Group #153266 — subgroups

Subgroups — #153266

Order 512.

313 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 84 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×#1416	C₂	1	2	yes	maximal
3	C₂×#1416	C₂	1	2	yes	maximal
4	#13165	C₂	1	1	yes	maximal, characteristic
5	#7465	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₃₂	—	2	1	no	—
7	C₂²×Q₃₂	C₂²	1	2	yes	—
8	C₂×#494	—	2	1	no	—
9	C₂²×C₃₂	C₂²	1	2	yes	—
10	#2247	—	2	4	no	—
11	C₂×#366	C₂²	1	1	yes	characteristic
12	C₂×#366	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1942	—	2	1	no	—
15	#1416	—	2	4	no	—
16	#1416	—	2	4	no	—
17	#1380	—	2	1	no	—
18	C₂²×Q₁₆	—	2	2	no	—
19	C₂²×Q₁₆	—	4	1	no	—
20	C₂×C₄⋊Q₈	—	4	1	no	—
21	C₂×Q₃₂	—	2	4	no	—
22	C₂×Q₃₂	—	2	2	no	—
23	C₂×Q₃₂	—	2	2	no	—
24	C₂×Q₃₂	—	2	2	no	—
25	C₂×Q₃₂	—	2	2	no	—
26	C₂×Q₃₂	—	2	2	no	—
27	C₂×Q₃₂	—	4	2	no	—
28	C₂×Q₃₂	—	4	2	no	—
29	C₂×Q₃₂	D₄	1	4	yes	—
30	C₂²×C₁₆	D₄	1	1	yes	characteristic
31	C₂²×C₁₆	C₂³	1	1	yes	characteristic
32	#494	—	2	4	no	—
33	#494	—	4	2	no	—
34	C₂×#107	—	2	1	no	—
35	C₂×#107	—	2	1	no	—
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	#389	—	4	1	no	—
38	C₂×C₃₂	—	2	2	no	—
39	C₂×C₃₂	—	2	2	no	—
40	C₂×C₃₂	—	2	2	no	—
41	#366	—	2	1	no	—
42	#366	—	2	2	no	—
43	#366	D₄	1	2	yes	—
44	C₄×C₁₆	—	2	2	no	—
45	#340	—	4	1	no	—
46	C₂²×Q₈	—	4	2	no	—
47	C₂×Q₁₆	—	2	4	no	—
48	C₂×Q₁₆	—	4	4	no	—
49	C₂×Q₁₆	—	4	2	no	—
50	C₂×Q₁₆	—	4	2	no	—
51	C₂×Q₁₆	—	4	2	no	—
52	C₂×Q₁₆	—	4	2	no	—
53	C₂×Q₁₆	—	4	2	no	—
54	C₂×Q₁₆	—	8	2	no	—
55	C₂×Q₁₆	—	8	2	no	—
56	C₂²×C₈	D₈	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
58	C₄⋊Q₈	—	4	4	no	—
59	C₄⋊Q₈	—	8	2	no	—
60	C₂×C₄⋊C₄	—	4	1	no	—
61	C₂×C₄⋊C₄	—	4	1	no	—
62	C₂×C₄⋊C₄	—	8	1	no	—
63	C₂×C₄⋊C₄	—	8	1	no	—
64	C₂×C₄⋊C₄	—	8	1	no	—
65	C₂×C₄²	D₈	1	1	yes	characteristic
66	Q₃₂	—	2	4	no	—
67	Q₃₂	—	2	4	no	—
68	Q₃₂	—	4	8	no	—
69	Q₃₂	—	4	2	no	—
70	Q₃₂	—	4	2	no	—
71	C₂×C₁₆	—	2	1	no	—
72	C₂×C₁₆	—	2	1	no	—
73	C₂×C₁₆	SD₁₆	1	2	yes	—
74	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
75	C₂×C₁₆	C₂×D₄	1	2	yes	—
76	C₂×C₁₆	#41	1	2	yes	—
77	C₂×C₁₆	#41	1	1	yes	characteristic
78	#107	—	2	2	no	—
79	#107	—	4	1	no	—
80	#107	—	4	1	no	—
81	#107	—	4	1	no	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	—	2	1	no	—
84	#95	—	8	1	no	—
85	C₃₂	—	4	2	no	—
86	C₂×Q₈	—	4	4	no	—
87	C₂×Q₈	—	8	2	no	—
88	C₂×Q₈	—	8	2	no	—
89	C₂×Q₈	—	8	2	no	—
90	C₂×Q₈	—	8	2	no	—
91	C₂×Q₈	—	8	2	no	—
92	C₂²×C₄	—	8	1	no	—
93	C₂²×C₄	—	8	2	no	—
94	C₂²×C₄	—	8	1	no	—
95	C₂²×C₄	D₁₆	1	1	yes	characteristic
96	C₂²×C₄	D₁₆	1	1	yes	characteristic
97	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	8	8	no	—
101	Q₁₆	—	8	2	no	—
102	Q₁₆	—	8	2	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	SD₃₂	1	2	yes	—
106	C₂×C₈	#120	1	1	yes	characteristic
107	C₂×C₈	#121	1	1	yes	characteristic
108	C₂×C₈	#124	1	1	yes	characteristic
109	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
111	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	8	4	no	—
114	C₄⋊C₄	—	8	4	no	—
115	C₄⋊C₄	—	8	1	no	—
116	C₄⋊C₄	—	8	1	no	—
117	C₄⋊C₄	—	8	1	no	—
118	C₄⋊C₄	—	16	2	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₁₆	—	2	2	no	—
122	C₁₆	#121	1	2	yes	—
123	C₁₆	#123	1	2	yes	—
124	C₂³	C₂×D₁₆	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	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	8	4	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	16	2	no	—
136	C₂×C₄	—	16	2	no	—
137	C₂×C₄	—	16	1	no	—
138	C₂×C₄	—	16	1	no	—
139	C₂×C₄	—	16	1	no	—
140	C₂×C₄	—	16	1	no	—
141	C₂×C₄	—	16	1	no	—
142	C₂×C₄	SD₆₄	1	2	yes	—
143	C₂×C₄	Q₆₄	1	2	yes	—
144	C₂×C₄	#447	1	1	yes	characteristic
145	C₂×C₄	#459	1	1	yes	characteristic
146	C₂×C₄	#486	1	1	yes	characteristic
147	C₂×C₄	C₂×SD₃₂	1	1	yes	characteristic
148	C₂×C₄	#509	1	1	yes	characteristic
149	C₂×C₄	#509	1	1	yes	characteristic
150	C₈	—	2	2	no	—
151	C₈	#394	1	1	yes	characteristic
152	C₈	#450	1	1	yes	characteristic
153	C₈	#460	1	1	yes	characteristic
154	C₈	#490	1	1	yes	characteristic
155	C₂²	#2184	1	1	yes	characteristic
156	C₂²	#2206	1	1	yes	characteristic
157	C₂²	#2240	1	1	yes	characteristic
158	C₂²	C₂×SD₆₄	1	1	yes	characteristic
159	C₂²	C₂×Q₆₄	1	1	yes	characteristic
160	C₂²	#2264	1	1	yes	characteristic
161	C₂²	#2264	1	1	yes	characteristic
162	C₄	—	2	2	no	—
163	C₄	—	2	2	no	—
164	C₄	—	16	4	no	—
165	C₄	—	16	2	no	—
166	C₄	—	32	1	no	—
167	C₄	#2041	1	1	yes	characteristic
168	C₄	#2187	1	1	yes	characteristic
169	C₄	#2207	1	1	yes	characteristic
170	C₄	#2244	1	1	yes	characteristic
171	C₂	#13407	1	1	yes	characteristic
172	C₂	#13662	1	1	yes	characteristic
173	C₂	#13665	1	1	yes	characteristic
174	C₂	#13687	1	1	yes	characteristic
175	C₂	#13688	1	1	yes	characteristic
176	C₂	#13720	1	1	yes	characteristic
177	C₂	#13724	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2247

C₂

yes

maximal, characteristic

C₂×#1416

C₂

yes

maximal

C₂×#1416

C₂

yes

maximal

#13165

C₂

yes

maximal, characteristic

#7465

C₂

yes

maximal, characteristic

C₂²×Q₃₂

—

C₂²×Q₃₂

C₂²

yes

—

C₂×#494

—

C₂²×C₃₂

C₂²

yes

—

#2247

—

C₂×#366

C₂²

yes

characteristic

C₂×#366

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

#389

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

#366

—

#366

—

#366

D₄

yes

—

C₄×C₁₆

—

#340

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

characteristic

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

#41

yes

—

C₂×C₁₆

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₃₂

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

D₁₆

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

Q₁₆

—

Q₁₆

—

100

Q₁₆

—

101

Q₁₆

—

102

Q₁₆

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

SD₃₂

yes

—

106

C₂×C₈

#120

yes

characteristic

107

C₂×C₈

#121

yes

characteristic

108

C₂×C₈

#124

yes

characteristic

109

C₂×C₈

C₂×SD₁₆

yes

characteristic

110

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

111

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₁₆

—

122

C₁₆

#121

yes

—

123

C₁₆

#123

yes

—

124

C₂³

C₂×D₁₆

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

—

141

C₂×C₄

—

142

C₂×C₄

SD₆₄

yes

—

143

C₂×C₄

Q₆₄

yes

—

144

C₂×C₄

#447

yes

characteristic

145

C₂×C₄

#459

yes

characteristic

146

C₂×C₄

#486

yes

characteristic

147

C₂×C₄

C₂×SD₃₂

yes

characteristic

148

C₂×C₄

#509

yes

characteristic

149

C₂×C₄

#509

yes

characteristic

150

C₈

—

151

C₈

#394

yes

characteristic

152

C₈

#450

yes

characteristic

153

C₈

#460

yes

characteristic

154

C₈

#490

yes

characteristic

155

C₂²

#2184

yes

characteristic

156

C₂²

#2206

yes

characteristic

157

C₂²

#2240

yes

characteristic

158

C₂²

C₂×SD₆₄

yes

characteristic

159

C₂²

C₂×Q₆₄

yes

characteristic

160

C₂²

#2264

yes

characteristic

161

C₂²

#2264

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

#2041

yes

characteristic

168

C₄

#2187

yes

characteristic

169

C₄

#2207

yes

characteristic

170

C₄

#2244

yes

characteristic

171

C₂

#13407

yes

characteristic

172

C₂

#13662

yes

characteristic

173

C₂

#13665

yes

characteristic

174

C₂

#13687

yes

characteristic

175

C₂

#13688

yes

characteristic

176

C₂

#13720

yes

characteristic

177

C₂

#13724

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×D₁₆

commutator, Frattini, cosocle

C₂²×C₁₆

C₂³

LCS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

subgroup	index
C₂×#2247	2
C₂×#1416	2
#13165	2
#7465	2

subgroup

index

Subgroups — #153266

Conjugacy classes of subgroups

Maximal subgroups