Group #140372 — subgroups

Subgroups — #140372

Order 512.

211 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 88 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	#13164	C₂	1	2	yes	maximal
2	#12668	C₂	1	1	yes	maximal, characteristic
3	#12011	C₂	1	1	yes	maximal, characteristic
4	#11402	C₂	1	1	yes	maximal, characteristic
5	#7179	C₂	1	2	yes	maximal
6	#2155	C₄	1	2	yes	—
7	#2153	—	2	2	no	—
8	C₂×#364	C₂²	1	2	yes	—
9	C₄×M₃₂	C₂²	1	1	yes	characteristic
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	#1772	C₂²	1	2	yes	—
12	C₂×C₈²	C₂²	1	1	yes	characteristic
13	#1385	C₄	1	1	yes	characteristic
14	#1384	C₄	1	1	yes	characteristic
15	#1378	—	2	2	no	—
16	#1373	—	2	2	no	—
17	#1312	—	2	2	no	—
18	C₂×M₃₂	—	2	2	no	—
19	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
20	C₂×M₃₂	C₂×C₄	1	1	yes	characteristic
21	C₂×M₃₂	D₄	1	1	yes	characteristic
22	C₂×M₃₂	D₄	1	1	yes	characteristic
23	C₂×#108	C₂×C₄	1	2	yes	—
24	#427	C₂×C₄	1	2	yes	—
25	#405	—	2	2	no	—
26	C₂×C₄×C₈	—	2	1	no	—
27	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
28	#365	—	2	4	no	—
29	#364	—	2	2	no	—
30	#364	—	2	2	no	—
31	#364	—	4	4	no	—
32	#346	—	2	1	no	—
33	#346	Q₈	1	2	yes	—
34	C₄×C₁₆	—	2	1	no	—
35	C₄×C₁₆	D₄	1	2	yes	—
36	#339	—	2	2	no	—
37	C₈²	—	2	1	no	—
38	C₈²	Q₈	1	1	yes	characteristic
39	C₈²	Q₈	1	1	yes	characteristic
40	C₂×M₁₆	—	2	2	no	—
41	C₂²×C₈	—	2	2	no	—
42	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
44	#117	—	2	2	no	—
45	C₂×C₄²	#41	1	1	yes	characteristic
46	M₃₂	—	2	2	no	—
47	M₃₂	—	2	2	no	—
48	M₃₂	—	2	2	no	—
49	M₃₂	—	2	2	no	—
50	M₃₂	—	4	2	no	—
51	M₃₂	—	4	2	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	2	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	#108	—	2	2	no	—
58	#108	M₁₆	1	4	yes	—
59	#107	—	2	2	no	—
60	#106	—	2	2	no	—
61	#97	—	4	2	no	—
62	C₄×C₈	—	2	2	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₄×C₈	—	4	2	no	—
66	C₄×C₈	C₂×D₄	1	1	yes	characteristic
67	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
68	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₄×C₈	#41	1	1	yes	characteristic
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	C₄×D₄	1	1	yes	characteristic
72	M₁₆	—	2	4	no	—
73	M₁₆	—	4	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	4	4	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
80	C₂×C₈	#121	1	1	yes	characteristic
81	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₈	C₂×M₁₆	1	2	yes	—
83	C₂×C₈	#131	1	2	yes	—
84	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
85	C₂×C₈	#137	1	2	yes	—
86	C₄⋊C₄	—	4	2	no	—
87	C₂²⋊C₄	—	4	2	no	—
88	C₄²	—	2	1	no	—
89	C₄²	—	4	2	no	—
90	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
91	C₄²	#125	1	1	yes	characteristic
92	C₁₆	—	2	2	no	—
93	C₁₆	—	2	2	no	—
94	C₁₆	—	4	1	no	—
95	C₁₆	—	4	1	no	—
96	C₁₆	—	8	2	no	—
97	C₂³	#443	1	1	yes	characteristic
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	#389	1	1	yes	characteristic
103	C₂×C₄	#435	1	2	yes	—
104	C₂×C₄	#440	1	2	yes	—
105	C₂×C₄	#443	1	1	yes	characteristic
106	C₈	—	2	4	no	—
107	C₈	—	4	2	no	—
108	C₈	—	4	4	no	—
109	C₈	#423	1	1	yes	characteristic
110	C₈	#424	1	1	yes	characteristic
111	C₈	#446	1	2	yes	—
112	C₈	#469	1	1	yes	characteristic
113	C₈	#470	1	1	yes	characteristic
114	C₈	#501	1	2	yes	—
115	C₂²	—	4	1	no	—
116	C₂²	#1588	1	2	yes	—
117	C₂²	#1945	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	4	1	no	—
120	C₄	—	16	2	no	—
121	C₄	#1585	1	2	yes	—
122	C₄	#1940	1	1	yes	characteristic
123	C₄	#1944	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	2	2	no	—
126	C₂	#9600	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13164

C₂

yes

maximal

#12668

C₂

yes

maximal, characteristic

#12011

C₂

yes

maximal, characteristic

#11402

C₂

yes

maximal, characteristic

#7179

C₂

yes

maximal

#2155

C₄

yes

—

#2153

—

C₂×#364

C₂²

yes

—

C₄×M₃₂

C₂²

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

#1772

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1385

C₄

yes

characteristic

#1384

C₄

yes

—

—

—

C₂×M₃₂

—

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

C₂×C₄

yes

characteristic

C₂×M₃₂

D₄

yes

characteristic

C₂×M₃₂

D₄

yes

characteristic

C₂×#108

C₂×C₄

yes

—

#427

C₂×C₄

yes

—

#405

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

Q₈

yes

—

C₄×C₁₆

—

C₄×C₁₆

D₄

yes

—

#339

—

C₈²

—

C₈²

Q₈

yes

characteristic

C₈²

Q₈

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

—

C₂×C₄²

#41

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

#108

—

#108

M₁₆

yes

—

#107

—

#106

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₄

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

—

C₂×C₈

#131

yes

—

C₂×C₈

AGL(1,ℤ/8)

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊Q₈

yes

characteristic

C₄²

#125

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#443

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#389

yes

characteristic

103

C₂×C₄

#435

yes

—

104

C₂×C₄

#440

yes

—

105

C₂×C₄

#443

yes

characteristic

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

#423

yes

characteristic

110

C₈

#424

yes

characteristic

111

C₈

#446

yes

—

112

C₈

#469

yes

characteristic

113

C₈

#470

yes

characteristic

114

C₈

#501

yes

—

115

C₂²

—

116

C₂²

#1588

yes

—

117

C₂²

#1945

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#1585

yes

—

122

C₄

#1940

yes

characteristic

123

C₄

#1944

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

#9600

yes

characteristic

	subgroup	G/H
center	C₈	#469
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#9600
prime radical	C₂³	#443
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₈

#469

commutator

C₂²×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂

C₂³

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄×M₃₂ ◁ 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
#13164	2
#12668	2
#12011	2
#11402	2
#7179	2

subgroup

index

Subgroups — #140372

Conjugacy classes of subgroups

Maximal subgroups