	1	2	yes	—
18	C₂×M₃₂	—	2	2	no	—
19	C₂²×C₁₆	—	2	2	no	—
20	#500	C₈	1	2	yes	—
21	#499	—	2	2	no	—
22	#498	C₈	1	2	yes	—
23	#446	—	2	4	no	—
24	C₈×Q₈	—	2	4	no	—
25	C₂×#108	—	2	2	no	—
26	#427	—	2	2	no	—
27	#424	—	2	2	no	—
28	#423	C₂×C₄	1	2	yes	—
29	#405	—	2	2	no	—
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	2	yes	—
32	C₂×C₄×C₈	D₄	1	2	yes	—
33	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
34	C₄³	C₂×C₄	1	1	yes	characteristic
35	#364	—	4	4	no	—
36	C₄⋊C₁₆	—	2	4	no	—
37	C₄×C₁₆	—	2	8	no	—
38	C₄×C₁₆	—	2	2	no	—
39	C₄×C₁₆	D₄	1	4	yes	—
40	#339	—	2	4	no	—
41	C₂×M₁₆	—	2	2	no	—
42	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Q₈	C₂×C₈	1	4	yes	—
46	#125	—	2	2	no	—
47	C₂²⋊Q₈	—	2	4	no	—
48	C₄×Q₈	—	2	4	no	—
49	#117	—	2	2	no	—
50	C₂×C₄²	—	2	2	no	—
51	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
53	M₃₂	—	4	4	no	—
54	C₂×C₁₆	—	2	8	no	—
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	—	4	2	no	—
57	#108	—	4	4	no	—
58	#107	—	2	4	no	—
59	#106	—	2	4	no	—
60	C₄⋊C₈	—	2	4	no	—
61	C₄⋊C₈	—	4	4	no	—
62	#97	—	4	2	no	—
63	C₄×C₈	—	2	4	no	—
64	C₄×C₈	—	4	2	no	—
65	C₄×C₈	C₂×C₈	1	1	yes	characteristic
66	C₄×C₈	C₂×C₈	1	1	yes	characteristic
67	C₄×C₈	M₁₆	1	2	yes	—
68	C₄×C₈	SD₁₆	1	4	yes	—
69	C₄×C₈	Q₁₆	1	4	yes	—
70	C₄×C₈	C₂×D₄	1	2	yes	—
71	C₄×C₈	#41	1	2	yes	—
72	C₂×Q₈	—	2	4	no	—
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄	#131	1	2	yes	—
76	M₁₆	—	4	4	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	C₂²⋊C₈	1	2	yes	—
80	C₂×C₈	C₂²⋊C₈	1	2	yes	—
81	C₂×C₈	#103	1	4	yes	—
82	C₂×C₈	#104	1	4	yes	—
83	C₂×C₈	#121	1	2	yes	—
84	C₂×C₈	#124	1	1	yes	characteristic
85	C₂×C₈	#127	1	1	yes	characteristic
86	C₂×C₈	C₂×SD₁₆	1	2	yes	—
87	C₂×C₈	C₂×Q₁₆	1	2	yes	—
88	C₄⋊C₄	—	2	4	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₂²⋊C₄	—	4	2	no	—
92	C₄²	—	2	8	no	—
93	C₄²	—	2	2	no	—
94	C₄²	—	4	2	no	—
95	C₄²	#103	1	4	yes	—
96	C₄²	C₄×D₄	1	2	yes	—
97	C₄²	C₂²×C₈	1	1	yes	characteristic
98	C₄²	C₂×M₁₆	1	1	yes	characteristic
99	C₁₆	—	2	8	no	—
100	C₁₆	—	8	2	no	—
101	C₂³	#408	1	1	yes	characteristic
102	Q₈	—	4	8	no	—
103	C₂×C₄	—	2	8	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	4	4	no	—
107	C₂×C₄	#390	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
109	C₂×C₄	C₂×#103	1	2	yes	—
110	C₂×C₄	C₂×#104	1	2	yes	—
111	C₂×C₄	C₈×D₄	1	2	yes	—
112	C₂×C₄	#436	1	2	yes	—
113	C₈	—	2	2	no	—
114	C₈	—	8	2	no	—
115	C₈	Q₈⋊C₈	1	4	yes	—
116	C₈	Q₈⋊C₈	1	4	yes	—
117	C₈	#487	1	2	yes	—
118	C₈	#492	1	1	yes	characteristic
119	C₈	#494	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#1493	1	2	yes	—
122	C₂²	#1954	1	1	yes	characteristic
123	C₄	—	2	8	no	—
124	C₄	—	2	2	no	—
125	C₄	—	8	4	no	—
126	C₄	C₂×Q₈⋊C₈	1	2	yes	—
127	C₄	#1956	1	1	yes	characteristic
128	C₄	#1962	1	1	yes	characteristic
129	C₄	#2170	1	4	yes	—
130	C₂	—	2	2	no	—
131	C₂	#9637	1	1	yes	characteristic
132	C₂	#13195	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21732

C₂

yes

maximal, characteristic

#12900

C₂

yes

maximal, characteristic

C₄²×C₁₆

C₂

yes

maximal, characteristic

#11608

C₂

yes

maximal

#3160

C₄

yes

characteristic

#2960

C₂²

yes

—

C₂×#364

—

#2150

C₂²

yes

—

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

C₂²

yes

—

#1994

C₄

yes

—

#1945

—

#1848

C₄

yes

characteristic

#1772

—

C₄²×C₈

C₂²

yes

characteristic

Q₈⋊C₁₆

—

#533

C₂×C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#500

C₈

yes

—

#499

—

#498

C₈

yes

—

#446

—

C₈×Q₈

—

—

—

—

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄³

C₂×C₄

yes

characteristic

#364

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

C₂×C₈

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

—

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#108

—

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

C₂×C₈

yes

characteristic

C₄×C₈

M₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

#131

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

C₂²⋊C₈

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

#104

yes

—

C₂×C₈

#121

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×SD₁₆

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#103

yes

—

C₄²

C₄×D₄

yes

—

C₄²

C₂²×C₈

yes

characteristic

C₄²

C₂×M₁₆

yes

characteristic

C₁₆

—

100

C₁₆

—

101

C₂³

#408

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

#390

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

109

C₂×C₄

C₂×#103

yes

—

110

C₂×C₄

C₂×#104

yes

—

111

C₂×C₄

C₈×D₄

yes

—

112

C₂×C₄

#436

yes

—

113

C₈

—

114

C₈

—

115

C₈

Q₈⋊C₈

yes

—

116

C₈

Q₈⋊C₈

yes

—

117

C₈

#487

yes

—

118

C₈

#492

yes

characteristic

119

C₈

#494

yes

characteristic

120

C₂²

—

121

C₂²

#1493

yes

—

122

C₂²

#1954

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂×Q₈⋊C₈

yes

—

127

C₄

#1956

yes

characteristic

128

C₄

#1962

yes

characteristic

129

C₄

#2170

yes

—

130

C₂

—

131

C₂

#9637

yes

characteristic

132

C₂

#13195

yes

—

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1954
prime radical	C₂³	#408
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₈

#127

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₈ ◁ 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
#21732	2
#12900	2
C₄²×C₁₆	2
#11608	2

subgroup

index

#21732

#12900

C₄²×C₁₆

#11608

Subgroups — #140509

Conjugacy classes of subgroups

Maximal subgroups