Group #144025 — subgroups

Subgroups — #144025

Order 512.

389 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 389 conjugacy classes of subgroups — too many to draw. Only its 100 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	C₂×#1674	C₂	1	2	yes	maximal
3	#13263	C₂	1	1	yes	maximal, characteristic
4	C₂×#1364	C₂	1	2	yes	maximal
5	#7154	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	2	yes	—
7	C₂×#462	—	2	2	no	—
8	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
9	#2252	—	2	4	no	—
10	C₂×#366	—	2	2	no	—
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1674	—	2	4	no	—
14	#1674	—	2	4	no	—
15	C₂×#335	C₂²	1	2	yes	—
16	C₂×#332	—	2	2	no	—
17	#1374	—	2	2	no	—
18	#1364	—	2	4	no	—
19	C₂²×Q₁₆	—	4	2	no	—
20	C₂×C₄⋊Q₈	—	2	2	no	—
21	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
22	C₂²×C₁₆	—	2	1	no	—
23	C₂²×C₁₆	D₄	1	2	yes	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	D₄	1	4	yes	—
28	#462	—	2	8	no	—
29	#462	—	4	4	no	—
30	C₈×Q₈	—	2	2	no	—
31	C₈×Q₈	—	2	2	no	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×C₄⋊C₈	—	2	2	no	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
37	C₂×#103	—	4	2	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#366	—	2	8	no	—
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄×C₁₆	—	2	2	no	—
43	#335	—	2	2	no	—
44	#335	D₄	1	4	yes	—
45	#332	—	2	8	no	—
46	C₂²×Q₈	—	4	2	no	—
47	C₂²×Q₈	D₈	1	1	yes	characteristic
48	C₂×Q₁₆	—	4	8	no	—
49	C₂×Q₁₆	—	8	4	no	—
50	C₂×Q₁₆	—	8	4	no	—
51	C₂²×C₈	—	2	1	no	—
52	C₂²×C₈	—	4	2	no	—
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₈	#41	1	1	yes	characteristic
55	C₄⋊Q₈	—	2	4	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₄⋊Q₈	—	4	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	C₄×Q₈	—	2	2	no	—
61	C₂×C₄⋊C₄	—	2	1	no	—
62	C₂×C₄⋊C₄	—	4	2	no	—
63	C₂×C₄⋊C₄	—	4	2	no	—
64	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
65	C₂×C₄²	—	2	1	no	—
66	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
67	C₂×C₁₆	—	2	2	no	—
68	C₂×C₁₆	—	2	2	no	—
69	C₂×C₁₆	—	4	1	no	—
70	C₂×C₁₆	—	4	2	no	—
71	C₂×C₁₆	Q₁₆	1	4	yes	complemented
72	#107	—	2	4	no	—
73	#107	—	2	4	no	—
74	#107	—	4	2	no	—
75	#107	—	4	2	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	—	2	4	no	—
78	C₄⋊C₈	—	4	2	no	—
79	C₄⋊C₈	—	4	2	no	—
80	#103	—	4	8	no	—
81	C₄×C₈	—	4	1	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	C₂×D₄	1	2	yes	—
84	C₄×C₈	C₂×D₄	1	2	yes	—
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	4	4	no	—
89	C₂×Q₈	—	8	2	no	—
90	C₂×Q₈	—	8	2	no	—
91	C₂×Q₈	—	8	2	no	—
92	C₂×Q₈	—	8	2	no	—
93	C₂×Q₈	—	8	2	no	—
94	C₂×Q₈	D₁₆	1	2	yes	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	8	2	no	—
98	C₂²×C₄	#121	1	1	yes	characteristic
99	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
100	C₂²×C₄	#135	1	1	yes	characteristic
101	Q₁₆	—	8	16	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	4	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	8	2	no	—
107	C₂×C₈	—	8	2	no	—
108	C₂×C₈	#120	1	1	yes	characteristic
109	C₂×C₈	#121	1	2	yes	—
110	C₂×C₈	#127	1	1	yes	characteristic
111	C₂×C₈	C₂×Q₁₆	1	2	yes	—
112	C₂×C₈	#137	1	2	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	4	no	—
116	C₄⋊C₄	—	4	4	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₄⋊C₄	—	8	2	no	—
119	C₄²	—	4	1	no	—
120	C₄²	—	4	1	no	—
121	C₄²	#120	1	2	yes	—
122	C₄²	AGL(1,ℤ/8)	1	2	yes	—
123	C₁₆	—	2	4	no	—
124	C₁₆	—	8	1	no	—
125	C₂³	#466	1	1	yes	characteristic
126	Q₈	—	2	4	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	8	4	no	—
129	Q₈	—	8	4	no	—
130	Q₈	—	16	2	no	—
131	Q₈	—	16	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	—	4	1	no	—
138	C₂×C₄	—	8	4	no	—
139	C₂×C₄	—	16	2	no	—
140	C₂×C₄	—	16	2	no	—
141	C₂×C₄	#392	1	1	yes	characteristic
142	C₂×C₄	#447	1	2	yes	—
143	C₂×C₄	#449	1	2	yes	—
144	C₂×C₄	#462	1	2	yes	—
145	C₂×C₄	#469	1	1	yes	characteristic
146	C₂×C₄	#473	1	2	yes	—
147	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
148	C₂×C₄	#508	1	1	yes	characteristic
149	C₂×C₄	#509	1	2	yes	—
150	C₈	—	2	2	no	—
151	C₈	—	8	4	no	—
152	C₈	—	8	1	no	—
153	C₈	#451	1	2	yes	—
154	C₈	#494	1	1	yes	characteristic
155	C₈	#497	1	1	yes	characteristic
156	C₂²	#1674	1	2	yes	—
157	C₂²	#1999	1	1	yes	characteristic
158	C₂²	#2215	1	1	yes	characteristic
159	C₂²	#2218	1	1	yes	characteristic
160	C₂²	#2222	1	2	yes	—
161	C₄	—	2	2	no	—
162	C₄	—	4	2	no	—
163	C₄	—	8	1	no	—
164	C₄	—	16	4	no	—
165	C₄	#1694	1	2	yes	—
166	C₄	#2016	1	1	yes	characteristic
167	C₄	#2020	1	1	yes	characteristic
168	C₄	#2184	1	2	yes	—
169	C₄	#2186	1	2	yes	—
170	C₂	#10265	1	1	yes	characteristic
171	C₂	#12457	1	2	yes	—
172	C₂	#12460	1	2	yes	—
173	C₂	#13304	1	1	yes	characteristic
174	C₂	#13315	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

C₂×#1674

C₂

yes

maximal

#13263

C₂

yes

maximal, characteristic

C₂×#1364

C₂

yes

maximal

#7154

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

—

C₂×#462

—

C₂×C₈×Q₈

C₂²

yes

characteristic

#2252

—

C₂×#366

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

C₂²

yes

—

C₂×#332

—

#1374

—

#1364

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

—

—

—

D₄

yes

—

#462

—

#462

—

C₈×Q₈

—

C₈×Q₈

—

C₂×#107

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

#366

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

—

#335

D₄

yes

—

#332

—

C₂²×Q₈

—

C₂²×Q₈

D₈

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

100

C₂²×C₄

#135

yes

characteristic

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

#120

yes

characteristic

109

C₂×C₈

#121

yes

—

110

C₂×C₈

#127

yes

characteristic

111

C₂×C₈

C₂×Q₁₆

yes

—

112

C₂×C₈

#137

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

#120

yes

—

122

C₄²

AGL(1,ℤ/8)

yes

—

123

C₁₆

—

124

C₁₆

—

125

C₂³

#466

yes

characteristic

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

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

—

141

C₂×C₄

#392

yes

characteristic

142

C₂×C₄

#447

yes

—

143

C₂×C₄

#449

yes

—

144

C₂×C₄

#462

yes

—

145

C₂×C₄

#469

yes

characteristic

146

C₂×C₄

#473

yes

—

147

C₂×C₄

C₂×D₁₆

yes

characteristic

148

C₂×C₄

#508

yes

characteristic

149

C₂×C₄

#509

yes

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

#451

yes

—

154

C₈

#494

yes

characteristic

155

C₈

#497

yes

characteristic

156

C₂²

#1674

yes

—

157

C₂²

#1999

yes

characteristic

158

C₂²

#2215

yes

characteristic

159

C₂²

#2218

yes

characteristic

160

C₂²

#2222

yes

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#1694

yes

—

166

C₄

#2016

yes

characteristic

167

C₄

#2020

yes

characteristic

168

C₄

#2184

yes

—

169

C₄

#2186

yes

—

170

C₂

#10265

yes

characteristic

171

C₂

#12457

yes

—

172

C₂

#12460

yes

—

173

C₂

#13304

yes

characteristic

174

C₂

#13315

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 ◁ C₂×C₈×Q₈ ◁ 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 ◁ C₂×C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
71. C₂×C₁₆	Q₁₆	semidirect

H ≅ G/N

type

71. C₂×C₁₆

Q₁₆

semidirect

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

subgroup	index
C₂×#2252	2
C₂×#1674	2
#13263	2
C₂×#1364	2
#7154	2

subgroup

index

Subgroups — #144025

Conjugacy classes of subgroups

Complements

Maximal subgroups