Group #146140 — subgroups

Subgroups — #146140

Order 512.

252 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 98 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	#13176	C₂	1	2	yes	maximal
2	#9594	C₂	1	1	yes	maximal, characteristic
3	#8818	C₂	1	1	yes	maximal, characteristic
4	#7274	C₂	1	2	yes	maximal
5	#7151	C₂	1	1	yes	maximal, characteristic
6	C₂×#366	—	2	2	no	—
7	C₂×C₄⋊C₁₆	C₂²	1	2	yes	—
8	#1939	—	2	1	no	—
9	#1845	—	2	1	no	—
10	#1845	—	2	1	no	—
11	#1845	C₂²	1	2	yes	—
12	C₄×#107	C₂²	1	1	yes	characteristic
13	C₂×#335	C₂²	1	1	yes	characteristic
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1380	—	2	2	no	—
16	#1279	—	2	1	no	—
17	C₂²×C₁₆	—	2	2	no	—
18	C₂×#107	—	2	2	no	—
19	C₂×#107	—	2	2	no	—
20	C₂×#107	—	4	2	no	—
21	C₂×#107	—	4	1	no	—
22	C₂×#107	—	4	1	no	—
23	C₂×#107	D₄	1	1	yes	characteristic
24	C₂×#107	D₄	1	1	yes	characteristic
25	C₂×C₄⋊C₈	—	2	1	no	—
26	C₂×C₄⋊C₈	—	4	1	no	—
27	C₂×C₄×C₈	D₄	1	1	yes	characteristic
28	C₂×C₄×C₈	D₄	1	1	yes	characteristic
29	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
30	#384	—	2	2	no	—
31	C₄×C₄⋊C₄	—	2	1	no	—
32	#366	—	4	4	no	—
33	C₄⋊C₁₆	—	2	2	no	—
34	C₄⋊C₁₆	Q₈	1	4	yes	—
35	#340	—	2	1	no	—
36	#336	—	4	1	no	—
37	#335	—	2	2	no	—
38	C₈²	—	2	2	no	—
39	C₂²×C₈	—	2	1	no	—
40	C₂²×C₈	—	4	1	no	—
41	C₂²×C₈	D₈	1	1	yes	characteristic
42	C₂²×C₈	D₈	1	1	yes	characteristic
43	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₈	#41	1	1	yes	characteristic
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄⋊C₄	—	4	2	no	—
48	C₂×C₄⋊C₄	—	4	2	no	—
49	C₂×C₄⋊C₄	—	4	2	no	—
50	C₂×C₄²	—	4	1	no	—
51	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
52	C₂×C₁₆	—	2	4	no	—
53	C₂×C₁₆	—	4	2	no	—
54	C₂×C₁₆	—	4	2	no	—
55	#107	—	2	1	no	—
56	#107	—	2	2	no	—
57	#107	—	4	2	no	—
58	#107	—	4	2	no	—
59	#107	—	8	4	no	—
60	#107	—	8	2	no	—
61	#107	—	8	2	no	—
62	#107	—	8	2	no	—
63	#107	D₈	1	2	yes	—
64	C₄⋊C₈	—	4	1	no	—
65	C₄⋊C₈	—	4	1	no	—
66	C₄⋊C₈	—	8	2	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	Q₁₆	1	2	yes	—
71	C₄×C₈	C₂×Q₈	1	2	yes	—
72	C₄×C₈	#41	1	2	yes	—
73	#95	—	4	1	no	—
74	C₂²×C₄	—	4	1	no	—
75	C₂²×C₄	—	4	1	no	—
76	C₂²×C₄	—	8	2	no	—
77	C₂²×C₄	#121	1	1	yes	characteristic
78	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
79	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	8	1	no	—
90	C₂×C₈	—	8	1	no	—
91	C₂×C₈	—	8	1	no	—
92	C₂×C₈	D₁₆	1	2	yes	—
93	C₂×C₈	Q₃₂	1	2	yes	—
94	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₈	#125	1	2	yes	—
96	C₂×C₈	C₂×D₈	1	1	yes	characteristic
97	C₂×C₈	#135	1	1	yes	characteristic
98	C₂×C₈	#137	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	4	1	no	—
103	C₄⋊C₄	—	8	4	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₄⋊C₄	—	8	2	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄⋊C₄	—	8	2	no	—
108	C₄²	—	8	1	no	—
109	C₄²	—	8	1	no	—
110	C₄²	#123	1	1	yes	characteristic
111	C₄²	C₄⋊Q₈	1	1	yes	characteristic
112	C₄²	C₂×Q₁₆	1	1	yes	characteristic
113	C₄²	#137	1	1	yes	characteristic
114	C₁₆	—	4	4	no	—
115	C₂³	#459	1	1	yes	characteristic
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	8	1	no	—
120	C₂×C₄	—	8	1	no	—
121	C₂×C₄	—	8	1	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	16	2	no	—
125	C₂×C₄	—	16	2	no	—
126	C₂×C₄	—	16	2	no	—
127	C₂×C₄	#398	1	1	yes	characteristic
128	C₂×C₄	#462	1	1	yes	characteristic
129	C₂×C₄	#466	1	1	yes	characteristic
130	C₂×C₄	#470	1	1	yes	characteristic
131	C₂×C₄	#472	1	1	yes	characteristic
132	C₂×C₄	#473	1	1	yes	characteristic
133	C₂×C₄	#480	1	1	yes	characteristic
134	C₂×C₄	#484	1	1	yes	characteristic
135	C₂×C₄	#500	1	1	yes	characteristic
136	C₂×C₄	#501	1	1	yes	characteristic
137	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
138	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
139	C₂×C₄	#509	1	1	yes	characteristic
140	C₂×C₄	#510	1	1	yes	characteristic
141	C₈	—	2	4	no	—
142	C₈	—	2	2	no	—
143	C₈	—	2	2	no	—
144	C₈	—	2	2	no	—
145	C₈	—	16	1	no	—
146	C₈	#474	1	2	yes	—
147	C₈	#479	1	2	yes	—
148	C₂²	#1667	1	1	yes	characteristic
149	C₂²	#1689	1	1	yes	characteristic
150	C₂²	#2083	1	1	yes	characteristic
151	C₂²	#2206	1	1	yes	characteristic
152	C₂²	#2209	1	1	yes	characteristic
153	C₂²	#2213	1	1	yes	characteristic
154	C₂²	#2214	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	8	2	no	—
158	C₄	—	16	1	no	—
159	C₄	—	32	2	no	—
160	C₄	#1674	1	1	yes	characteristic
161	C₄	#1696	1	1	yes	characteristic
162	C₄	#2086	1	1	yes	characteristic
163	C₄	#2089	1	1	yes	characteristic
164	C₄	#2232	1	1	yes	characteristic
165	C₄	#2236	1	1	yes	characteristic
166	C₄	#2252	1	1	yes	characteristic
167	C₄	#2255	1	1	yes	characteristic
168	C₂	#10649	1	1	yes	characteristic
169	C₂	#12424	1	1	yes	characteristic
170	C₂	#12427	1	1	yes	characteristic
171	C₂	#12497	1	1	yes	characteristic
172	C₂	#12500	1	1	yes	characteristic
173	C₂	#13505	1	1	yes	characteristic
174	C₂	#13508	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13176

C₂

yes

maximal

#9594

C₂

yes

maximal, characteristic

#8818

C₂

yes

maximal, characteristic

#7274

C₂

yes

maximal

#7151

C₂

yes

maximal, characteristic

C₂×#366

—

C₂×C₄⋊C₁₆

C₂²

yes

—

—

—

—

C₂²

yes

—

C₄×#107

C₂²

yes

characteristic

C₂×#335

C₂²

yes

characteristic

C₂×C₈²

C₂²

yes

characteristic

#1380

—

#1279

—

C₂²×C₁₆

—

—

—

—

—

—

D₄

yes

characteristic

C₂×#107

D₄

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

C₄×C₄⋊C₄

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

Q₈

yes

—

#340

—

#336

—

#335

—

C₈²

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

D₈

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#125

yes

—

C₂×C₈

C₂×D₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₄²

#123

yes

characteristic

111

C₄²

C₄⋊Q₈

yes

characteristic

112

C₄²

C₂×Q₁₆

yes

characteristic

113

C₄²

#137

yes

characteristic

114

C₁₆

—

115

C₂³

#459

yes

characteristic

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

#398

yes

characteristic

128

C₂×C₄

#462

yes

characteristic

129

C₂×C₄

#466

yes

characteristic

130

C₂×C₄

#470

yes

characteristic

131

C₂×C₄

#472

yes

characteristic

132

C₂×C₄

#473

yes

characteristic

133

C₂×C₄

#480

yes

characteristic

134

C₂×C₄

#484

yes

characteristic

135

C₂×C₄

#500

yes

characteristic

136

C₂×C₄

#501

yes

characteristic

137

C₂×C₄

C₂×D₁₆

yes

characteristic

138

C₂×C₄

C₂×Q₃₂

yes

characteristic

139

C₂×C₄

#509

yes

characteristic

140

C₂×C₄

#510

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#474

yes

—

147

C₈

#479

yes

—

148

C₂²

#1667

yes

characteristic

149

C₂²

#1689

yes

characteristic

150

C₂²

#2083

yes

characteristic

151

C₂²

#2206

yes

characteristic

152

C₂²

#2209

yes

characteristic

153

C₂²

#2213

yes

characteristic

154

C₂²

#2214

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#1674

yes

characteristic

161

C₄

#1696

yes

characteristic

162

C₄

#2086

yes

characteristic

163

C₄

#2089

yes

characteristic

164

C₄

#2232

yes

characteristic

165

C₄

#2236

yes

characteristic

166

C₄

#2252

yes

characteristic

167

C₄

#2255

yes

characteristic

168

C₂

#10649

yes

characteristic

169

C₂

#12424

yes

characteristic

170

C₂

#12427

yes

characteristic

171

C₂

#12497

yes

characteristic

172

C₂

#12500

yes

characteristic

173

C₂

#13505

yes

characteristic

174

C₂

#13508

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#459
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

#459

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
#13176	2
#9594	2
#8818	2
#7274	2
#7151	2

subgroup

index

Subgroups — #146140

Conjugacy classes of subgroups

Maximal subgroups