Group #142144 — subgroups

Subgroups — #142144

Order 512.

255 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 90 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	#33261	C₂	1	1	yes	maximal, characteristic
2	#12901	C₂	1	1	yes	maximal, characteristic
3	#12660	C₂	1	1	yes	maximal, characteristic
4	#7271	C₂	1	2	yes	maximal
5	#7269	C₂	1	2	yes	maximal
6	#2647	C₄	1	2	yes	—
7	C₂×#364	—	2	1	no	—
8	C₂×#364	C₂²	1	2	yes	—
9	C₂×#364	C₂²	1	2	yes	—
10	#2150	—	2	1	no	—
11	C₂×#347	—	2	1	no	—
12	#1944	—	2	1	no	—
13	#1943	—	2	1	no	—
14	#1942	—	2	1	no	—
15	#1942	—	2	1	no	—
16	#1846	C₂²	1	1	yes	characteristic
17	#1845	C₂²	1	1	yes	characteristic
18	#1836	C₄	1	2	yes	—
19	#1726	C₂²	1	1	yes	characteristic
20	#1377	—	2	1	no	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	C₂×#107	—	2	1	no	—
24	C₂×#107	—	2	1	no	—
25	C₂×#107	—	2	1	no	—
26	C₂×#106	—	2	1	no	—
27	C₂×#97	C₂×C₄	1	1	yes	characteristic
28	C₂×#97	D₄	1	1	yes	characteristic
29	C₂×#97	D₄	1	1	yes	characteristic
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
33	#400	—	4	2	no	—
34	#398	—	4	2	no	—
35	#389	—	2	2	no	—
36	#389	—	2	2	no	—
37	#384	C₂×C₄	1	2	yes	—
38	#382	C₂×C₄	1	2	yes	—
39	#376	C₂×C₄	1	1	yes	characteristic
40	#364	—	2	2	no	—
41	#364	—	2	4	no	—
42	#364	—	4	2	no	—
43	#364	Q₈	1	4	yes	—
44	C₄⋊C₁₆	—	4	2	no	—
45	#347	—	4	2	no	—
46	#340	—	2	2	no	—
47	#340	—	2	2	no	—
48	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄⋊C₄	—	4	2	no	—
55	C₂×C₄⋊C₄	—	4	2	no	—
56	C₂×C₄⋊C₄	—	4	2	no	—
57	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
60	M₃₂	—	2	4	no	—
61	M₃₂	—	4	2	no	—
62	M₃₂	—	4	2	no	—
63	M₃₂	—	4	2	no	—
64	C₂×C₁₆	—	4	2	no	—
65	C₂×C₁₆	—	4	2	no	—
66	#107	—	4	2	no	—
67	#107	—	4	1	no	—
68	#107	—	4	1	no	—
69	#107	—	4	1	no	—
70	#107	—	4	1	no	—
71	#106	—	4	1	no	—
72	#106	—	4	1	no	—
73	#97	—	2	2	no	—
74	#97	—	2	2	no	—
75	#97	—	2	1	no	—
76	#97	—	2	1	no	—
77	C₄×C₈	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	C₂×Q₈	1	2	yes	—
82	C₄×C₈	#41	1	2	yes	—
83	#95	—	2	1	no	—
84	#95	—	2	1	no	—
85	#95	—	4	2	no	—
86	#95	—	4	2	no	—
87	#95	—	4	2	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	4	2	no	—
91	C₂²×C₄	—	4	2	no	—
92	C₂²×C₄	#100	1	1	yes	characteristic
93	C₂²×C₄	#100	1	1	yes	characteristic
94	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	1	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	#101	1	2	yes	—
101	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
102	C₂×C₈	#124	1	1	yes	characteristic
103	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₈	#135	1	2	yes	—
105	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
106	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
107	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
108	C₂×C₈	#137	1	1	yes	characteristic
109	C₂×C₈	#137	1	1	yes	characteristic
110	C₂×C₈	#137	1	1	yes	characteristic
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	16	2	no	—
116	C₄⋊C₄	—	16	2	no	—
117	C₄⋊C₄	—	16	2	no	—
118	C₄²	—	2	1	no	—
119	C₄²	—	2	1	no	—
120	C₄²	—	2	1	no	—
121	C₄²	—	2	1	no	—
122	C₄²	—	2	1	no	—
123	C₄²	C₄×Q₈	1	2	yes	—
124	C₁₆	—	4	4	no	—
125	C₁₆	—	8	2	no	—
126	C₂³	C₂×#100	1	1	yes	characteristic
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	#391	1	1	yes	characteristic
144	C₂×C₄	C₂×#101	1	1	yes	characteristic
145	C₂×C₄	#416	1	1	yes	characteristic
146	C₂×C₄	#417	1	1	yes	characteristic
147	C₂×C₄	#417	1	1	yes	characteristic
148	C₂×C₄	#417	1	1	yes	characteristic
149	C₈	—	2	2	no	—
150	C₈	—	4	1	no	—
151	C₈	#478	1	2	yes	—
152	C₈	#479	1	2	yes	—
153	C₈	#490	1	1	yes	characteristic
154	C₈	#491	1	1	yes	characteristic
155	C₈	#501	1	1	yes	characteristic
156	C₈	#501	1	1	yes	characteristic
157	C₂²	—	2	1	no	—
158	C₂²	—	2	1	no	—
159	C₂²	#1551	1	1	yes	characteristic
160	C₂²	#1560	1	1	yes	characteristic
161	C₂²	#1997	1	1	yes	characteristic
162	C₄	—	4	2	no	—
163	C₄	—	4	1	no	—
164	C₄	—	4	1	no	—
165	C₄	—	8	1	no	—
166	C₄	—	16	2	no	—
167	C₄	—	16	2	no	—
168	C₄	#1552	1	1	yes	characteristic
169	C₄	#1561	1	1	yes	characteristic
170	C₄	#1989	1	1	yes	characteristic
171	C₄	#1992	1	1	yes	characteristic
172	C₂	—	2	1	no	—
173	C₂	—	2	1	no	—
174	C₂	#9906	1	1	yes	characteristic
175	C₂	#11989	1	1	yes	characteristic
176	C₂	#11990	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33261

C₂

yes

maximal, characteristic

#12901

C₂

yes

maximal, characteristic

#12660

C₂

yes

maximal, characteristic

#7271

C₂

yes

maximal

#7269

C₂

yes

maximal

#2647

C₄

yes

—

C₂×#364

—

C₂×#364

C₂²

yes

—

C₂×#364

C₂²

yes

—

—

—

—

—

—

—

C₂²

yes

characteristic

#1845

C₂²

yes

characteristic

#1836

C₄

yes

—

#1726

C₂²

yes

characteristic

#1377

—

C₂×M₃₂

—

C₂×M₃₂

—

—

—

—

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₂×C₄

yes

—

#382

C₂×C₄

yes

—

#376

C₂×C₄

yes

—

—

—

Q₈

yes

—

C₄⋊C₁₆

—

#347

—

#340

—

#340

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

—

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

#101

yes

—

101

C₂×C₈

C₂²⋊Q₈

yes

—

102

C₂×C₈

#124

yes

characteristic

103

C₂×C₈

C₄⋊Q₈

yes

characteristic

104

C₂×C₈

#135

yes

—

105

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

106

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

107

C₂×C₈

AGL(1,ℤ/8)

yes

characteristic

108

C₂×C₈

#137

yes

characteristic

109

C₂×C₈

#137

yes

characteristic

110

C₂×C₈

#137

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₄×Q₈

yes

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

C₂×#100

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

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

—

143

C₂×C₄

#391

yes

characteristic

144

C₂×C₄

C₂×#101

yes

characteristic

145

C₂×C₄

#416

yes

characteristic

146

C₂×C₄

#417

yes

characteristic

147

C₂×C₄

#417

yes

characteristic

148

C₂×C₄

#417

yes

characteristic

149

C₈

—

150

C₈

—

151

C₈

#478

yes

—

152

C₈

#479

yes

—

153

C₈

#490

yes

characteristic

154

C₈

#491

yes

characteristic

155

C₈

#501

yes

characteristic

156

C₈

#501

yes

characteristic

157

C₂²

—

158

C₂²

—

159

C₂²

#1551

yes

characteristic

160

C₂²

#1560

yes

characteristic

161

C₂²

#1997

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

#1552

yes

characteristic

169

C₄

#1561

yes

characteristic

170

C₄

#1989

yes

characteristic

171

C₄

#1992

yes

characteristic

172

C₂

—

173

C₂

—

174

C₂

#9906

yes

characteristic

175

C₂

#11989

yes

characteristic

176

C₂

#11990

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1551

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

C₂×#100

LCS

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ 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
#33261	2
#12901	2
#12660	2
#7271	2
#7269	2

subgroup

index

Subgroups — #142144

Conjugacy classes of subgroups

Maximal subgroups