Group #146030 — subgroups

Subgroups — #146030

Order 512.

247 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 247 conjugacy classes of subgroups — too many to draw. Only its 92 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	#13263	C₂	1	1	yes	maximal, characteristic
2	#12905	C₂	1	1	yes	maximal, characteristic
3	#8645	C₂	1	2	yes	maximal
4	#7274	C₂	1	2	yes	maximal
5	#7154	C₂	1	1	yes	maximal, characteristic
6	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
7	C₂×#366	—	2	1	no	—
8	C₂×#366	—	2	1	no	—
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1845	C₂²	1	2	yes	—
12	#1803	—	2	2	no	—
13	C₂×#335	C₂²	1	2	yes	—
14	#1374	—	2	1	no	—
15	#1374	—	2	1	no	—
16	#1296	—	2	2	no	—
17	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
18	C₂²×C₁₆	—	2	1	no	—
19	C₂²×C₁₆	D₄	1	1	yes	characteristic
20	C₂²×C₁₆	D₄	1	1	yes	characteristic
21	C₈×Q₈	—	2	2	no	—
22	C₈×Q₈	—	2	2	no	—
23	C₂×#107	—	2	2	no	—
24	C₂×#107	—	2	2	no	—
25	C₂×C₄⋊C₈	—	2	1	no	—
26	C₂×C₄⋊C₈	—	2	2	no	—
27	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
28	C₂×#103	—	4	2	no	—
29	C₂×C₄×C₈	—	2	1	no	—
30	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
31	#384	—	2	2	no	—
32	#366	—	4	2	no	—
33	#366	—	4	2	no	—
34	C₄⋊C₁₆	—	2	2	no	—
35	C₄×C₁₆	—	2	2	no	—
36	#340	—	4	2	no	—
37	#335	—	2	2	no	—
38	#335	—	2	2	no	—
39	C₂²×Q₈	D₈	1	1	yes	characteristic
40	C₂²×C₈	—	2	1	no	—
41	C₂²×C₈	—	4	2	no	—
42	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
43	C₂²×C₈	#41	1	1	yes	characteristic
44	C₄×Q₈	—	2	2	no	—
45	C₄×Q₈	—	2	2	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×C₄⋊C₄	D₈	1	1	yes	characteristic
51	C₂×C₄²	—	2	1	no	—
52	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
53	C₂×C₁₆	—	2	1	no	—
54	C₂×C₁₆	—	2	1	no	—
55	C₂×C₁₆	—	2	1	no	—
56	C₂×C₁₆	—	2	1	no	—
57	C₂×C₁₆	—	4	1	no	—
58	C₂×C₁₆	—	4	1	no	—
59	C₂×C₁₆	—	4	1	no	—
60	C₂×C₁₆	SD₁₆	1	2	yes	—
61	C₂×C₁₆	Q₁₆	1	2	yes	—
62	#107	—	4	2	no	—
63	#107	—	4	2	no	—
64	#107	—	8	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	4	2	no	—
67	C₄⋊C₈	—	4	2	no	—
68	C₄⋊C₈	—	4	2	no	—
69	#103	—	8	4	no	—
70	C₄×C₈	—	4	1	no	—
71	C₄×C₈	—	4	1	no	—
72	C₄×C₈	#41	1	2	yes	—
73	C₄×C₈	#41	1	2	yes	—
74	C₂×Q₈	—	2	1	no	—
75	C₂×Q₈	—	2	2	no	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	Q₃₂	1	2	yes	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	—	2	1	no	—
80	C₂²×C₄	—	8	2	no	—
81	C₂²×C₄	#121	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
83	C₂²×C₄	#135	1	1	yes	characteristic
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	—	8	2	no	—
90	C₂×C₈	—	8	2	no	—
91	C₂×C₈	—	8	2	no	—
92	C₂×C₈	#123	1	1	yes	characteristic
93	C₂×C₈	#124	1	1	yes	characteristic
94	C₂×C₈	#126	1	2	yes	—
95	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
97	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
98	C₂×C₈	#137	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	8	2	no	—
102	C₄⋊C₄	—	8	2	no	—
103	C₄⋊C₄	—	8	2	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₄⋊C₄	—	16	2	no	—
106	C₄²	—	4	1	no	—
107	C₄²	—	4	1	no	—
108	C₄²	#123	1	2	yes	—
109	C₄²	#137	1	2	yes	—
110	C₁₆	—	2	2	no	—
111	C₁₆	—	2	2	no	—
112	C₁₆	—	8	1	no	—
113	C₂³	#466	1	1	yes	characteristic
114	Q₈	—	2	4	no	—
115	Q₈	—	4	2	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	16	2	no	—
125	C₂×C₄	—	16	2	no	—
126	C₂×C₄	—	16	2	no	—
127	C₂×C₄	#397	1	1	yes	characteristic
128	C₂×C₄	#460	1	1	yes	characteristic
129	C₂×C₄	#462	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	2	yes	—
134	C₂×C₄	#485	1	2	yes	—
135	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
136	C₂×C₄	#508	1	1	yes	characteristic
137	C₂×C₄	#509	1	2	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	8	1	no	—
140	C₈	—	16	2	no	—
141	C₈	#483	1	1	yes	characteristic
142	C₈	#484	1	1	yes	characteristic
143	C₈	#487	1	1	yes	characteristic
144	C₈	#489	1	1	yes	characteristic
145	C₂²	#1668	1	1	yes	characteristic
146	C₂²	#1674	1	1	yes	characteristic
147	C₂²	#2075	1	1	yes	characteristic
148	C₂²	#2216	1	1	yes	characteristic
149	C₂²	#2218	1	1	yes	characteristic
150	C₂²	#2221	1	1	yes	characteristic
151	C₂²	#2222	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	8	1	no	—
155	C₄	—	32	2	no	—
156	C₄	#1690	1	1	yes	characteristic
157	C₄	#1696	1	1	yes	characteristic
158	C₄	#2072	1	1	yes	characteristic
159	C₄	#2078	1	1	yes	characteristic
160	C₄	#2234	1	2	yes	—
161	C₄	#2236	1	2	yes	—
162	C₂	#10617	1	1	yes	characteristic
163	C₂	#12440	1	1	yes	characteristic
164	C₂	#12442	1	1	yes	characteristic
165	C₂	#12458	1	1	yes	characteristic
166	C₂	#12460	1	1	yes	characteristic
167	C₂	#13493	1	1	yes	characteristic
168	C₂	#13495	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13263

C₂

yes

maximal, characteristic

#12905

C₂

yes

maximal, characteristic

#8645

C₂

yes

maximal

#7274

C₂

yes

maximal

#7154

C₂

yes

maximal, characteristic

C₂×C₈×Q₈

C₂²

yes

characteristic

C₂×#366

—

C₂×#366

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1845

C₂²

yes

—

#1803

—

C₂×#335

C₂²

yes

—

#1374

—

#1374

—

#1296

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₂×#107

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#340

—

#335

—

#335

—

C₂²×Q₈

D₈

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₈

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

#107

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Q₃₂

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×Q₁₆

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

C₄²

—

107

C₄²

—

108

C₄²

#123

yes

—

109

C₄²

#137

yes

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₂³

#466

yes

characteristic

114

Q₈

—

115

Q₈

—

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

#397

yes

characteristic

128

C₂×C₄

#460

yes

characteristic

129

C₂×C₄

#462

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

—

134

C₂×C₄

#485

yes

—

135

C₂×C₄

C₂×Q₃₂

yes

characteristic

136

C₂×C₄

#508

yes

characteristic

137

C₂×C₄

#509

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

#483

yes

characteristic

142

C₈

#484

yes

characteristic

143

C₈

#487

yes

characteristic

144

C₈

#489

yes

characteristic

145

C₂²

#1668

yes

characteristic

146

C₂²

#1674

yes

characteristic

147

C₂²

#2075

yes

characteristic

148

C₂²

#2216

yes

characteristic

149

C₂²

#2218

yes

characteristic

150

C₂²

#2221

yes

characteristic

151

C₂²

#2222

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#1690

yes

characteristic

157

C₄

#1696

yes

characteristic

158

C₄

#2072

yes

characteristic

159

C₄

#2078

yes

characteristic

160

C₄

#2234

yes

—

161

C₄

#2236

yes

—

162

C₂

#10617

yes

characteristic

163

C₂

#12440

yes

characteristic

164

C₂

#12442

yes

characteristic

165

C₂

#12458

yes

characteristic

166

C₂

#12460

yes

characteristic

167

C₂

#13493

yes

characteristic

168

C₂

#13495

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#466
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#466

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ 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
#13263	2
#12905	2
#8645	2
#7274	2
#7154	2

subgroup

index

Subgroups — #146030

Conjugacy classes of subgroups

Maximal subgroups