Group #141393 — subgroups

Subgroups — #141393

Order 512.

308 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 102 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×#1710	C₂	1	1	yes	maximal, characteristic
2	#11972	C₂	1	2	yes	maximal
3	#11971	C₂	1	2	yes	maximal
4	#11410	C₂	1	1	yes	maximal, characteristic
5	#8824	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	1	yes	characteristic
7	C₂×#500	C₂²	1	1	yes	characteristic
8	C₂×#487	—	2	1	no	—
9	C₂×#487	—	2	1	no	—
10	#2162	C₂²	1	2	yes	—
11	#1837	—	2	1	no	—
12	#1710	—	2	1	no	—
13	#1710	—	2	1	no	—
14	#1710	C₄	1	2	yes	—
15	#1710	C₄	1	2	yes	—
16	#1567	C₂²	1	2	yes	—
17	#1566	—	2	1	no	—
18	C₂×C₈²	C₂²	1	1	yes	characteristic
19	#1365	—	2	4	no	—
20	#1364	—	2	4	no	—
21	#1312	—	2	1	no	—
22	C₈×C₁₆	—	2	1	no	—
23	C₂×C₄⋊Q₈	—	2	1	no	—
24	C₂×C₄⋊Q₈	—	2	1	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	—	2	1	no	—
28	#500	—	2	1	no	—
29	#500	C₂×C₄	1	2	yes	—
30	#500	C₂×C₄	1	2	yes	—
31	#487	—	2	2	no	—
32	#487	—	2	2	no	—
33	#487	—	2	2	no	—
34	#487	—	2	2	no	—
35	#487	—	4	2	no	—
36	#487	—	4	2	no	—
37	C₂×#107	—	2	2	no	—
38	C₂×#107	—	2	2	no	—
39	#423	—	2	2	no	—
40	C₂×#103	—	4	2	no	—
41	C₂×#103	—	4	2	no	—
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	C₄⋊C₁₆	D₄	1	4	yes	—
46	C₂²⋊C₁₆	—	2	2	no	—
47	C₄×C₁₆	—	2	2	no	—
48	#341	—	4	2	no	—
49	#335	—	2	2	no	—
50	#335	—	2	2	no	—
51	#334	—	4	2	no	—
52	C₈²	C₂×C₄	1	1	yes	characteristic
53	C₈²	C₂×C₄	1	1	yes	characteristic
54	C₈²	D₄	1	1	yes	characteristic
55	C₈²	D₄	1	1	yes	characteristic
56	C₂²×Q₈	—	4	1	no	—
57	C₂²×Q₈	—	4	1	no	—
58	C₂×M₁₆	—	4	2	no	—
59	C₂²×C₈	D₈	1	2	yes	—
60	C₂²×C₈	SD₁₆	1	2	yes	—
61	C₂²×C₈	#41	1	2	yes	—
62	C₄⋊Q₈	—	2	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	4	1	no	—
65	C₄⋊Q₈	—	4	1	no	—
66	C₄⋊Q₈	—	4	1	no	—
67	C₄⋊Q₈	—	4	1	no	—
68	C₄⋊Q₈	—	4	1	no	—
69	C₄⋊Q₈	—	4	1	no	—
70	C₂×C₄⋊C₄	—	4	1	no	—
71	C₂×C₄⋊C₄	—	4	1	no	—
72	C₂×C₄⋊C₄	—	4	1	no	—
73	C₂×C₄⋊C₄	—	4	1	no	—
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₁₆	—	2	4	no	—
76	#107	—	2	4	no	—
77	#107	—	2	4	no	—
78	#107	—	4	2	no	—
79	#107	—	4	2	no	—
80	C₄⋊C₈	—	4	2	no	—
81	C₄⋊C₈	—	4	2	no	—
82	#103	—	4	4	no	—
83	#103	—	4	4	no	—
84	#103	—	4	4	no	—
85	#103	—	4	4	no	—
86	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
87	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
88	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
89	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
90	C₄×C₈	D₈	1	2	yes	—
91	C₄×C₈	SD₁₆	1	2	yes	—
92	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₄×C₈	C₂×D₄	1	1	yes	characteristic
94	C₄×C₈	C₂×D₄	1	2	yes	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×Q₈	—	8	4	no	—
98	C₂×Q₈	—	8	1	no	—
99	C₂×Q₈	—	8	1	no	—
100	C₂×Q₈	—	8	4	no	—
101	C₂²×C₄	—	8	1	no	—
102	C₂²×C₄	—	8	1	no	—
103	C₂²×C₄	#124	1	1	yes	characteristic
104	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
105	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
106	M₁₆	—	8	4	no	—
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	4	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	8	2	no	—
111	C₂×C₈	#102	1	2	yes	—
112	C₂×C₈	#102	1	2	yes	—
113	C₂×C₈	#102	1	2	yes	—
114	C₂×C₈	#102	1	2	yes	—
115	C₂×C₈	C₄×D₄	1	2	yes	—
116	C₂×C₈	#121	1	2	yes	—
117	C₄⋊C₄	—	4	2	no	—
118	C₄⋊C₄	—	4	2	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₄⋊C₄	—	8	1	no	—
123	C₄⋊C₄	—	8	1	no	—
124	C₄⋊C₄	—	8	1	no	—
125	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₄²	#127	1	1	yes	characteristic
127	C₄²	C₂×D₈	1	1	yes	characteristic
128	C₄²	C₂×SD₁₆	1	1	yes	characteristic
129	C₁₆	—	4	4	no	—
130	C₂³	#486	1	1	yes	characteristic
131	Q₈	—	8	4	no	—
132	Q₈	—	8	4	no	—
133	Q₈	—	16	2	no	—
134	Q₈	—	16	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	16	2	no	—
140	C₂×C₄	—	16	2	no	—
141	C₂×C₄	#390	1	1	yes	characteristic
142	C₂×C₄	C₂×#102	1	1	yes	characteristic
143	C₂×C₄	C₂×#102	1	1	yes	characteristic
144	C₂×C₄	#486	1	1	yes	characteristic
145	C₂×C₄	#492	1	1	yes	characteristic
146	C₂×C₄	#493	1	1	yes	characteristic
147	C₂×C₄	#508	1	4	yes	—
148	C₈	—	2	2	no	—
149	C₈	—	2	2	no	—
150	C₈	—	2	2	no	—
151	C₈	—	2	2	no	—
152	C₈	—	2	2	no	—
153	C₈	—	16	2	no	—
154	C₈	#473	1	4	yes	—
155	C₂²	—	2	1	no	—
156	C₂²	—	2	1	no	—
157	C₂²	#1957	1	1	yes	characteristic
158	C₂²	#2242	1	2	yes	—
159	C₄	—	2	1	no	—
160	C₄	—	2	1	no	—
161	C₄	—	16	2	no	—
162	C₄	—	16	2	no	—
163	C₄	#1979	1	2	yes	—
164	C₄	#2138	1	2	yes	—
165	C₄	#2148	1	2	yes	—
166	C₄	#2249	1	2	yes	—
167	C₂	—	2	2	no	—
168	C₂	#9772	1	1	yes	characteristic
169	C₂	#13204	1	1	yes	characteristic
170	C₂	#13214	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1710

C₂

yes

maximal, characteristic

#11972

C₂

yes

maximal

#11971

C₂

yes

maximal

#11410

C₂

yes

maximal, characteristic

#8824

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

characteristic

C₂×#500

C₂²

yes

—

—

C₂²

yes

—

—

—

—

C₄

yes

—

#1710

C₄

yes

—

#1567

C₂²

yes

—

#1566

—

C₂×C₈²

C₂²

yes

—

—

—

C₈×C₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

—

—

—

—

C₂×C₄

yes

—

#500

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

D₄

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

—

—

—

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

#41

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×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

—

—

—

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

#124

yes

characteristic

104

C₂²×C₄

C₂×D₈

yes

characteristic

105

C₂²×C₄

C₂×SD₁₆

yes

characteristic

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

#102

yes

—

112

C₂×C₈

#102

yes

—

113

C₂×C₈

#102

yes

—

114

C₂×C₈

#102

yes

—

115

C₂×C₈

C₄×D₄

yes

—

116

C₂×C₈

#121

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

C₂×C₂²⋊C₄

yes

characteristic

126

C₄²

#127

yes

characteristic

127

C₄²

C₂×D₈

yes

characteristic

128

C₄²

C₂×SD₁₆

yes

characteristic

129

C₁₆

—

130

C₂³

#486

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

#390

yes

characteristic

142

C₂×C₄

C₂×#102

yes

characteristic

143

C₂×C₄

C₂×#102

yes

characteristic

144

C₂×C₄

#486

yes

characteristic

145

C₂×C₄

#492

yes

characteristic

146

C₂×C₄

#493

yes

characteristic

147

C₂×C₄

#508

yes

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

—

153

C₈

—

154

C₈

#473

yes

—

155

C₂²

—

156

C₂²

—

157

C₂²

#1957

yes

characteristic

158

C₂²

#2242

yes

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#1979

yes

—

164

C₄

#2138

yes

—

165

C₄

#2148

yes

—

166

C₄

#2249

yes

—

167

C₂

—

168

C₂

#9772

yes

characteristic

169

C₂

#13204

yes

characteristic

170

C₂

#13214

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1957

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#486

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ 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×#1710	2
#11972	2
#11971	2
#11410	2
#8824	2

subgroup

index

Subgroups — #141393

Conjugacy classes of subgroups

Maximal subgroups