Group #145580 — subgroups

Subgroups — #145580

Order 512.

289 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 96 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×#2252	C₂	1	1	yes	maximal, characteristic
2	#13159	C₂	1	1	yes	maximal, characteristic
3	C₂×#1365	C₂	1	2	yes	maximal
4	#9730	C₂	1	2	yes	maximal
5	#7154	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	2	yes	—
7	#2252	—	2	4	no	—
8	C₂×#366	—	2	1	no	—
9	C₂×#366	—	2	1	no	—
10	C₂×C₄⋊C₁₆	—	2	1	no	—
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1987	—	2	2	no	—
14	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×#335	C₂²	1	2	yes	—
16	C₂×#333	—	2	2	no	—
17	#1374	—	2	1	no	—
18	#1365	—	2	4	no	—
19	C₂×C₄⋊Q₈	—	2	2	no	—
20	C₂²×C₁₆	—	2	1	no	—
21	C₂²×C₁₆	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	#500	—	2	2	no	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	D₄	1	4	yes	—
27	C₂×#107	—	2	2	no	—
28	C₂×#107	—	2	2	no	—
29	C₂×C₄⋊C₈	—	2	2	no	—
30	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
31	C₂×C₄×C₈	—	2	1	no	—
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
34	#366	—	2	4	no	—
35	#366	—	2	4	no	—
36	C₄⋊C₁₆	—	2	2	no	—
37	C₄⋊C₁₆	—	4	2	no	—
38	C₄×C₁₆	—	2	2	no	—
39	#340	—	4	2	no	—
40	#340	—	4	2	no	—
41	#336	—	2	1	no	—
42	#335	—	2	2	no	—
43	#335	—	2	2	no	—
44	#333	—	4	4	no	—
45	C₂²×Q₈	—	4	2	no	—
46	C₂²×C₈	—	2	1	no	—
47	C₂²×C₈	—	4	2	no	—
48	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₈	#41	1	1	yes	characteristic
50	C₄⋊Q₈	—	2	4	no	—
51	C₄⋊Q₈	—	4	2	no	—
52	C₄⋊Q₈	—	4	2	no	—
53	C₄⋊Q₈	—	4	2	no	—
54	C₂×C₄⋊C₄	—	4	2	no	—
55	C₂×C₄⋊C₄	—	4	2	no	—
56	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
57	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
58	C₂×C₄²	—	2	1	no	—
59	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
60	C₂×C₁₆	—	2	1	no	—
61	C₂×C₁₆	—	2	1	no	—
62	C₂×C₁₆	—	2	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	C₂×C₁₆	—	4	1	no	—
65	C₂×C₁₆	—	4	1	no	—
66	C₂×C₁₆	—	4	1	no	—
67	C₂×C₁₆	SD₁₆	1	2	yes	—
68	C₂×C₁₆	Q₁₆	1	2	yes	—
69	#107	—	2	4	no	—
70	#107	—	2	4	no	—
71	#107	—	4	2	no	—
72	#107	—	4	2	no	—
73	C₄⋊C₈	—	2	2	no	—
74	C₄⋊C₈	—	4	2	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₄×C₈	—	4	1	no	—
77	C₄×C₈	—	4	1	no	—
78	C₄×C₈	C₂×D₄	1	2	yes	—
79	C₄×C₈	#41	1	2	yes	—
80	#95	—	2	1	no	—
81	C₂×Q₈	—	4	4	no	—
82	C₂×Q₈	—	8	2	no	—
83	C₂×Q₈	—	16	2	no	—
84	C₂×Q₈	—	16	2	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	8	2	no	—
88	C₂²×C₄	#121	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
90	C₂²×C₄	#135	1	1	yes	characteristic
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	4	1	no	—
95	C₂×C₈	—	4	1	no	—
96	C₂×C₈	—	8	2	no	—
97	C₂×C₈	—	8	2	no	—
98	C₂×C₈	—	8	2	no	—
99	C₂×C₈	#120	1	1	yes	characteristic
100	C₂×C₈	#123	1	2	yes	—
101	C₂×C₈	#124	1	1	yes	characteristic
102	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₈	#137	1	1	yes	characteristic
105	C₂×C₈	#137	1	1	yes	characteristic
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	4	no	—
109	C₄⋊C₄	—	4	4	no	—
110	C₄⋊C₄	—	8	2	no	—
111	C₄⋊C₄	—	8	2	no	—
112	C₄⋊C₄	Q₃₂	1	2	yes	—
113	C₄²	—	4	1	no	—
114	C₄²	—	4	1	no	—
115	C₄²	#121	1	2	yes	—
116	C₄²	AGL(1,ℤ/8)	1	2	yes	—
117	C₁₆	—	2	2	no	—
118	C₁₆	—	2	2	no	—
119	C₁₆	—	8	1	no	—
120	C₂³	#466	1	1	yes	characteristic
121	Q₈	—	16	4	no	—
122	Q₈	—	16	4	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	8	4	no	—
132	C₂×C₄	—	16	2	no	—
133	C₂×C₄	—	16	2	no	—
134	C₂×C₄	#394	1	1	yes	characteristic
135	C₂×C₄	#459	1	2	yes	—
136	C₂×C₄	#461	1	1	yes	characteristic
137	C₂×C₄	#462	1	1	yes	characteristic
138	C₂×C₄	#463	1	2	yes	—
139	C₂×C₄	#469	1	1	yes	characteristic
140	C₂×C₄	#473	1	1	yes	characteristic
141	C₂×C₄	#473	1	1	yes	characteristic
142	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
143	C₂×C₄	#508	1	1	yes	characteristic
144	C₂×C₄	#510	1	2	yes	—
145	C₈	—	2	2	no	—
146	C₈	—	8	1	no	—
147	C₈	—	16	2	no	—
148	C₈	#448	1	1	yes	characteristic
149	C₈	#451	1	1	yes	characteristic
150	C₈	#487	1	1	yes	characteristic
151	C₈	#491	1	1	yes	characteristic
152	C₂²	#1673	1	1	yes	characteristic
153	C₂²	#1674	1	1	yes	characteristic
154	C₂²	#2054	1	1	yes	characteristic
155	C₂²	#2216	1	1	yes	characteristic
156	C₂²	#2219	1	1	yes	characteristic
157	C₂²	#2221	1	1	yes	characteristic
158	C₂²	#2221	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	8	1	no	—
162	C₄	—	16	4	no	—
163	C₄	#1693	1	1	yes	characteristic
164	C₄	#1694	1	1	yes	characteristic
165	C₄	#2043	1	1	yes	characteristic
166	C₄	#2047	1	1	yes	characteristic
167	C₄	#2209	1	2	yes	—
168	C₄	#2211	1	2	yes	—
169	C₂	#10545	1	1	yes	characteristic
170	C₂	#12452	1	1	yes	characteristic
171	C₂	#12455	1	1	yes	characteristic
172	C₂	#12458	1	1	yes	characteristic
173	C₂	#12461	1	1	yes	characteristic
174	C₂	#13439	1	1	yes	characteristic
175	C₂	#13450	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

#13159

C₂

yes

maximal, characteristic

C₂×#1365

C₂

yes

maximal

#9730

C₂

yes

maximal

#7154

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

—

#2252

—

C₂×#366

—

C₂×#366

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1987

—

C₈×C₄⋊C₄

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

C₂×#333

—

#1374

—

#1365

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

D₄

yes

—

C₂×#107

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

D₄

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

—

—

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

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×C₄⋊C₄

D₈

yes

characteristic

C₂×C₄⋊C₄

D₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#120

yes

characteristic

100

C₂×C₈

#123

yes

—

101

C₂×C₈

#124

yes

characteristic

102

C₂×C₈

C₂×SD₁₆

yes

characteristic

103

C₂×C₈

C₂×Q₁₆

yes

characteristic

104

C₂×C₈

#137

yes

characteristic

105

C₂×C₈

#137

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

Q₃₂

yes

—

113

C₄²

—

114

C₄²

—

115

C₄²

#121

yes

—

116

C₄²

AGL(1,ℤ/8)

yes

—

117

C₁₆

—

118

C₁₆

—

119

C₁₆

—

120

C₂³

#466

yes

characteristic

121

Q₈

—

122

Q₈

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#394

yes

characteristic

135

C₂×C₄

#459

yes

—

136

C₂×C₄

#461

yes

characteristic

137

C₂×C₄

#462

yes

characteristic

138

C₂×C₄

#463

yes

—

139

C₂×C₄

#469

yes

characteristic

140

C₂×C₄

#473

yes

characteristic

141

C₂×C₄

#473

yes

characteristic

142

C₂×C₄

C₂×Q₃₂

yes

characteristic

143

C₂×C₄

#508

yes

characteristic

144

C₂×C₄

#510

yes

—

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

#448

yes

characteristic

149

C₈

#451

yes

characteristic

150

C₈

#487

yes

characteristic

151

C₈

#491

yes

characteristic

152

C₂²

#1673

yes

characteristic

153

C₂²

#1674

yes

characteristic

154

C₂²

#2054

yes

characteristic

155

C₂²

#2216

yes

characteristic

156

C₂²

#2219

yes

characteristic

157

C₂²

#2221

yes

characteristic

158

C₂²

#2221

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#1693

yes

characteristic

164

C₄

#1694

yes

characteristic

165

C₄

#2043

yes

characteristic

166

C₄

#2047

yes

characteristic

167

C₄

#2209

yes

—

168

C₄

#2211

yes

—

169

C₂

#10545

yes

characteristic

170

C₂

#12452

yes

characteristic

171

C₂

#12455

yes

characteristic

172

C₂

#12458

yes

characteristic

173

C₂

#12461

yes

characteristic

174

C₂

#13439

yes

characteristic

175

C₂

#13450

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#466
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#466

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ 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
C₂×#2252	2
#13159	2
C₂×#1365	2
#9730	2
#7154	2

subgroup

index

Subgroups — #145580

Conjugacy classes of subgroups

Maximal subgroups