Group #146151 — subgroups

Subgroups — #146151

Order 512.

202 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 78 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	#13160	C₂	1	1	yes	maximal, characteristic
2	#12910	C₂	1	1	yes	maximal, characteristic
3	#9627	C₂	1	2	yes	maximal
4	#7276	C₂	1	2	yes	maximal
5	#7157	C₂	1	1	yes	maximal, characteristic
6	C₂×#368	—	2	1	no	—
7	C₂×#368	—	2	1	no	—
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×#346	C₂²	1	1	yes	characteristic
11	#1848	C₂²	1	2	yes	—
12	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
13	#1571	—	2	2	no	—
14	#1564	C₂²	1	2	yes	—
15	#1374	—	2	1	no	—
16	#1303	—	2	2	no	—
17	C₂²×C₁₆	—	2	1	no	—
18	C₂²×C₁₆	D₄	1	1	yes	characteristic
19	C₂²×C₁₆	D₄	1	1	yes	characteristic
20	#431	—	2	2	no	—
21	C₂×#108	—	2	2	no	—
22	C₂×#108	—	2	2	no	—
23	#423	—	2	2	no	—
24	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	—	2	1	no	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
28	#368	—	4	2	no	—
29	#368	—	4	2	no	—
30	C₄⋊C₁₆	—	2	2	no	—
31	C₄⋊C₁₆	—	4	2	no	—
32	#346	—	2	2	no	—
33	#336	—	2	1	no	—
34	#335	—	2	2	no	—
35	#335	—	4	4	no	—
36	#334	Q₈	1	4	yes	—
37	C₂×M₁₆	—	4	2	no	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₈	#41	1	1	yes	characteristic
41	C₂×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄²	—	2	1	no	—
43	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
44	C₂×C₁₆	—	2	1	no	—
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	—	2	1	no	—
47	C₂×C₁₆	—	2	1	no	—
48	C₂×C₁₆	—	4	1	no	—
49	C₂×C₁₆	—	4	1	no	—
50	C₂×C₁₆	—	4	1	no	—
51	C₂×C₁₆	D₈	1	2	yes	—
52	C₂×C₁₆	Q₁₆	1	2	yes	—
53	#108	—	4	2	no	—
54	#108	—	4	2	no	—
55	#108	—	8	2	no	—
56	C₄⋊C₈	—	2	4	no	—
57	C₄⋊C₈	—	2	2	no	—
58	C₄⋊C₈	—	4	2	no	—
59	C₄⋊C₈	—	4	2	no	—
60	C₂²⋊C₈	—	4	2	no	—
61	C₄×C₈	—	4	1	no	—
62	C₄×C₈	—	4	1	no	—
63	C₄×C₈	—	4	2	no	—
64	C₄×C₈	C₂×Q₈	1	2	yes	—
65	C₄×C₈	#41	1	2	yes	—
66	#95	Q₁₆	1	1	yes	characteristic
67	#95	Q₁₆	1	1	yes	characteristic
68	C₂²×C₄	—	2	1	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	#121	1	1	yes	characteristic
71	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
72	C₂²×C₄	#135	1	1	yes	characteristic
73	M₁₆	—	8	2	no	—
74	M₁₆	—	8	2	no	—
75	C₂×C₈	—	2	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	4	1	no	—
78	C₂×C₈	—	4	1	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	4	4	no	—
81	C₂×C₈	—	8	2	no	—
82	C₂×C₈	#123	1	1	yes	characteristic
83	C₂×C₈	#125	1	2	yes	—
84	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₈	C₂×D₈	1	1	yes	characteristic
86	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
87	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×C₈	#137	1	1	yes	characteristic
89	C₄⋊C₄	—	4	2	no	—
90	C₄²	—	4	1	no	—
91	C₄²	—	4	1	no	—
92	C₄²	C₂²⋊Q₈	1	2	yes	—
93	C₄²	AGL(1,ℤ/8)	1	2	yes	—
94	C₁₆	—	2	2	no	—
95	C₁₆	—	2	2	no	—
96	C₁₆	—	8	1	no	—
97	C₂³	#467	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	#398	1	1	yes	characteristic
107	C₂×C₄	#459	1	1	yes	characteristic
108	C₂×C₄	#462	1	1	yes	characteristic
109	C₂×C₄	#469	1	1	yes	characteristic
110	C₂×C₄	#472	1	1	yes	characteristic
111	C₂×C₄	#473	1	1	yes	characteristic
112	C₂×C₄	#477	1	2	yes	—
113	C₂×C₄	#478	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	8	4	no	—
116	C₈	—	8	1	no	—
117	C₈	—	16	2	no	—
118	C₈	#480	1	1	yes	characteristic
119	C₈	#484	1	1	yes	characteristic
120	C₈	#500	1	1	yes	characteristic
121	C₈	#501	1	1	yes	characteristic
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	#1669	1	1	yes	characteristic
125	C₂²	#1676	1	1	yes	characteristic
126	C₂²	#2087	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	2	no	—
129	C₄	—	4	2	no	—
130	C₄	—	8	1	no	—
131	C₄	#1687	1	1	yes	characteristic
132	C₄	#1694	1	1	yes	characteristic
133	C₄	#2083	1	1	yes	characteristic
134	C₄	#2089	1	1	yes	characteristic
135	C₂	—	2	1	no	—
136	C₂	—	2	1	no	—
137	C₂	#10650	1	1	yes	characteristic
138	C₂	#13519	1	1	yes	characteristic
139	C₂	#13520	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13160

C₂

yes

maximal, characteristic

#12910

C₂

yes

maximal, characteristic

#9627

C₂

yes

maximal

#7276

C₂

yes

maximal

#7157

C₂

yes

maximal, characteristic

C₂×#368

—

C₂×#368

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×#346

C₂²

yes

characteristic

#1848

C₂²

yes

—

C₈×C₄⋊C₄

C₂²

yes

characteristic

#1571

—

#1564

C₂²

yes

—

#1374

—

#1303

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

D₄

yes

characteristic

#368

—

#368

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

—

—

—

—

Q₈

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

D₈

yes

—

C₂×C₁₆

Q₁₆

yes

—

#108

—

#108

—

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

Q₁₆

yes

characteristic

#95

Q₁₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×Q₁₆

yes

characteristic

C₂²×C₄

#135

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#123

yes

characteristic

C₂×C₈

#125

yes

—

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊Q₈

yes

—

C₄²

AGL(1,ℤ/8)

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#467

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#398

yes

characteristic

107

C₂×C₄

#459

yes

characteristic

108

C₂×C₄

#462

yes

characteristic

109

C₂×C₄

#469

yes

characteristic

110

C₂×C₄

#472

yes

characteristic

111

C₂×C₄

#473

yes

characteristic

112

C₂×C₄

#477

yes

—

113

C₂×C₄

#478

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

#480

yes

characteristic

119

C₈

#484

yes

characteristic

120

C₈

#500

yes

characteristic

121

C₈

#501

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

#1669

yes

characteristic

125

C₂²

#1676

yes

characteristic

126

C₂²

#2087

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#1687

yes

characteristic

132

C₄

#1694

yes

characteristic

133

C₄

#2083

yes

characteristic

134

C₄

#2089

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

#10650

yes

characteristic

138

C₂

#13519

yes

characteristic

139

C₂

#13520

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2087
prime radical	C₂³	#467
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₂×C₄

#469

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

C₂²

C₂³

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ 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
#13160	2
#12910	2
#9627	2
#7276	2
#7157	2

subgroup

index

Subgroups — #146151

Conjugacy classes of subgroups

Maximal subgroups