Group #145578 — subgroups

Subgroups — #145578

Order 512.

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

normal subgroups only

This group has 305 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×#1364	C₂	1	2	yes	maximal
4	#9731	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	#1981	—	2	2	no	—
14	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×#335	C₂²	1	2	yes	—
16	C₂×#332	—	2	2	no	—
17	#1374	—	2	1	no	—
18	#1364	—	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×#103	—	4	2	no	—
32	C₂×#103	—	4	2	no	—
33	C₂×C₄×C₈	—	2	1	no	—
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
36	#366	—	2	4	no	—
37	#366	—	2	4	no	—
38	C₄⋊C₁₆	—	2	2	no	—
39	C₄⋊C₁₆	—	4	2	no	—
40	C₄×C₁₆	—	2	2	no	—
41	#336	—	2	1	no	—
42	#335	—	2	2	no	—
43	#335	—	2	2	no	—
44	#332	—	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	#103	—	8	4	no	—
77	#103	—	8	4	no	—
78	C₄×C₈	—	4	1	no	—
79	C₄×C₈	—	4	1	no	—
80	C₄×C₈	C₂×D₄	1	2	yes	—
81	C₄×C₈	#41	1	2	yes	—
82	#95	—	2	1	no	—
83	C₂×Q₈	—	4	4	no	—
84	C₂×Q₈	—	8	2	no	—
85	C₂×Q₈	—	8	2	no	—
86	C₂×Q₈	—	8	2	no	—
87	C₂×Q₈	—	8	2	no	—
88	C₂×Q₈	—	8	2	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	8	2	no	—
92	C₂²×C₄	#121	1	1	yes	characteristic
93	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
94	C₂²×C₄	#135	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	4	1	no	—
100	C₂×C₈	—	8	2	no	—
101	C₂×C₈	—	8	2	no	—
102	C₂×C₈	—	8	2	no	—
103	C₂×C₈	#120	1	1	yes	characteristic
104	C₂×C₈	#123	1	2	yes	—
105	C₂×C₈	#124	1	1	yes	characteristic
106	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
107	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
108	C₂×C₈	#137	1	1	yes	characteristic
109	C₂×C₈	#137	1	1	yes	characteristic
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	4	no	—
113	C₄⋊C₄	—	4	4	no	—
114	C₄⋊C₄	—	8	2	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₄⋊C₄	D₁₆	1	2	yes	—
117	C₄²	—	4	1	no	—
118	C₄²	—	4	1	no	—
119	C₄²	#121	1	2	yes	—
120	C₄²	AGL(1,ℤ/8)	1	2	yes	—
121	C₁₆	—	2	2	no	—
122	C₁₆	—	2	2	no	—
123	C₁₆	—	8	1	no	—
124	C₂³	#466	1	1	yes	characteristic
125	Q₈	—	8	4	no	—
126	Q₈	—	8	4	no	—
127	Q₈	—	16	2	no	—
128	Q₈	—	16	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	8	4	no	—
138	C₂×C₄	—	16	2	no	—
139	C₂×C₄	—	16	2	no	—
140	C₂×C₄	#394	1	1	yes	characteristic
141	C₂×C₄	#459	1	2	yes	—
142	C₂×C₄	#461	1	1	yes	characteristic
143	C₂×C₄	#462	1	1	yes	characteristic
144	C₂×C₄	#463	1	2	yes	—
145	C₂×C₄	#469	1	1	yes	characteristic
146	C₂×C₄	#473	1	1	yes	characteristic
147	C₂×C₄	#473	1	1	yes	characteristic
148	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
149	C₂×C₄	#508	1	1	yes	characteristic
150	C₂×C₄	#509	1	2	yes	—
151	C₈	—	2	2	no	—
152	C₈	—	8	1	no	—
153	C₈	—	16	2	no	—
154	C₈	#448	1	1	yes	characteristic
155	C₈	#451	1	1	yes	characteristic
156	C₈	#487	1	1	yes	characteristic
157	C₈	#491	1	1	yes	characteristic
158	C₂²	#1673	1	1	yes	characteristic
159	C₂²	#1674	1	1	yes	characteristic
160	C₂²	#2054	1	1	yes	characteristic
161	C₂²	#2215	1	1	yes	characteristic
162	C₂²	#2218	1	1	yes	characteristic
163	C₂²	#2222	1	1	yes	characteristic
164	C₂²	#2222	1	1	yes	characteristic
165	C₄	—	2	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	8	1	no	—
168	C₄	—	16	4	no	—
169	C₄	#1693	1	1	yes	characteristic
170	C₄	#1694	1	1	yes	characteristic
171	C₄	#2043	1	1	yes	characteristic
172	C₄	#2047	1	1	yes	characteristic
173	C₄	#2206	1	2	yes	—
174	C₄	#2210	1	2	yes	—
175	C₂	#10545	1	1	yes	characteristic
176	C₂	#12451	1	1	yes	characteristic
177	C₂	#12454	1	1	yes	characteristic
178	C₂	#12457	1	1	yes	characteristic
179	C₂	#12460	1	1	yes	characteristic
180	C₂	#13441	1	1	yes	characteristic
181	C₂	#13448	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

#13159

C₂

yes

maximal, characteristic

C₂×#1364

C₂

yes

maximal

#9731

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

#1981

—

C₈×C₄⋊C₄

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

C₂×#332

—

#1374

—

#1364

—

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×#103

—

C₂×#103

—

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

—

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

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

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

#120

yes

characteristic

104

C₂×C₈

#123

yes

—

105

C₂×C₈

#124

yes

characteristic

106

C₂×C₈

C₂×SD₁₆

yes

characteristic

107

C₂×C₈

C₂×Q₁₆

yes

characteristic

108

C₂×C₈

#137

yes

characteristic

109

C₂×C₈

#137

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

D₁₆

yes

—

117

C₄²

—

118

C₄²

—

119

C₄²

#121

yes

—

120

C₄²

AGL(1,ℤ/8)

yes

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

#466

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#394

yes

characteristic

141

C₂×C₄

#459

yes

—

142

C₂×C₄

#461

yes

characteristic

143

C₂×C₄

#462

yes

characteristic

144

C₂×C₄

#463

yes

—

145

C₂×C₄

#469

yes

characteristic

146

C₂×C₄

#473

yes

characteristic

147

C₂×C₄

#473

yes

characteristic

148

C₂×C₄

C₂×D₁₆

yes

characteristic

149

C₂×C₄

#508

yes

characteristic

150

C₂×C₄

#509

yes

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

#448

yes

characteristic

155

C₈

#451

yes

characteristic

156

C₈

#487

yes

characteristic

157

C₈

#491

yes

characteristic

158

C₂²

#1673

yes

characteristic

159

C₂²

#1674

yes

characteristic

160

C₂²

#2054

yes

characteristic

161

C₂²

#2215

yes

characteristic

162

C₂²

#2218

yes

characteristic

163

C₂²

#2222

yes

characteristic

164

C₂²

#2222

yes

characteristic

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

#1693

yes

characteristic

170

C₄

#1694

yes

characteristic

171

C₄

#2043

yes

characteristic

172

C₄

#2047

yes

characteristic

173

C₄

#2206

yes

—

174

C₄

#2210

yes

—

175

C₂

#10545

yes

characteristic

176

C₂

#12451

yes

characteristic

177

C₂

#12454

yes

characteristic

178

C₂

#12457

yes

characteristic

179

C₂

#12460

yes

characteristic

180

C₂

#13441

yes

characteristic

181

C₂

#13448

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×#1364	2
#9731	2
#7154	2

subgroup

index

Subgroups — #145578

Conjugacy classes of subgroups

Maximal subgroups