Group #141253 — subgroups

Subgroups — #141253

Order 512.

364 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 106 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×#1713	C₂	1	1	yes	maximal, characteristic
2	#13256	C₂	1	2	yes	maximal
3	#12004	C₂	1	1	yes	maximal, characteristic
4	#11932	C₂	1	2	yes	maximal
5	C₈×#108	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	—	2	1	no	—
7	C₂×#500	—	2	1	no	—
8	C₂×#494	C₂²	1	2	yes	—
9	#2150	C₂²	1	2	yes	—
10	C₂×#362	—	2	4	no	—
11	C₄×#108	C₂²	1	1	yes	characteristic
12	#1713	—	2	2	no	—
13	#1713	C₄	1	4	yes	—
14	#1564	C₂²	1	1	yes	characteristic
15	C₈×M₁₆	—	2	1	no	—
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	#1370	—	2	2	no	—
18	#1362	—	2	4	no	—
19	#1278	—	2	1	no	—
20	C₂²×Q₁₆	—	2	2	no	—
21	C₂²×Q₁₆	—	2	2	no	—
22	C₂×C₄⋊Q₈	—	2	2	no	—
23	C₂×M₃₂	—	2	2	no	—
24	#500	—	2	4	no	—
25	#500	—	2	4	no	—
26	#500	—	4	2	no	—
27	#500	—	4	1	no	—
28	#500	—	4	1	no	—
29	#494	—	2	2	no	—
30	#494	—	2	2	no	—
31	#494	—	2	2	no	—
32	#494	C₂×C₄	1	4	yes	—
33	#431	—	2	1	no	—
34	C₂×#108	D₄	1	2	yes	—
35	C₂×#107	—	4	2	no	—
36	C₂×#107	—	4	2	no	—
37	C₄×M₁₆	—	2	1	no	—
38	C₂×C₄×C₈	D₄	1	1	yes	characteristic
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄⋊C₁₆	—	2	2	no	—
43	#362	—	4	8	no	—
44	#341	—	2	1	no	—
45	#335	D₄	1	2	yes	—
46	#334	—	2	1	no	—
47	#322	—	4	1	no	—
48	C₈²	—	4	1	no	—
49	C₈²	C₂×C₄	1	2	yes	—
50	C₈²	D₄	1	2	yes	—
51	C₂²×Q₈	—	4	2	no	—
52	C₂×Q₁₆	—	2	4	no	—
53	C₂×Q₁₆	—	2	4	no	—
54	C₂×Q₁₆	—	4	2	no	—
55	C₂×Q₁₆	—	4	2	no	—
56	C₂×Q₁₆	—	4	4	no	—
57	C₂×Q₁₆	—	4	4	no	—
58	C₂×Q₁₆	—	8	4	no	—
59	C₂×M₁₆	—	2	2	no	—
60	C₂²×C₈	—	2	1	no	—
61	C₂²×C₈	SD₁₆	1	2	yes	—
62	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₈	#41	1	1	yes	characteristic
64	C₄⋊Q₈	—	2	4	no	—
65	C₄⋊Q₈	—	4	2	no	—
66	C₄⋊Q₈	—	4	2	no	—
67	C₄⋊Q₈	—	4	2	no	—
68	C₂×C₄⋊C₄	—	4	2	no	—
69	C₂×C₄⋊C₄	—	4	2	no	—
70	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
71	M₃₂	—	4	4	no	—
72	C₂×C₁₆	—	4	2	no	—
73	#108	—	2	2	no	—
74	#108	D₈	1	4	yes	—
75	#107	—	4	8	no	—
76	#107	—	4	4	no	—
77	#107	—	4	4	no	—
78	C₄⋊C₈	—	2	2	no	—
79	C₂²⋊C₈	—	4	1	no	—
80	#97	—	4	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₄×C₈	—	4	1	no	—
83	C₄×C₈	—	4	1	no	—
84	C₄×C₈	C₂²⋊C₄	1	2	yes	—
85	C₄×C₈	C₂²⋊C₄	1	2	yes	—
86	C₄×C₈	D₈	1	1	yes	characteristic
87	C₄×C₈	D₈	1	1	yes	characteristic
88	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
89	C₄×C₈	C₂×D₄	1	1	yes	characteristic
90	C₄×C₈	C₂×D₄	1	1	yes	characteristic
91	C₄×C₈	#41	1	1	yes	characteristic
92	C₂×Q₈	—	4	4	no	—
93	C₂×Q₈	—	8	2	no	—
94	C₂×Q₈	—	8	4	no	—
95	C₂×Q₈	—	8	4	no	—
96	C₂²×C₄	—	8	2	no	—
97	C₂²×C₄	#121	1	1	yes	characteristic
98	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
99	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
100	Q₁₆	—	4	8	no	—
101	Q₁₆	—	8	4	no	—
102	Q₁₆	—	8	4	no	—
103	Q₁₆	—	8	2	no	—
104	Q₁₆	—	8	2	no	—
105	M₁₆	—	2	4	no	—
106	M₁₆	—	4	2	no	—
107	C₂×C₈	—	2	4	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	4	no	—
111	C₂×C₈	—	4	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	#102	1	2	yes	—
114	C₂×C₈	#102	1	2	yes	—
115	C₂×C₈	C₄×D₄	1	1	yes	characteristic
116	C₂×C₈	C₄×D₄	1	1	yes	characteristic
117	C₂×C₈	#124	1	1	yes	characteristic
118	C₂×C₈	#127	1	1	yes	characteristic
119	C₂×C₈	C₂×D₈	1	2	yes	—
120	C₂×C₈	#135	1	2	yes	—
121	C₄⋊C₄	—	4	4	no	—
122	C₄⋊C₄	—	4	4	no	—
123	C₄⋊C₄	—	8	2	no	—
124	C₄⋊C₄	—	8	2	no	—
125	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₄²	#121	1	1	yes	characteristic
127	C₄²	C₂×D₈	1	1	yes	characteristic
128	C₄²	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₁₆	—	8	2	no	—
130	C₂³	#460	1	1	yes	characteristic
131	Q₈	—	8	4	no	—
132	Q₈	—	8	4	no	—
133	Q₈	—	16	4	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	8	4	no	—
137	C₂×C₄	—	16	4	no	—
138	C₂×C₄	#390	1	1	yes	characteristic
139	C₂×C₄	C₂×#102	1	1	yes	characteristic
140	C₂×C₄	#418	1	1	yes	characteristic
141	C₂×C₄	#459	1	1	yes	characteristic
142	C₂×C₄	#459	1	1	yes	characteristic
143	C₂×C₄	#460	1	1	yes	characteristic
144	C₂×C₄	#466	1	2	yes	—
145	C₂×C₄	#472	1	2	yes	—
146	C₈	—	2	4	no	—
147	C₈	—	2	2	no	—
148	C₈	—	2	1	no	—
149	C₈	—	2	1	no	—
150	C₈	—	4	4	no	—
151	C₈	—	8	2	no	—
152	C₈	#362	1	4	yes	complemented
153	C₈	C₄×D₈	1	2	yes	—
154	C₈	#493	1	1	yes	characteristic
155	C₈	#495	1	1	yes	characteristic
156	C₂²	—	2	1	no	—
157	C₂²	—	2	1	no	—
158	C₂²	#1668	1	2	yes	—
159	C₂²	#1982	1	1	yes	characteristic
160	C₄	—	2	1	no	—
161	C₄	—	2	1	no	—
162	C₄	—	16	4	no	—
163	C₄	#1667	1	2	yes	—
164	C₄	#1970	1	1	yes	characteristic
165	C₄	#1978	1	1	yes	characteristic
166	C₄	C₂×#362	1	2	yes	—
167	C₄	#2214	1	2	yes	—
168	C₂	—	2	2	no	—
169	C₂	#9734	1	1	yes	characteristic
170	C₂	#13256	1	1	yes	characteristic
171	C₂	#13262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1713

C₂

yes

maximal, characteristic

#13256

C₂

yes

maximal

#12004

C₂

yes

maximal, characteristic

#11932

C₂

yes

maximal

C₈×#108

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

—

C₂×#494

C₂²

yes

—

#2150

C₂²

yes

—

C₂×#362

—

C₄×#108

C₂²

yes

characteristic

#1713

—

#1713

C₄

yes

—

#1564

C₂²

yes

characteristic

C₈×M₁₆

—

C₂×C₈²

C₂²

yes

—

—

—

C₂²×Q₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

—

—

—

—

—

—

—

—

—

C₂×C₄

yes

—

#431

—

C₂×#108

D₄

yes

—

C₂×#107

—

C₂×#107

—

C₄×M₁₆

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#362

—

#341

—

#335

D₄

yes

—

#334

—

#322

—

C₈²

—

C₈²

C₂×C₄

yes

—

C₈²

D₄

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

#108

—

#108

D₈

yes

—

#107

—

#107

—

#107

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

C₂²⋊C₄

yes

—

C₄×C₈

D₈

yes

characteristic

C₄×C₈

D₈

yes

characteristic

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

#41

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

characteristic

100

Q₁₆

—

101

Q₁₆

—

102

Q₁₆

—

103

Q₁₆

—

104

Q₁₆

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

#102

yes

—

114

C₂×C₈

#102

yes

—

115

C₂×C₈

C₄×D₄

yes

characteristic

116

C₂×C₈

C₄×D₄

yes

characteristic

117

C₂×C₈

#124

yes

characteristic

118

C₂×C₈

#127

yes

characteristic

119

C₂×C₈

C₂×D₈

yes

—

120

C₂×C₈

#135

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄²

C₂×C₂²⋊C₄

yes

characteristic

126

C₄²

#121

yes

characteristic

127

C₄²

C₂×D₈

yes

characteristic

128

C₄²

AGL(1,ℤ/8)

yes

characteristic

129

C₁₆

—

130

C₂³

#460

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#390

yes

characteristic

139

C₂×C₄

C₂×#102

yes

characteristic

140

C₂×C₄

#418

yes

characteristic

141

C₂×C₄

#459

yes

characteristic

142

C₂×C₄

#459

yes

characteristic

143

C₂×C₄

#460

yes

characteristic

144

C₂×C₄

#466

yes

—

145

C₂×C₄

#472

yes

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₈

—

150

C₈

—

151

C₈

—

152

C₈

#362

yes

complemented

153

C₈

C₄×D₈

yes

—

154

C₈

#493

yes

characteristic

155

C₈

#495

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

#1668

yes

—

159

C₂²

#1982

yes

characteristic

160

C₄

—

161

C₄

—

162

C₄

—

163

C₄

#1667

yes

—

164

C₄

#1970

yes

characteristic

165

C₄

#1978

yes

characteristic

166

C₄

C₂×#362

yes

—

167

C₄

#2214

yes

—

168

C₂

—

169

C₂

#9734

yes

characteristic

170

C₂

#13256

yes

characteristic

171

C₂

#13262

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1982
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#460
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

#1982

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#460

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

N	H ≅ G/N	type
152. C₈	#362	semidirect

H ≅ G/N

type

152. C₈

#362

semidirect

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

subgroup	index
C₂×#1713	2
#13256	2
#12004	2
#11932	2
C₈×#108	2

subgroup

index

Subgroups — #141253

Conjugacy classes of subgroups

Complements

Maximal subgroups