Group #124503 — subgroups

Subgroups — #124503

Order 512.

286 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 94 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	#21724	C₂	1	1	yes	maximal, characteristic
2	#12881	C₂	1	1	yes	maximal, characteristic
3	#8137	C₂	1	1	yes	maximal, characteristic
4	#7127	C₂	1	2	yes	maximal
5	#7126	C₂	1	2	yes	maximal
6	C₂×#498	C₄	1	1	yes	characteristic
7	#2331	—	2	1	no	—
8	C₂×#364	—	2	1	no	—
9	#1993	—	2	1	no	—
10	#1944	C₂²	1	2	yes	—
11	#1941	—	2	1	no	—
12	#1846	C₄	1	1	yes	characteristic
13	#1776	C₄	1	2	yes	—
14	C₄×#106	—	2	1	no	—
15	#1726	C₂²	1	1	yes	characteristic
16	#1450	—	2	1	no	—
17	#1377	C₂²	1	1	yes	characteristic
18	#1377	C₂²	1	1	yes	characteristic
19	#1375	—	2	1	no	—
20	#1275	C₂²	1	2	yes	—
21	#1274	—	2	1	no	—
22	C₂×C₄⋊Q₈	—	2	1	no	—
23	C₂×M₃₂	—	2	1	no	—
24	C₂×M₃₂	—	2	1	no	—
25	#498	—	2	2	no	—
26	#498	C₈	1	4	yes	complemented
27	#431	—	2	2	no	—
28	C₂×#107	C₂×C₄	1	2	yes	—
29	C₂×#106	—	2	1	no	—
30	C₂×#106	—	2	1	no	—
31	C₂×#106	C₂×C₄	1	2	yes	—
32	C₂×#97	C₂×C₄	1	1	yes	characteristic
33	C₂×#97	D₄	1	1	yes	characteristic
34	C₂×#97	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	D₄	1	1	yes	characteristic
38	#391	—	4	2	no	—
39	#389	—	2	2	no	—
40	#387	—	4	2	no	—
41	#382	—	2	1	no	—
42	C₄×C₄⋊C₄	—	2	2	no	—
43	#376	D₄	1	1	yes	characteristic
44	#364	—	4	2	no	—
45	#340	—	2	2	no	—
46	#340	—	2	2	no	—
47	#322	—	2	2	no	—
48	C₈²	—	2	2	no	—
49	C₂²×Q₈	—	4	1	no	—
50	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄⋊Q₈	—	2	4	no	—
55	C₄⋊Q₈	—	4	2	no	—
56	C₂×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	—	4	1	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄²	—	4	2	no	—
61	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	M₃₂	—	4	1	no	—
65	M₃₂	—	4	1	no	—
66	M₃₂	—	4	1	no	—
67	M₃₂	—	4	1	no	—
68	C₂×C₁₆	—	4	1	no	—
69	C₂×C₁₆	—	4	1	no	—
70	#107	—	4	2	no	—
71	#106	—	2	4	no	—
72	#106	—	2	2	no	—
73	#106	—	4	1	no	—
74	#106	—	4	1	no	—
75	#106	C₂×C₈	1	4	yes	—
76	C₄⋊C₈	—	4	2	no	—
77	C₂²⋊C₈	—	4	2	no	—
78	#97	—	2	2	no	—
79	#97	—	4	1	no	—
80	#97	—	4	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	2	1	no	—
83	C₄×C₈	—	2	4	no	—
84	C₄×C₈	C₂×C₈	1	2	yes	—
85	C₄×C₈	M₁₆	1	2	yes	—
86	C₄×C₈	SD₁₆	1	2	yes	—
87	C₄×C₈	Q₁₆	1	2	yes	—
88	#95	—	2	2	no	—
89	#95	—	4	2	no	—
90	#95	—	4	1	no	—
91	#95	—	4	1	no	—
92	#95	—	4	2	no	—
93	C₂×Q₈	—	4	4	no	—
94	C₂×Q₈	—	16	2	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	4	2	no	—
98	C₂²×C₄	—	4	2	no	—
99	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	4	4	no	—
108	C₂×C₈	C₂²⋊C₈	1	2	yes	—
109	C₂×C₈	#101	1	2	yes	—
110	C₂×C₈	#103	1	2	yes	—
111	C₂×C₈	#104	1	1	yes	characteristic
112	C₂×C₈	#104	1	1	yes	characteristic
113	C₂×C₈	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₈	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
117	C₄⋊C₄	—	2	4	no	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	4	2	no	—
122	C₄⋊C₄	—	16	2	no	—
123	C₄²	—	2	1	no	—
124	C₄²	—	2	1	no	—
125	C₄²	—	2	1	no	—
126	C₄²	—	2	1	no	—
127	C₄²	—	16	2	no	—
128	C₄²	C₂²⋊C₈	1	2	yes	—
129	C₄²	#103	1	2	yes	—
130	C₁₆	—	8	1	no	—
131	C₁₆	—	8	1	no	—
132	C₂³	#418	1	1	yes	characteristic
133	Q₈	—	16	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	4	2	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	—	8	2	no	—
142	C₂×C₄	—	8	2	no	—
143	C₂×C₄	—	8	2	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂×C₄	#410	1	1	yes	characteristic
148	C₂×C₄	C₂×#101	1	1	yes	characteristic
149	C₂×C₄	C₂×#103	1	1	yes	characteristic
150	C₂×C₄	C₂×#104	1	1	yes	characteristic
151	C₂×C₄	#421	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	2	1	no	—
155	C₈	—	2	1	no	—
156	C₈	—	2	2	no	—
157	C₈	—	8	4	no	—
158	C₈	Q₈⋊C₈	1	2	yes	—
159	C₈	Q₈⋊C₈	1	2	yes	—
160	C₂²	—	2	2	no	—
161	C₂²	#1478	1	1	yes	characteristic
162	C₂²	#1505	1	1	yes	characteristic
163	C₂²	#1549	1	1	yes	characteristic
164	C₄	—	4	2	no	—
165	C₄	—	4	2	no	—
166	C₄	—	4	2	no	—
167	C₄	—	8	4	no	—
168	C₄	—	16	2	no	—
169	C₄	#1461	1	1	yes	characteristic
170	C₄	C₂×Q₈⋊C₈	1	1	yes	characteristic
171	C₄	#1518	1	1	yes	characteristic
172	C₄	#1531	1	1	yes	characteristic
173	C₂	—	2	1	no	—
174	C₂	—	2	1	no	—
175	C₂	#7591	1	1	yes	characteristic
176	C₂	#11769	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21724

C₂

yes

maximal, characteristic

#12881

C₂

yes

maximal, characteristic

#8137

C₂

yes

maximal, characteristic

#7127

C₂

yes

maximal

#7126

C₂

yes

maximal

C₂×#498

C₄

yes

—

—

—

C₂²

yes

—

#1941

—

#1846

C₄

yes

characteristic

#1776

C₄

yes

—

C₄×#106

—

#1726

C₂²

yes

characteristic

#1450

—

#1377

C₂²

yes

characteristic

#1377

C₂²

yes

characteristic

#1375

—

#1275

C₂²

yes

—

#1274

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

C₂×M₃₂

—

#498

—

#498

C₈

yes

complemented

#431

—

C₂×#107

C₂×C₄

yes

—

C₂×#106

—

C₂×#106

—

C₂×#106

C₂×C₄

yes

—

C₂×#97

C₂×C₄

yes

characteristic

C₂×#97

D₄

yes

characteristic

C₂×#97

C₂³

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

—

—

—

C₄×C₄⋊C₄

—

#376

D₄

yes

—

—

—

—

C₈²

—

C₂²×Q₈

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²×C₄

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×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

C₂×C₈

yes

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

100

C₂²×C₄

AGL(1,ℤ/8)

yes

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊C₈

yes

—

109

C₂×C₈

#101

yes

—

110

C₂×C₈

#103

yes

—

111

C₂×C₈

#104

yes

characteristic

112

C₂×C₈

#104

yes

characteristic

113

C₂×C₈

C₂²×C₈

yes

characteristic

114

C₂×C₈

C₂×M₁₆

yes

characteristic

115

C₂×C₈

C₂×SD₁₆

yes

characteristic

116

C₂×C₈

C₂×Q₁₆

yes

characteristic

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂²⋊C₈

yes

—

129

C₄²

#103

yes

—

130

C₁₆

—

131

C₁₆

—

132

C₂³

#418

yes

characteristic

133

Q₈

—

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

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

147

C₂×C₄

#410

yes

characteristic

148

C₂×C₄

C₂×#101

yes

characteristic

149

C₂×C₄

C₂×#103

yes

characteristic

150

C₂×C₄

C₂×#104

yes

characteristic

151

C₂×C₄

#421

yes

characteristic

152

C₈

—

153

C₈

—

154

C₈

—

155

C₈

—

156

C₈

—

157

C₈

—

158

C₈

Q₈⋊C₈

yes

—

159

C₈

Q₈⋊C₈

yes

—

160

C₂²

—

161

C₂²

#1478

yes

characteristic

162

C₂²

#1505

yes

characteristic

163

C₂²

#1549

yes

characteristic

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

169

C₄

#1461

yes

characteristic

170

C₄

C₂×Q₈⋊C₈

yes

characteristic

171

C₄

#1518

yes

characteristic

172

C₄

#1531

yes

characteristic

173

C₂

—

174

C₂

—

175

C₂

#7591

yes

characteristic

176

C₂

#11769

yes

—

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1505

commutator

C₂×C₈

C₂²×C₈

Frattini, cosocle

C₂×#97

C₂³

prime radical

C₂³

#418

LCS

G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G

LECS

G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
26. #498	C₈	semidirect

H ≅ G/N

type

26. #498

C₈

semidirect

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

subgroup	index
#21724	2
#12881	2
#8137	2
#7127	2
#7126	2

subgroup

index

Subgroups — #124503

Conjugacy classes of subgroups

Complements

Maximal subgroups