Group #141245 — subgroups

Subgroups — #141245

Order 512.

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

normal subgroups only

This group has 348 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	C₂×#1713	C₂	1	1	yes	maximal, characteristic
2	#13279	C₂	1	2	yes	maximal
3	#12004	C₂	1	1	yes	maximal, characteristic
4	#11941	C₂	1	2	yes	maximal
5	C₈×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	—	2	1	no	—
7	C₂×#500	C₂²	1	2	yes	—
8	C₂×#494	—	2	1	no	—
9	#2150	C₂²	1	2	yes	—
10	C₄×#108	C₂²	1	1	yes	characteristic
11	#1713	—	2	2	no	—
12	#1713	C₄	1	4	yes	—
13	#1564	C₂²	1	1	yes	characteristic
14	C₈×M₁₆	—	2	1	no	—
15	C₂×C₈²	C₂²	1	1	yes	characteristic
16	#1387	—	2	4	no	—
17	#1370	—	2	2	no	—
18	#1365	—	2	4	no	—
19	#1278	—	2	1	no	—
20	C₂²×Q₁₆	—	4	2	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	#500	—	2	4	no	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	—	4	2	no	—
28	#500	C₂×C₄	1	4	yes	—
29	#494	—	2	4	no	—
30	#494	—	4	1	no	—
31	#494	—	4	1	no	—
32	#431	—	2	1	no	—
33	C₂×#108	D₄	1	2	yes	—
34	C₂×#107	—	2	2	no	—
35	C₂×#107	—	2	2	no	—
36	C₂×#107	—	4	2	no	—
37	C₄×M₁₆	—	2	1	no	—
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄⋊C₁₆	—	2	2	no	—
43	#341	—	2	1	no	—
44	#335	D₄	1	2	yes	—
45	#334	—	2	1	no	—
46	#322	—	4	1	no	—
47	C₈²	—	4	1	no	—
48	C₈²	C₂×C₄	1	2	yes	—
49	C₈²	D₄	1	2	yes	—
50	C₂²×Q₈	—	4	2	no	—
51	C₂×Q₁₆	—	4	4	no	—
52	C₂×Q₁₆	—	4	4	no	—
53	C₂×Q₁₆	—	8	8	no	—
54	C₂×M₁₆	—	2	2	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₂²×C₈	SD₁₆	1	2	yes	—
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₈	#41	1	1	yes	characteristic
59	C₄⋊Q₈	—	2	4	no	—
60	C₄⋊Q₈	—	4	2	no	—
61	C₄⋊Q₈	—	4	2	no	—
62	C₄⋊Q₈	—	4	2	no	—
63	C₂×C₄⋊C₄	—	4	2	no	—
64	C₂×C₄⋊C₄	—	4	2	no	—
65	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
66	M₃₂	—	4	4	no	—
67	C₂×C₁₆	—	4	2	no	—
68	#108	—	2	2	no	—
69	#108	Q₁₆	1	4	yes	complemented
70	#107	—	2	4	no	—
71	#107	—	2	4	no	—
72	#107	—	4	8	no	—
73	#107	—	4	2	no	—
74	#107	—	4	2	no	—
75	C₄⋊C₈	—	2	2	no	—
76	C₂²⋊C₈	—	4	1	no	—
77	#97	—	4	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	4	1	no	—
80	C₄×C₈	—	4	1	no	—
81	C₄×C₈	C₂²⋊C₄	1	2	yes	—
82	C₄×C₈	C₂²⋊C₄	1	2	yes	—
83	C₄×C₈	D₈	1	1	yes	characteristic
84	C₄×C₈	D₈	1	1	yes	characteristic
85	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₄×C₈	C₂×D₄	1	1	yes	characteristic
87	C₄×C₈	C₂×D₄	1	1	yes	characteristic
88	C₄×C₈	#41	1	1	yes	characteristic
89	C₂×Q₈	—	4	4	no	—
90	C₂×Q₈	—	8	8	no	—
91	C₂×Q₈	—	8	2	no	—
92	C₂²×C₄	—	8	2	no	—
93	C₂²×C₄	#121	1	1	yes	characteristic
94	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
95	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
96	Q₁₆	—	8	8	no	—
97	Q₁₆	—	8	8	no	—
98	M₁₆	—	2	4	no	—
99	M₁₆	—	4	2	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	#102	1	2	yes	—
107	C₂×C₈	#102	1	2	yes	—
108	C₂×C₈	C₄×D₄	1	1	yes	characteristic
109	C₂×C₈	C₄×D₄	1	1	yes	characteristic
110	C₂×C₈	#124	1	1	yes	characteristic
111	C₂×C₈	#127	1	1	yes	characteristic
112	C₂×C₈	C₂×Q₁₆	1	2	yes	—
113	C₂×C₈	#135	1	2	yes	—
114	C₄⋊C₄	—	4	4	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₄²	#121	1	1	yes	characteristic
120	C₄²	C₂×D₈	1	1	yes	characteristic
121	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₁₆	—	8	2	no	—
123	C₂³	#460	1	1	yes	characteristic
124	Q₈	—	8	8	no	—
125	Q₈	—	16	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	8	4	no	—
129	C₂×C₄	—	16	4	no	—
130	C₂×C₄	#390	1	1	yes	characteristic
131	C₂×C₄	C₂×#102	1	1	yes	characteristic
132	C₂×C₄	#418	1	1	yes	characteristic
133	C₂×C₄	#459	1	1	yes	characteristic
134	C₂×C₄	#459	1	1	yes	characteristic
135	C₂×C₄	#460	1	1	yes	characteristic
136	C₂×C₄	#467	1	2	yes	—
137	C₂×C₄	#473	1	2	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	2	2	no	—
140	C₈	—	2	1	no	—
141	C₈	—	2	1	no	—
142	C₈	—	4	4	no	—
143	C₈	—	8	2	no	—
144	C₈	#361	1	4	yes	—
145	C₈	C₄×Q₁₆	1	2	yes	—
146	C₈	#494	1	1	yes	characteristic
147	C₈	#495	1	1	yes	characteristic
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#1670	1	2	yes	—
151	C₂²	#1982	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	—	16	4	no	—
155	C₄	#1669	1	2	yes	—
156	C₄	#1969	1	1	yes	characteristic
157	C₄	#1979	1	1	yes	characteristic
158	C₄	C₂×#361	1	2	yes	—
159	C₄	#2214	1	2	yes	—
160	C₂	—	2	2	no	—
161	C₂	#9733	1	1	yes	characteristic
162	C₂	#13257	1	1	yes	characteristic
163	C₂	#13262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1713

C₂

yes

maximal, characteristic

#13279

C₂

yes

maximal

#12004

C₂

yes

maximal, characteristic

#11941

C₂

yes

maximal

C₈×#108

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

C₂²

yes

—

C₂×#494

—

#2150

C₂²

yes

—

C₄×#108

C₂²

yes

characteristic

#1713

—

#1713

C₄

yes

—

#1564

C₂²

yes

characteristic

C₈×M₁₆

—

C₂×C₈²

C₂²

yes

—

—

—

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

—

—

—

—

—

C₂×C₄

yes

—

—

—

—

—

D₄

yes

—

C₂×#107

—

C₂×#107

—

C₂×#107

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#341

—

#335

D₄

yes

—

#334

—

#322

—

C₈²

—

C₈²

C₂×C₄

yes

—

C₈²

D₄

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#108

—

#108

Q₁₆

yes

—

—

—

—

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

#102

yes

—

107

C₂×C₈

#102

yes

—

108

C₂×C₈

C₄×D₄

yes

characteristic

109

C₂×C₈

C₄×D₄

yes

characteristic

110

C₂×C₈

#124

yes

characteristic

111

C₂×C₈

#127

yes

characteristic

112

C₂×C₈

C₂×Q₁₆

yes

—

113

C₂×C₈

#135

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

C₂×C₂²⋊C₄

yes

characteristic

119

C₄²

#121

yes

characteristic

120

C₄²

C₂×D₈

yes

characteristic

121

C₄²

AGL(1,ℤ/8)

yes

characteristic

122

C₁₆

—

123

C₂³

#460

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#390

yes

characteristic

131

C₂×C₄

C₂×#102

yes

characteristic

132

C₂×C₄

#418

yes

characteristic

133

C₂×C₄

#459

yes

characteristic

134

C₂×C₄

#459

yes

characteristic

135

C₂×C₄

#460

yes

characteristic

136

C₂×C₄

#467

yes

—

137

C₂×C₄

#473

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

#361

yes

—

145

C₈

C₄×Q₁₆

yes

—

146

C₈

#494

yes

characteristic

147

C₈

#495

yes

characteristic

148

C₂²

—

149

C₂²

—

150

C₂²

#1670

yes

—

151

C₂²

#1982

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

#1669

yes

—

156

C₄

#1969

yes

characteristic

157

C₄

#1979

yes

characteristic

158

C₄

C₂×#361

yes

—

159

C₄

#2214

yes

—

160

C₂

—

161

C₂

#9733

yes

characteristic

162

C₂

#13257

yes

characteristic

163

C₂

#13262

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1982
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#460
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1982

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#460

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
69. #108	Q₁₆	semidirect

H ≅ G/N

type

69. #108

Q₁₆

semidirect

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

subgroup	index
C₂×#1713	2
#13279	2
#12004	2
#11941	2
C₈×#108	2

subgroup

index

Subgroups — #141245

Conjugacy classes of subgroups

Complements

Maximal subgroups