Group #140560 — subgroups

Subgroups — #140560

Order 512.

306 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 106 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	#21735	C₂	1	1	yes	maximal, characteristic
2	#12899	C₂	1	1	yes	maximal, characteristic
3	#12593	C₂	1	1	yes	maximal, characteristic
4	#11606	C₂	1	4	yes	maximal
5	#3162	C₄	1	1	yes	characteristic
6	#2960	C₂²	1	2	yes	—
7	C₂×C₄⋊C₁₆	—	2	2	no	—
8	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
9	C₄×M₃₂	C₂²	1	2	yes	—
10	#1995	C₄	1	2	yes	—
11	#1946	—	2	2	no	—
12	#1849	C₄	1	1	yes	characteristic
13	#1772	—	2	2	no	—
14	#1726	C₂²	1	1	yes	characteristic
15	#1375	—	2	2	no	—
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	#501	—	2	4	no	—
21	#446	—	2	4	no	—
22	C₈×Q₈	—	2	4	no	—
23	C₂×#108	—	2	2	no	—
24	#427	—	2	2	no	—
25	#424	—	2	2	no	—
26	#423	C₂×C₄	1	2	yes	—
27	#405	—	2	2	no	—
28	C₂×#97	C₂×C₄	1	1	yes	characteristic
29	C₂×#97	D₄	1	2	yes	—
30	C₂×C₄×C₈	D₄	1	2	yes	—
31	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
32	#376	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₁₆	—	2	4	no	—
34	C₄⋊C₁₆	—	4	4	no	—
35	#346	—	2	2	no	—
36	C₄×C₁₆	D₄	1	4	yes	—
37	#339	—	2	4	no	—
38	C₂×M₁₆	—	2	2	no	—
39	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
40	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
42	C₄⋊Q₈	M₁₆	1	4	yes	—
43	#125	—	2	2	no	—
44	C₂²⋊Q₈	—	2	4	no	—
45	C₄×Q₈	—	2	4	no	—
46	#117	—	2	2	no	—
47	C₂×C₄²	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
49	M₃₂	—	4	4	no	—
50	C₂×C₁₆	—	2	4	no	—
51	C₂×C₁₆	—	4	2	no	—
52	C₂×C₁₆	—	4	4	no	—
53	#108	—	4	4	no	—
54	#107	—	2	4	no	—
55	#106	—	2	4	no	—
56	C₄⋊C₈	—	2	4	no	—
57	C₄⋊C₈	—	4	4	no	—
58	#97	—	2	2	no	—
59	#97	—	4	2	no	—
60	#97	—	4	2	no	—
61	C₄×C₈	—	4	2	no	—
62	C₄×C₈	SD₁₆	1	4	yes	—
63	C₄×C₈	Q₁₆	1	4	yes	—
64	C₄×C₈	C₂×D₄	1	2	yes	—
65	C₄×C₈	#41	1	2	yes	—
66	#95	—	2	2	no	—
67	C₂×Q₈	—	2	4	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄	#131	1	2	yes	—
71	M₁₆	—	4	4	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	4	2	no	—
76	C₂×C₈	#103	1	4	yes	—
77	C₂×C₈	#121	1	2	yes	—
78	C₂×C₈	#124	1	1	yes	characteristic
79	C₂×C₈	#127	1	1	yes	characteristic
80	C₂×C₈	C₂×SD₁₆	1	2	yes	—
81	C₂×C₈	C₂×Q₁₆	1	2	yes	—
82	C₄⋊C₄	—	2	4	no	—
83	C₄⋊C₄	—	2	4	no	—
84	C₄⋊C₄	—	4	4	no	—
85	C₂²⋊C₄	—	4	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	4	2	no	—
88	C₄²	#103	1	4	yes	—
89	C₄²	C₄×D₄	1	2	yes	—
90	C₄²	C₂×M₁₆	1	2	yes	—
91	C₁₆	—	4	4	no	—
92	C₁₆	—	8	2	no	—
93	C₂³	#408	1	1	yes	characteristic
94	Q₈	—	4	8	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	#390	1	1	yes	characteristic
100	C₂×C₄	#407	1	1	yes	characteristic
101	C₂×C₄	C₂×#103	1	2	yes	—
102	C₂×C₄	#421	1	2	yes	—
103	C₂×C₄	#435	1	4	yes	—
104	C₈	—	2	2	no	—
105	C₈	—	2	4	no	—
106	C₈	—	8	2	no	—
107	C₈	#487	1	2	yes	—
108	C₈	#492	1	1	yes	characteristic
109	C₈	#494	1	1	yes	characteristic
110	C₂²	—	2	2	no	—
111	C₂²	#1481	1	2	yes	—
112	C₂²	#1955	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	4	4	no	—
115	C₄	—	8	4	no	—
116	C₄	#1487	1	2	yes	—
117	C₄	#1956	1	1	yes	characteristic
118	C₄	#1963	1	1	yes	characteristic
119	C₂	—	2	2	no	—
120	C₂	#9643	1	1	yes	characteristic
121	C₂	#13198	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21735

C₂

yes

maximal, characteristic

#12899

C₂

yes

maximal, characteristic

#12593

C₂

yes

maximal, characteristic

#11606

C₂

yes

maximal

#3162

C₄

yes

characteristic

#2960

C₂²

yes

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

—

C₄×M₃₂

C₂²

yes

—

#1995

C₄

yes

—

#1946

—

#1849

C₄

yes

characteristic

#1772

—

#1726

C₂²

yes

characteristic

#1375

—

Q₈⋊C₁₆

—

#533

C₂×C₄

yes

—

C₂×M₃₂

—

C₂²×C₁₆

—

#501

—

#446

—

C₈×Q₈

—

—

—

—

C₂×C₄

yes

—

#405

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

C₂³

yes

characteristic

#376

C₂×C₄

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

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

M₁₆

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

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

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

#131

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#103

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

#103

yes

—

C₄²

C₄×D₄

yes

—

C₄²

C₂×M₁₆

yes

—

C₁₆

—

C₁₆

—

C₂³

#408

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#390

yes

characteristic

100

C₂×C₄

#407

yes

characteristic

101

C₂×C₄

C₂×#103

yes

—

102

C₂×C₄

#421

yes

—

103

C₂×C₄

#435

yes

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

#487

yes

—

108

C₈

#492

yes

characteristic

109

C₈

#494

yes

characteristic

110

C₂²

—

111

C₂²

#1481

yes

—

112

C₂²

#1955

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#1487

yes

—

117

C₄

#1956

yes

characteristic

118

C₄

#1963

yes

characteristic

119

C₂

—

120

C₂

#9643

yes

characteristic

121

C₂

#13198

yes

—

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1955
prime radical	C₂³	#408
LCS	G ▷ C₂×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

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

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
#21735	2
#12899	2
#12593	2
#11606	2

subgroup

index

Subgroups — #140560

Conjugacy classes of subgroups

Maximal subgroups