Group #144156 — subgroups

Subgroups — #144156

Order 512.

441 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 120 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×#2241	C₂	1	2	yes	maximal
2	C₂×#1713	C₂	1	1	yes	maximal, characteristic
3	#13282	C₂	1	2	yes	maximal
4	#9825	C₂	1	1	yes	maximal, characteristic
5	#7136	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	C₂×#494	C₂²	1	1	yes	characteristic
10	#2241	—	2	4	no	—
11	#2241	—	2	4	no	—
12	C₂×#358	—	2	4	no	—
13	C₂×C₄×C₁₆	C₂²	1	2	yes	—
14	#1990	—	2	1	no	—
15	#1990	—	2	1	no	—
16	#1845	C₂²	1	1	yes	characteristic
17	#1713	—	2	4	no	—
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1380	—	2	4	no	—
20	C₂²×Q₁₆	—	2	2	no	—
21	C₂²×Q₁₆	—	4	2	no	—
22	C₂×C₄⋊Q₈	—	2	2	no	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂²×C₁₆	D₄	1	4	yes	—
25	#500	—	2	4	no	—
26	#500	—	2	2	no	—
27	#500	—	2	2	no	—
28	#500	—	2	2	no	—
29	#500	—	4	2	no	—
30	#500	D₄	1	4	yes	—
31	#494	—	2	4	no	—
32	#494	—	2	2	no	—
33	#494	—	2	1	no	—
34	#494	—	4	2	no	—
35	#494	D₄	1	2	yes	—
36	C₂×#107	—	2	2	no	—
37	C₂×#107	—	2	2	no	—
38	C₂×#107	—	2	2	no	—
39	C₂×#107	—	4	2	no	—
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	#384	—	2	1	no	—
44	#358	—	2	16	no	—
45	C₄×C₁₆	—	2	2	no	—
46	C₄×C₁₆	—	2	2	no	—
47	#340	—	4	2	no	—
48	#340	—	4	2	no	—
49	C₈²	—	2	2	no	—
50	C₂²×Q₈	—	4	1	no	—
51	C₂²×Q₈	—	4	2	no	—
52	C₂×Q₁₆	—	2	4	no	—
53	C₂×Q₁₆	—	4	8	no	—
54	C₂×Q₁₆	—	4	2	no	—
55	C₂×Q₁₆	—	4	4	no	—
56	C₂×Q₁₆	—	4	4	no	—
57	C₂×Q₁₆	—	8	4	no	—
58	C₂×Q₁₆	—	8	4	no	—
59	C₂²×C₈	D₈	1	2	yes	—
60	C₂²×C₈	D₈	1	2	yes	—
61	C₂²×C₈	C₂×D₄	1	2	yes	—
62	C₄⋊Q₈	—	2	4	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	4	1	no	—
65	C₄⋊Q₈	—	4	2	no	—
66	C₄⋊Q₈	—	4	1	no	—
67	C₄⋊Q₈	—	4	1	no	—
68	C₄⋊Q₈	—	4	2	no	—
69	C₄⋊Q₈	—	4	2	no	—
70	C₂×C₄⋊C₄	—	4	2	no	—
71	C₂×C₄⋊C₄	—	4	1	no	—
72	C₂×C₄⋊C₄	—	4	1	no	—
73	C₂×C₄⋊C₄	—	4	2	no	—
74	C₂×C₄⋊C₄	—	4	1	no	—
75	C₂×C₄⋊C₄	—	4	1	no	—
76	C₂×C₄⋊C₄	—	4	1	no	—
77	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
78	C₂×C₁₆	—	2	4	no	—
79	C₂×C₁₆	—	2	4	no	—
80	C₂×C₁₆	—	2	4	no	—
81	#107	—	2	4	no	—
82	#107	—	2	4	no	—
83	#107	—	4	8	no	—
84	#107	—	4	2	no	—
85	#107	—	4	2	no	—
86	#107	—	4	4	no	—
87	C₄×C₈	—	2	1	no	—
88	C₄×C₈	—	2	1	no	—
89	C₄×C₈	—	2	1	no	—
90	C₄×C₈	—	2	1	no	—
91	C₄×C₈	C₂×D₄	1	2	yes	—
92	C₄×C₈	C₂×D₄	1	1	yes	characteristic
93	C₄×C₈	#41	1	1	yes	characteristic
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	4	4	no	—
96	C₂×Q₈	—	8	2	no	—
97	C₂×Q₈	—	8	1	no	—
98	C₂×Q₈	—	8	2	no	—
99	C₂×Q₈	—	8	2	no	—
100	C₂×Q₈	—	8	4	no	—
101	C₂×Q₈	—	8	4	no	—
102	C₂²×C₄	—	8	1	no	—
103	C₂²×C₄	—	8	2	no	—
104	C₂²×C₄	—	8	1	no	—
105	C₂²×C₄	#127	1	1	yes	characteristic
106	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
107	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
108	Q₁₆	—	4	8	no	—
109	Q₁₆	—	8	16	no	—
110	Q₁₆	—	8	4	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	—	2	2	no	—
115	C₂×C₈	SD₃₂	1	4	yes	—
116	C₂×C₈	Q₃₂	1	4	yes	—
117	C₂×C₈	#120	1	2	yes	—
118	C₂×C₈	#121	1	2	yes	—
119	C₂×C₈	#135	1	4	yes	—
120	C₂×C₈	#137	1	4	yes	—
121	C₄⋊C₄	—	4	4	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₄⋊C₄	—	4	2	no	—
125	C₄⋊C₄	—	8	1	no	—
126	C₄⋊C₄	—	8	1	no	—
127	C₄⋊C₄	—	8	2	no	—
128	C₄⋊C₄	—	8	2	no	—
129	C₄⋊C₄	—	8	2	no	—
130	C₄⋊C₄	—	8	2	no	—
131	C₄⋊C₄	—	8	2	no	—
132	C₄²	#120	1	1	yes	characteristic
133	C₄²	#121	1	1	yes	characteristic
134	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₁₆	—	4	4	no	—
137	C₂³	#493	1	1	yes	characteristic
138	Q₈	—	8	8	no	—
139	Q₈	—	8	4	no	—
140	Q₈	—	16	4	no	—
141	Q₈	—	16	2	no	—
142	C₂×C₄	—	8	4	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	16	1	no	—
145	C₂×C₄	—	16	1	no	—
146	C₂×C₄	—	16	2	no	—
147	C₂×C₄	—	16	2	no	—
148	C₂×C₄	—	16	1	no	—
149	C₂×C₄	—	16	1	no	—
150	C₂×C₄	—	16	1	no	—
151	C₂×C₄	#392	1	1	yes	characteristic
152	C₂×C₄	#447	1	1	yes	characteristic
153	C₂×C₄	#447	1	1	yes	characteristic
154	C₂×C₄	#459	1	1	yes	characteristic
155	C₂×C₄	#459	1	1	yes	characteristic
156	C₂×C₄	#495	1	2	yes	—
157	C₂×C₄	#496	1	1	yes	characteristic
158	C₂×C₄	#497	1	2	yes	—
159	C₂×C₄	C₂×SD₃₂	1	2	yes	—
160	C₂×C₄	C₂×Q₃₂	1	2	yes	—
161	C₂×C₄	#509	1	2	yes	—
162	C₂×C₄	#510	1	2	yes	—
163	C₈	—	2	4	no	—
164	C₈	—	2	4	no	—
165	C₈	#452	1	4	yes	—
166	C₈	#464	1	4	yes	—
167	C₂²	#1709	1	1	yes	characteristic
168	C₂²	#1722	1	1	yes	characteristic
169	C₂²	#2011	1	1	yes	characteristic
170	C₂²	#2247	1	1	yes	characteristic
171	C₂²	#2248	1	1	yes	characteristic
172	C₂²	#2250	1	1	yes	characteristic
173	C₂²	#2251	1	1	yes	characteristic
174	C₄	—	16	4	no	—
175	C₄	—	16	2	no	—
176	C₄	—	32	1	no	—
177	C₄	#1720	1	2	yes	—
178	C₄	#2015	1	2	yes	—
179	C₄	#2187	1	2	yes	—
180	C₄	#2188	1	2	yes	—
181	C₄	#2207	1	2	yes	—
182	C₄	#2209	1	2	yes	—
183	C₂	#10306	1	1	yes	characteristic
184	C₂	#12510	1	1	yes	characteristic
185	C₂	#12512	1	1	yes	characteristic
186	C₂	#12534	1	1	yes	characteristic
187	C₂	#12535	1	1	yes	characteristic
188	C₂	#13328	1	1	yes	characteristic
189	C₂	#13330	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2241

C₂

yes

maximal

C₂×#1713

C₂

yes

maximal, characteristic

#13282

C₂

yes

maximal

#9825

C₂

yes

maximal, characteristic

#7136

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

C₂²

yes

—

C₂×#494

—

C₂×#494

C₂²

yes

—

—

—

C₂×C₄×C₁₆

C₂²

yes

—

#1990

—

#1990

—

#1845

C₂²

yes

characteristic

#1713

—

C₂×C₈²

C₂²

yes

characteristic

#1380

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

—

—

—

—

—

—

D₄

yes

—

—

—

—

—

D₄

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#358

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

#340

—

C₈²

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

#127

yes

characteristic

106

C₂²×C₄

C₂×D₈

yes

characteristic

107

C₂²×C₄

C₂×D₈

yes

characteristic

108

Q₁₆

—

109

Q₁₆

—

110

Q₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

SD₃₂

yes

—

116

C₂×C₈

Q₃₂

yes

—

117

C₂×C₈

#120

yes

—

118

C₂×C₈

#121

yes

—

119

C₂×C₈

#135

yes

—

120

C₂×C₈

#137

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

—

132

C₄²

#120

yes

characteristic

133

C₄²

#121

yes

characteristic

134

C₄²

AGL(1,ℤ/8)

yes

characteristic

135

C₄²

AGL(1,ℤ/8)

yes

characteristic

136

C₁₆

—

137

C₂³

#493

yes

characteristic

138

Q₈

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

151

C₂×C₄

#392

yes

characteristic

152

C₂×C₄

#447

yes

characteristic

153

C₂×C₄

#447

yes

characteristic

154

C₂×C₄

#459

yes

characteristic

155

C₂×C₄

#459

yes

characteristic

156

C₂×C₄

#495

yes

—

157

C₂×C₄

#496

yes

characteristic

158

C₂×C₄

#497

yes

—

159

C₂×C₄

C₂×SD₃₂

yes

—

160

C₂×C₄

C₂×Q₃₂

yes

—

161

C₂×C₄

#509

yes

—

162

C₂×C₄

#510

yes

—

163

C₈

—

164

C₈

—

165

C₈

#452

yes

—

166

C₈

#464

yes

—

167

C₂²

#1709

yes

characteristic

168

C₂²

#1722

yes

characteristic

169

C₂²

#2011

yes

characteristic

170

C₂²

#2247

yes

characteristic

171

C₂²

#2248

yes

characteristic

172

C₂²

#2250

yes

characteristic

173

C₂²

#2251

yes

characteristic

174

C₄

—

175

C₄

—

176

C₄

—

177

C₄

#1720

yes

—

178

C₄

#2015

yes

—

179

C₄

#2187

yes

—

180

C₄

#2188

yes

—

181

C₄

#2207

yes

—

182

C₄

#2209

yes

—

183

C₂

#10306

yes

characteristic

184

C₂

#12510

yes

characteristic

185

C₂

#12512

yes

characteristic

186

C₂

#12534

yes

characteristic

187

C₂

#12535

yes

characteristic

188

C₂

#13328

yes

characteristic

189

C₂

#13330

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#493

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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×#2241	2
C₂×#1713	2
#13282	2
#9825	2
#7136	2

subgroup

index

Subgroups — #144156

Conjugacy classes of subgroups

Maximal subgroups