Group #135010 — subgroups

Subgroups — #135010

Order 512.

292 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 122 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×Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
2	#12888	C₂	1	1	yes	maximal, characteristic
3	C₂×#1372	C₂	1	1	yes	maximal, characteristic
4	#8645	C₂	1	2	yes	maximal
5	#7274	C₂	1	2	yes	maximal
6	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
7	Q₈×C₁₆	—	2	2	no	—
8	Q₈×C₁₆	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₁₆	—	2	1	no	—
10	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	—	2	1	no	—
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1845	C₂²	1	2	yes	—
14	#1803	—	2	2	no	—
15	C₂×#335	C₂²	1	2	yes	—
16	#1380	—	2	1	no	—
17	#1380	—	2	1	no	—
18	#1372	—	2	2	no	—
19	#1296	—	2	2	no	—
20	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	—	2	1	no	—
22	C₂²×C₁₆	D₄	1	1	yes	characteristic
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	C₈×Q₈	—	2	1	no	—
25	C₈×Q₈	—	2	1	no	—
26	C₈×Q₈	C₂×C₄	1	2	yes	—
27	C₈×Q₈	Q₈	1	2	yes	—
28	C₂×#107	—	2	2	no	—
29	C₂×#107	—	2	2	no	—
30	C₂×C₄⋊C₈	—	2	1	no	—
31	C₂×C₄⋊C₈	—	2	2	no	—
32	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
33	C₂×#103	—	4	2	no	—
34	C₂×C₄×C₈	—	2	1	no	—
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	#384	—	2	2	no	—
37	C₄⋊C₁₆	—	2	4	no	—
38	C₄⋊C₁₆	—	2	1	no	—
39	C₄⋊C₁₆	C₂×C₄	1	2	yes	—
40	C₄×C₁₆	—	2	4	no	—
41	C₄×C₁₆	C₂×C₄	1	2	yes	—
42	C₄×C₁₆	D₄	1	2	yes	—
43	#340	—	4	2	no	—
44	#335	—	2	2	no	—
45	#335	—	2	2	no	—
46	C₂²×Q₈	D₈	1	1	yes	characteristic
47	C₂²×C₈	—	2	1	no	—
48	C₂²×C₈	—	4	2	no	—
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₈	#41	1	1	yes	characteristic
51	C₄×Q₈	—	2	1	no	—
52	C₄×Q₈	—	2	1	no	—
53	C₄×Q₈	C₄⋊C₄	1	2	yes	—
54	C₄×Q₈	Q₁₆	1	2	yes	—
55	C₂×C₄⋊C₄	—	2	1	no	—
56	C₂×C₄⋊C₄	—	4	2	no	—
57	C₂×C₄⋊C₄	—	4	2	no	—
58	C₂×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
60	C₂×C₄²	—	2	1	no	—
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	4	2	no	—
66	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
67	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
68	#107	—	4	2	no	—
69	#107	—	4	2	no	—
70	#107	—	8	2	no	—
71	C₄⋊C₈	—	2	2	no	—
72	C₄⋊C₈	—	2	2	no	—
73	C₄⋊C₈	—	2	1	no	—
74	C₄⋊C₈	—	4	2	no	—
75	C₄⋊C₈	—	4	2	no	—
76	C₄⋊C₈	C₄⋊C₄	1	2	yes	—
77	#103	—	8	4	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₄×C₈	C₂×D₄	1	1	yes	characteristic
82	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	C₂×Q₈	—	2	1	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	#107	1	2	yes	—
87	C₂×Q₈	SD₃₂	1	4	yes	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	2	1	no	—
90	C₂²×C₄	—	8	2	no	—
91	C₂²×C₄	#121	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
93	C₂²×C₄	#135	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	—	8	2	no	—
98	C₂×C₈	—	8	2	no	—
99	C₂×C₈	—	8	2	no	—
100	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₈	#117	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₈	#135	1	2	yes	—
105	C₂×C₈	#137	1	2	yes	—
106	C₄⋊C₄	—	2	2	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₄⋊C₄	—	8	2	no	—
111	C₄⋊C₄	—	8	2	no	—
112	C₄⋊C₄	—	8	2	no	—
113	C₄⋊C₄	—	16	2	no	—
114	C₄⋊C₄	#107	1	2	yes	—
115	C₄²	—	2	2	no	—
116	C₄²	—	2	2	no	—
117	C₄²	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₄²	#121	1	1	yes	characteristic
119	C₄²	C₂×Q₁₆	1	1	yes	characteristic
120	C₄²	#135	1	1	yes	characteristic
121	C₁₆	—	2	2	no	—
122	C₁₆	—	2	2	no	—
123	C₁₆	—	4	2	no	—
124	C₂³	#466	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	2	4	no	—
127	Q₈	#367	1	4	yes	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	16	2	no	—
136	C₂×C₄	—	16	2	no	—
137	C₂×C₄	—	16	2	no	—
138	C₂×C₄	#380	1	1	yes	characteristic
139	C₂×C₄	C₂×#107	1	1	yes	characteristic
140	C₂×C₄	#427	1	1	yes	characteristic
141	C₂×C₄	#464	1	2	yes	—
142	C₂×C₄	#466	1	1	yes	characteristic
143	C₂×C₄	#467	1	1	yes	characteristic
144	C₂×C₄	#467	1	1	yes	characteristic
145	C₂×C₄	#471	1	2	yes	—
146	C₂×C₄	C₂×SD₃₂	1	2	yes	—
147	C₂×C₄	#508	1	2	yes	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	2	no	—
151	C₈	—	16	2	no	—
152	C₈	#416	1	1	yes	characteristic
153	C₈	#419	1	1	yes	characteristic
154	C₈	#479	1	2	yes	—
155	C₂²	#1682	1	2	yes	—
156	C₂²	#1808	1	1	yes	characteristic
157	C₂²	#2217	1	2	yes	—
158	C₂²	#2221	1	1	yes	characteristic
159	C₂²	#2222	1	1	yes	characteristic
160	C₄	—	2	4	no	—
161	C₄	—	2	1	no	—
162	C₄	—	2	1	no	—
163	C₄	—	4	2	no	—
164	C₄	—	32	2	no	—
165	C₄	#1684	1	2	yes	—
166	C₄	#1805	1	1	yes	characteristic
167	C₄	#1815	1	1	yes	characteristic
168	C₄	C₂×#367	1	1	yes	characteristic
169	C₄	#2158	1	1	yes	characteristic
170	C₄	#2217	1	2	yes	—
171	C₂	#8649	1	1	yes	characteristic
172	C₂	#12476	1	2	yes	—
173	C₂	#12478	1	2	yes	—
174	C₂	#12835	1	1	yes	characteristic
175	C₂	#12842	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×C₁₆

C₂

yes

maximal, characteristic

#12888

C₂

yes

maximal, characteristic

C₂×#1372

C₂

yes

maximal, characteristic

#8645

C₂

yes

maximal

#7274

C₂

yes

maximal

C₂×C₈×Q₈

C₂²

yes

characteristic

Q₈×C₁₆

—

Q₈×C₁₆

C₄

yes

complemented

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1845

C₂²

yes

—

#1803

—

C₂×#335

C₂²

yes

—

—

—

—

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

characteristic

C₈×Q₈

—

C₈×Q₈

—

C₈×Q₈

C₂×C₄

yes

—

C₈×Q₈

Q₈

yes

—

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

characteristic

#384

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂×C₄

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

#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

—

C₄×Q₈

C₄⋊C₄

yes

—

C₄×Q₈

Q₁₆

yes

—

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

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

#107

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₄⋊C₄

yes

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×Q₈

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#107

yes

—

C₂×Q₈

SD₃₂

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

—

100

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

101

C₂×C₈

#117

yes

characteristic

102

C₂×C₈

C₂²⋊Q₈

yes

characteristic

103

C₂×C₈

C₂²⋊Q₈

yes

characteristic

104

C₂×C₈

#135

yes

—

105

C₂×C₈

#137

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

#107

yes

—

115

C₄²

—

116

C₄²

—

117

C₄²

C₂×C₄⋊C₄

yes

characteristic

118

C₄²

#121

yes

characteristic

119

C₄²

C₂×Q₁₆

yes

characteristic

120

C₄²

#135

yes

characteristic

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

#466

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

#367

yes

—

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

#380

yes

characteristic

139

C₂×C₄

C₂×#107

yes

characteristic

140

C₂×C₄

#427

yes

characteristic

141

C₂×C₄

#464

yes

—

142

C₂×C₄

#466

yes

characteristic

143

C₂×C₄

#467

yes

characteristic

144

C₂×C₄

#467

yes

characteristic

145

C₂×C₄

#471

yes

—

146

C₂×C₄

C₂×SD₃₂

yes

—

147

C₂×C₄

#508

yes

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

#416

yes

characteristic

153

C₈

#419

yes

characteristic

154

C₈

#479

yes

—

155

C₂²

#1682

yes

—

156

C₂²

#1808

yes

characteristic

157

C₂²

#2217

yes

—

158

C₂²

#2221

yes

characteristic

159

C₂²

#2222

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

#1684

yes

—

166

C₄

#1805

yes

characteristic

167

C₄

#1815

yes

characteristic

168

C₄

C₂×#367

yes

characteristic

169

C₄

#2158

yes

characteristic

170

C₄

#2217

yes

—

171

C₂

#8649

yes

characteristic

172

C₂

#12476

yes

—

173

C₂

#12478

yes

—

174

C₂

#12835

yes

characteristic

175

C₂

#12842

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ 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

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
8. Q₈×C₁₆	C₄	semidirect

H ≅ G/N

type

8. Q₈×C₁₆

C₄

semidirect

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

subgroup	index
C₂×Q₈×C₁₆	2
#12888	2
C₂×#1372	2
#8645	2
#7274	2

subgroup

index

C₂×Q₈×C₁₆

Subgroups — #135010

Conjugacy classes of subgroups

Complements

Maximal subgroups