Group #140355 — subgroups

Subgroups — #140355

Order 512.

210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 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	#12676	C₂	1	1	yes	maximal, characteristic
2	C₂×#1368	C₂	1	1	yes	maximal, characteristic
3	#11405	C₂	1	1	yes	maximal, characteristic
4	#9607	C₂	1	2	yes	maximal
5	#7173	C₂	1	2	yes	maximal
6	C₂×#368	C₄	1	1	yes	characteristic
7	C₂×#368	C₄	1	1	yes	characteristic
8	C₄×M₃₂	C₂²	1	1	yes	characteristic
9	C₂×#346	C₂²	1	1	yes	characteristic
10	C₄×#108	C₂²	1	2	yes	—
11	#1565	—	2	2	no	—
12	C₂×#334	C₂²	1	2	yes	—
13	#1450	C₂²	1	1	yes	characteristic
14	#1381	C₄	1	2	yes	—
15	#1368	—	2	2	no	—
16	#1313	—	2	2	no	—
17	#1283	—	2	2	no	—
18	C₂×M₃₂	—	2	1	no	—
19	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
21	#431	D₄	1	2	yes	—
22	C₂×#108	C₂×C₄	1	2	yes	—
23	C₂×#108	C₂×C₄	1	2	yes	—
24	C₂×C₄⋊C₈	—	2	2	no	—
25	C₄×M₁₆	—	2	2	no	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	#368	—	2	2	no	—
28	#368	C₈	1	4	yes	complemented
29	#346	—	2	1	no	—
30	#346	D₄	1	2	yes	—
31	#346	Q₈	1	2	yes	—
32	C₄×C₁₆	Q₈	1	2	yes	—
33	#341	—	2	2	no	—
34	#335	—	4	2	no	—
35	#334	—	2	2	no	—
36	#334	—	2	2	no	—
37	#334	—	4	2	no	—
38	#322	—	2	1	no	—
39	C₈²	—	2	1	no	—
40	C₂×M₁₆	—	2	2	no	—
41	C₂×M₁₆	—	2	2	no	—
42	C₂²×C₈	—	4	2	no	—
43	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₈	#41	1	1	yes	characteristic
45	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
46	M₃₂	—	4	1	no	—
47	M₃₂	—	4	1	no	—
48	C₂×C₁₆	—	2	1	no	—
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	2	no	—
51	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
52	C₂×C₁₆	C₄⋊C₄	1	2	yes	—
53	C₂×C₁₆	C₂×C₈	1	2	yes	—
54	C₂×C₁₆	M₁₆	1	2	yes	—
55	#108	—	2	2	no	—
56	#108	—	2	2	no	—
57	#108	—	2	2	no	—
58	#108	C₂×C₈	1	4	yes	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	4	2	no	—
61	C₄⋊C₈	—	4	2	no	—
62	C₂²⋊C₈	—	2	2	no	—
63	#97	—	4	2	no	—
64	C₄×C₈	—	2	2	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	C₄×D₄	1	1	yes	characteristic
71	C₂²×C₄	#137	1	2	yes	—
72	M₁₆	—	2	4	no	—
73	M₁₆	—	4	2	no	—
74	M₁₆	—	4	2	no	—
75	M₁₆	—	4	2	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	4	2	no	—
79	C₂×C₈	—	4	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	8	2	no	—
82	C₂×C₈	—	16	2	no	—
83	C₂×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₈	C₄×Q₈	1	1	yes	characteristic
85	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₈	#125	1	1	yes	characteristic
87	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
88	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₈	#131	1	1	yes	characteristic
90	C₂×C₈	#131	1	1	yes	characteristic
91	C₄²	#121	1	1	yes	characteristic
92	C₄²	C₄⋊Q₈	1	1	yes	characteristic
93	C₄²	AGL(1,ℤ/8)	1	2	yes	—
94	C₁₆	—	2	2	no	—
95	C₁₆	—	4	2	no	—
96	C₁₆	C₄⋊C₈	1	4	yes	—
97	C₂³	#441	1	1	yes	characteristic
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	#389	1	1	yes	characteristic
100	C₂×C₄	C₈×D₄	1	1	yes	characteristic
101	C₂×C₄	#436	1	1	yes	characteristic
102	C₂×C₄	#442	1	1	yes	characteristic
103	C₂×C₄	#444	1	1	yes	characteristic
104	C₂×C₄	#444	1	1	yes	characteristic
105	C₂×C₄	#468	1	2	yes	—
106	C₂×C₄	#501	1	2	yes	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	4	4	no	—
110	C₈	—	8	2	no	—
111	C₈	—	8	2	no	—
112	C₈	—	16	2	no	—
113	C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
114	C₈	#424	1	1	yes	characteristic
115	C₈	C₈×Q₈	1	1	yes	characteristic
116	C₈	#446	1	1	yes	characteristic
117	C₂²	—	2	2	no	—
118	C₂²	#1579	1	1	yes	characteristic
119	C₂²	#1593	1	1	yes	characteristic
120	C₂²	#1943	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	#1580	1	1	yes	characteristic
124	C₄	#1594	1	1	yes	characteristic
125	C₄	#1939	1	1	yes	characteristic
126	C₄	#1946	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	2	1	no	—
129	C₂	#9595	1	1	yes	characteristic
130	C₂	#13175	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#12676

C₂

yes

maximal, characteristic

C₂×#1368

C₂

yes

maximal, characteristic

#11405

C₂

yes

maximal, characteristic

#9607

C₂

yes

maximal

#7173

C₂

yes

maximal

C₂×#368

C₄

yes

characteristic

C₂×#368

C₄

yes

characteristic

C₄×M₃₂

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

C₄×#108

C₂²

yes

—

#1565

—

C₂×#334

C₂²

yes

—

#1450

C₂²

yes

characteristic

#1381

C₄

yes

—

#1368

—

#1313

—

#1283

—

C₂×M₃₂

—

C₂²×C₁₆

C₂×C₄

yes

characteristic

C₂²×C₁₆

C₂×C₄

yes

characteristic

#431

D₄

yes

—

C₂×#108

C₂×C₄

yes

—

C₂×#108

C₂×C₄

yes

—

C₂×C₄⋊C₈

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂³

yes

characteristic

#368

—

#368

C₈

yes

complemented

#346

—

#346

D₄

yes

—

#346

Q₈

yes

—

C₄×C₁₆

Q₈

yes

—

—

—

—

—

—

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×C₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₄⋊C₄

yes

—

C₂×C₁₆

C₂×C₈

yes

—

C₂×C₁₆

M₁₆

yes

—

—

—

—

C₂×C₈

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

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

C₄×D₄

yes

characteristic

C₂²×C₄

#137

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₄⋊C₄

yes

characteristic

C₂×C₈

C₄×Q₈

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

#125

yes

characteristic

C₂×C₈

C₂²×C₈

yes

characteristic

C₂×C₈

C₂×M₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₄²

C₄⋊Q₈

yes

characteristic

C₄²

AGL(1,ℤ/8)

yes

—

C₁₆

—

C₁₆

—

C₁₆

C₄⋊C₈

yes

—

C₂³

#441

yes

characteristic

C₂×C₄

—

C₂×C₄

#389

yes

characteristic

100

C₂×C₄

C₈×D₄

yes

characteristic

101

C₂×C₄

#436

yes

characteristic

102

C₂×C₄

#442

yes

characteristic

103

C₂×C₄

#444

yes

characteristic

104

C₂×C₄

#444

yes

characteristic

105

C₂×C₄

#468

yes

—

106

C₂×C₄

#501

yes

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

C₂×C₄⋊C₈

yes

characteristic

114

C₈

#424

yes

characteristic

115

C₈

C₈×Q₈

yes

characteristic

116

C₈

#446

yes

characteristic

117

C₂²

—

118

C₂²

#1579

yes

characteristic

119

C₂²

#1593

yes

characteristic

120

C₂²

#1943

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

#1580

yes

characteristic

124

C₄

#1594

yes

characteristic

125

C₄

#1939

yes

characteristic

126

C₄

#1946

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

#9595

yes

characteristic

130

C₂

#13175

yes

—

	subgroup	G/H
center, socle	C₂²	#1943
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#441
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1943

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#441

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
28. #368	C₈	semidirect

H ≅ G/N

type

28. #368

C₈

semidirect

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

subgroup	index
#12676	2
C₂×#1368	2
#11405	2
#9607	2
#7173	2

subgroup

index

Subgroups — #140355

Conjugacy classes of subgroups

Complements

Maximal subgroups