Group #143985 — subgroups

Subgroups — #143985

Order 512.

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

normal subgroups only

This group has 421 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×#2247	C₂	1	1	yes	maximal, characteristic
2	C₂×#1670	C₂	1	2	yes	maximal
3	#13263	C₂	1	1	yes	maximal, characteristic
4	C₂×#1362	C₂	1	2	yes	maximal
5	#7154	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	—	2	2	no	—
7	C₂×#494	C₂²	1	2	yes	—
8	C₂×#475	—	2	2	no	—
9	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
10	#2247	—	2	4	no	—
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1670	—	2	4	no	—
14	#1670	—	2	4	no	—
15	C₂×#335	C₂²	1	2	yes	—
16	C₂×#332	—	2	2	no	—
17	#1374	—	2	2	no	—
18	#1362	—	2	4	no	—
19	C₂²×Q₁₆	—	2	2	no	—
20	C₂²×Q₁₆	—	2	2	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
23	C₂×Q₃₂	—	2	8	no	—
24	C₂×Q₃₂	—	4	4	no	—
25	C₂×Q₃₂	—	4	4	no	—
26	C₂²×C₁₆	—	2	1	no	—
27	C₂²×C₁₆	D₄	1	2	yes	—
28	#494	—	2	2	no	—
29	#494	—	2	2	no	—
30	#494	—	2	2	no	—
31	#494	D₄	1	4	yes	—
32	#475	—	2	8	no	—
33	#475	—	4	4	no	—
34	C₈×Q₈	—	2	2	no	—
35	C₈×Q₈	—	2	2	no	—
36	C₂×#106	—	4	2	no	—
37	C₂×C₄⋊C₈	—	2	2	no	—
38	C₂×C₄⋊C₈	—	2	1	no	—
39	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
40	C₂×#103	—	4	2	no	—
41	C₂×C₄×C₈	—	2	1	no	—
42	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
43	C₄⋊C₁₆	—	2	2	no	—
44	C₄×C₁₆	—	2	2	no	—
45	#335	—	2	2	no	—
46	#335	D₄	1	4	yes	—
47	#332	—	2	8	no	—
48	C₂²×Q₈	—	4	2	no	—
49	C₂²×Q₈	D₈	1	1	yes	characteristic
50	C₂×Q₁₆	—	2	4	no	—
51	C₂×Q₁₆	—	2	4	no	—
52	C₂×Q₁₆	—	4	2	no	—
53	C₂×Q₁₆	—	4	2	no	—
54	C₂×Q₁₆	—	4	4	no	—
55	C₂×Q₁₆	—	4	4	no	—
56	C₂×Q₁₆	—	8	4	no	—
57	C₂²×C₈	—	2	1	no	—
58	C₂²×C₈	—	4	2	no	—
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	#41	1	1	yes	characteristic
61	C₄⋊Q₈	—	2	4	no	—
62	C₄⋊Q₈	—	4	2	no	—
63	C₄⋊Q₈	—	4	2	no	—
64	C₄⋊Q₈	—	4	2	no	—
65	C₄×Q₈	—	2	2	no	—
66	C₄×Q₈	—	2	2	no	—
67	C₂×C₄⋊C₄	—	2	1	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄⋊C₄	—	4	2	no	—
70	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
71	C₂×C₄²	—	2	1	no	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	Q₃₂	—	4	16	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	—	4	1	no	—
77	C₂×C₁₆	—	4	2	no	—
78	C₂×C₁₆	SD₁₆	1	4	yes	—
79	#106	—	4	8	no	—
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	2	4	no	—
82	C₄⋊C₈	—	4	2	no	—
83	C₄⋊C₈	—	4	2	no	—
84	#103	—	4	8	no	—
85	C₄×C₈	—	4	1	no	—
86	C₄×C₈	—	4	1	no	—
87	C₄×C₈	C₂×D₄	1	2	yes	—
88	C₄×C₈	C₂×D₄	1	2	yes	—
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	4	4	no	—
93	C₂×Q₈	—	8	2	no	—
94	C₂×Q₈	—	8	4	no	—
95	C₂×Q₈	—	8	4	no	—
96	C₂×Q₈	Q₃₂	1	2	yes	complemented
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	8	2	no	—
100	C₂²×C₄	#121	1	1	yes	characteristic
101	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
102	C₂²×C₄	#135	1	1	yes	characteristic
103	Q₁₆	—	4	8	no	—
104	Q₁₆	—	8	4	no	—
105	Q₁₆	—	8	4	no	—
106	Q₁₆	—	8	2	no	—
107	Q₁₆	—	8	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	4	1	no	—
110	C₂×C₈	—	4	4	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	8	2	no	—
113	C₂×C₈	—	8	2	no	—
114	C₂×C₈	#120	1	1	yes	characteristic
115	C₂×C₈	#121	1	2	yes	—
116	C₂×C₈	#127	1	1	yes	characteristic
117	C₂×C₈	C₂×SD₁₆	1	2	yes	—
118	C₂×C₈	AGL(1,ℤ/8)	1	2	yes	—
119	C₄⋊C₄	—	2	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	4	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	8	2	no	—
124	C₄⋊C₄	—	8	2	no	—
125	C₄²	—	4	1	no	—
126	C₄²	—	4	1	no	—
127	C₄²	#120	1	2	yes	—
128	C₄²	AGL(1,ℤ/8)	1	2	yes	—
129	C₁₆	—	2	4	no	—
130	C₁₆	—	8	1	no	—
131	C₂³	#466	1	1	yes	characteristic
132	Q₈	—	2	4	no	—
133	Q₈	—	4	2	no	—
134	Q₈	—	8	4	no	—
135	Q₈	—	8	4	no	—
136	Q₈	—	16	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	1	no	—
140	C₂×C₄	—	4	1	no	—
141	C₂×C₄	—	4	2	no	—
142	C₂×C₄	—	4	1	no	—
143	C₂×C₄	—	8	4	no	—
144	C₂×C₄	—	16	4	no	—
145	C₂×C₄	#392	1	1	yes	characteristic
146	C₂×C₄	#447	1	2	yes	—
147	C₂×C₄	#449	1	2	yes	—
148	C₂×C₄	#460	1	2	yes	—
149	C₂×C₄	#469	1	1	yes	characteristic
150	C₂×C₄	#472	1	2	yes	—
151	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
152	C₂×C₄	#508	1	1	yes	characteristic
153	C₂×C₄	#510	1	2	yes	—
154	C₈	—	2	2	no	—
155	C₈	—	8	4	no	—
156	C₈	—	8	1	no	—
157	C₈	#450	1	2	yes	—
158	C₈	#492	1	1	yes	characteristic
159	C₈	#496	1	1	yes	characteristic
160	C₂²	#1668	1	2	yes	—
161	C₂²	#1999	1	1	yes	characteristic
162	C₂²	#2216	1	1	yes	characteristic
163	C₂²	#2219	1	1	yes	characteristic
164	C₂²	#2222	1	2	yes	—
165	C₄	—	2	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	8	1	no	—
168	C₄	—	16	4	no	—
169	C₄	#1688	1	2	yes	—
170	C₄	#2014	1	1	yes	characteristic
171	C₄	#2018	1	1	yes	characteristic
172	C₄	#2188	1	2	yes	—
173	C₄	#2189	1	2	yes	—
174	C₂	#10262	1	1	yes	characteristic
175	C₂	#12440	1	2	yes	—
176	C₂	#12443	1	2	yes	—
177	C₂	#13306	1	1	yes	characteristic
178	C₂	#13315	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2247

C₂

yes

maximal, characteristic

C₂×#1670

C₂

yes

maximal

#13263

C₂

yes

maximal, characteristic

C₂×#1362

C₂

yes

maximal

#7154

C₂

yes

maximal, characteristic

C₂²×Q₃₂

—

C₂×#494

C₂²

yes

—

C₂×#475

—

C₂×C₈×Q₈

C₂²

yes

characteristic

#2247

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

C₂²

yes

—

C₂×#332

—

#1374

—

#1362

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

—

—

—

D₄

yes

—

#475

—

#475

—

C₈×Q₈

—

C₈×Q₈

—

C₂×#106

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

—

#335

D₄

yes

—

#332

—

C₂²×Q₈

—

C₂²×Q₈

D₈

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

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

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

#106

—

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

Q₃₂

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

#121

yes

characteristic

101

C₂²×C₄

C₂×D₈

yes

characteristic

102

C₂²×C₄

#135

yes

characteristic

103

Q₁₆

—

104

Q₁₆

—

105

Q₁₆

—

106

Q₁₆

—

107

Q₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

#120

yes

characteristic

115

C₂×C₈

#121

yes

—

116

C₂×C₈

#127

yes

characteristic

117

C₂×C₈

C₂×SD₁₆

yes

—

118

C₂×C₈

AGL(1,ℤ/8)

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

#120

yes

—

128

C₄²

AGL(1,ℤ/8)

yes

—

129

C₁₆

—

130

C₁₆

—

131

C₂³

#466

yes

characteristic

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

#392

yes

characteristic

146

C₂×C₄

#447

yes

—

147

C₂×C₄

#449

yes

—

148

C₂×C₄

#460

yes

—

149

C₂×C₄

#469

yes

characteristic

150

C₂×C₄

#472

yes

—

151

C₂×C₄

C₂×Q₃₂

yes

characteristic

152

C₂×C₄

#508

yes

characteristic

153

C₂×C₄

#510

yes

—

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

#450

yes

—

158

C₈

#492

yes

characteristic

159

C₈

#496

yes

characteristic

160

C₂²

#1668

yes

—

161

C₂²

#1999

yes

characteristic

162

C₂²

#2216

yes

characteristic

163

C₂²

#2219

yes

characteristic

164

C₂²

#2222

yes

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

#1688

yes

—

170

C₄

#2014

yes

characteristic

171

C₄

#2018

yes

characteristic

172

C₄

#2188

yes

—

173

C₄

#2189

yes

—

174

C₂

#10262

yes

characteristic

175

C₂

#12440

yes

—

176

C₂

#12443

yes

—

177

C₂

#13306

yes

characteristic

178

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
96. C₂×Q₈	Q₃₂	semidirect

H ≅ G/N

type

96. C₂×Q₈

Q₃₂

semidirect

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

subgroup	index
C₂×#2247	2
C₂×#1670	2
#13263	2
C₂×#1362	2
#7154	2

subgroup

index

Subgroups — #143985

Conjugacy classes of subgroups

Complements

Maximal subgroups