Group #144032 — subgroups

Subgroups — #144032

Order 512.

326 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 82 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	#33728	C₂	1	1	yes	maximal, characteristic
2	#19210	C₂	1	2	yes	maximal
3	#13267	C₂	1	1	yes	maximal, characteristic
4	#11936	C₂	1	2	yes	maximal
5	#7156	C₂	1	1	yes	maximal, characteristic
6	#3249	—	2	2	no	—
7	#3160	C₂²	1	2	yes	—
8	#2960	C₂²	1	1	yes	characteristic
9	#2255	—	2	2	no	—
10	#2255	—	2	2	no	—
11	#2158	—	2	2	no	—
12	#2150	C₂²	1	1	yes	characteristic
13	C₂×#346	C₂²	1	1	yes	characteristic
14	#1696	—	2	2	no	—
15	#1694	—	2	2	no	—
16	#1672	—	2	4	no	—
17	C₂×#334	C₂²	1	2	yes	—
18	#1550	—	2	2	no	—
19	#1379	—	2	2	no	—
20	#1363	—	2	4	no	—
21	#533	—	2	2	no	—
22	#533	D₄	1	1	yes	characteristic
23	C₂×M₃₂	—	2	1	no	—
24	C₂²×C₁₆	D₄	1	2	yes	—
25	#500	D₄	1	2	yes	—
26	#500	D₄	1	2	yes	—
27	#499	—	2	2	no	—
28	#499	—	2	2	no	—
29	#498	—	2	2	no	—
30	#477	—	2	4	no	—
31	#477	—	4	2	no	—
32	#467	—	4	4	no	—
33	#462	—	2	4	no	—
34	#462	—	4	2	no	—
35	#446	—	2	2	no	—
36	C₈×Q₈	—	2	2	no	—
37	#427	—	2	2	no	—
38	#427	—	2	2	no	—
39	#424	—	2	1	no	—
40	#423	D₄	1	1	yes	characteristic
41	C₂×C₄⋊C₈	—	2	2	no	—
42	#405	—	2	1	no	—
43	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
44	#367	—	2	2	no	—
45	#367	—	2	2	no	—
46	#366	—	2	4	no	—
47	C₄⋊C₁₆	—	2	2	no	—
48	#346	—	2	2	no	—
49	#334	—	2	2	no	—
50	#334	D₄	1	4	yes	—
51	#333	—	2	4	no	—
52	#332	—	2	4	no	—
53	C₂×Q₁₆	—	4	4	no	—
54	C₂×M₁₆	—	2	1	no	—
55	C₂²×C₈	—	4	2	no	—
56	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₈	#41	1	1	yes	characteristic
58	C₄⋊Q₈	—	2	2	no	—
59	C₄⋊Q₈	—	2	2	no	—
60	C₄⋊Q₈	D₈	1	2	yes	—
61	#125	—	2	1	no	—
62	#125	—	4	2	no	—
63	C₂²⋊Q₈	—	2	2	no	—
64	C₂²⋊Q₈	—	4	2	no	—
65	C₂²⋊Q₈	—	4	2	no	—
66	C₄×Q₈	—	2	2	no	—
67	C₄×Q₈	—	4	2	no	—
68	C₄×Q₈	—	4	2	no	—
69	#117	—	2	1	no	—
70	#117	—	4	2	no	—
71	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
72	M₃₂	—	4	2	no	—
73	C₂×C₁₆	—	2	2	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	—	4	1	no	—
76	C₂×C₁₆	Q₁₆	1	4	yes	complemented
77	#107	—	2	2	no	—
78	#107	—	2	2	no	—
79	#107	—	2	2	no	—
80	#107	—	2	2	no	—
81	#107	—	4	4	no	—
82	#106	—	4	2	no	—
83	#106	—	4	2	no	—
84	C₄⋊C₈	—	2	2	no	—
85	C₄⋊C₈	—	2	4	no	—
86	C₄⋊C₈	—	4	2	no	—
87	C₄⋊C₈	—	4	2	no	—
88	#103	—	4	4	no	—
89	#103	—	4	4	no	—
90	#97	—	4	1	no	—
91	C₄×C₈	—	4	1	no	—
92	C₄×C₈	C₂×D₄	1	2	yes	—
93	C₄×C₈	C₂×D₄	1	2	yes	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂²×C₄	#121	1	1	yes	characteristic
98	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
99	Q₁₆	—	16	4	no	—
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	4	1	no	—
103	C₂×C₈	—	4	4	no	—
104	C₂×C₈	—	16	2	no	—
105	C₂×C₈	#120	1	1	yes	characteristic
106	C₂×C₈	#121	1	2	yes	—
107	C₂×C₈	#127	1	1	yes	characteristic
108	C₂×C₈	C₂×Q₁₆	1	2	yes	—
109	C₂×C₈	#137	1	2	yes	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	—	4	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	4	2	no	—
117	C₄⋊C₄	—	8	2	no	—
118	C₄⋊C₄	—	8	2	no	—
119	C₂²⋊C₄	—	4	1	no	—
120	C₂²⋊C₄	—	8	2	no	—
121	C₄²	—	4	1	no	—
122	C₄²	—	8	2	no	—
123	C₄²	#120	1	2	yes	—
124	C₄²	C₂×D₈	1	1	yes	characteristic
125	C₄²	#135	1	1	yes	characteristic
126	C₁₆	—	2	4	no	—
127	C₁₆	—	8	1	no	—
128	C₂³	#469	1	1	yes	characteristic
129	Q₈	—	4	4	no	—
130	Q₈	—	8	2	no	—
131	Q₈	—	8	2	no	—
132	Q₈	—	16	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	8	2	no	—
136	C₂×C₄	—	8	2	no	—
137	C₂×C₄	#392	1	1	yes	characteristic
138	C₂×C₄	#447	1	2	yes	—
139	C₂×C₄	#449	1	2	yes	—
140	C₂×C₄	#462	1	2	yes	—
141	C₂×C₄	#466	1	1	yes	characteristic
142	C₂×C₄	#473	1	2	yes	—
143	C₈	—	2	2	no	—
144	C₈	—	8	4	no	—
145	C₈	—	8	1	no	—
146	C₈	#451	1	2	yes	—
147	C₈	#494	1	1	yes	characteristic
148	C₈	#497	1	1	yes	characteristic
149	C₂²	—	2	2	no	—
150	C₂²	#1694	1	2	yes	—
151	C₂²	#1999	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	2	2	no	—
154	C₄	—	8	2	no	—
155	C₄	—	16	2	no	—
156	C₄	—	16	2	no	—
157	C₄	#1674	1	2	yes	—
158	C₄	#2016	1	1	yes	characteristic
159	C₄	#2020	1	1	yes	characteristic
160	C₂	—	2	2	no	—
161	C₂	#10265	1	1	yes	characteristic
162	C₂	#13318	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33728

C₂

yes

maximal, characteristic

#19210

C₂

yes

maximal

#13267

C₂

yes

maximal, characteristic

#11936

C₂

yes

maximal

#7156

C₂

yes

maximal, characteristic

#3249

—

#3160

C₂²

yes

—

#2960

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#346

C₂²

yes

—

—

—

C₂²

yes

—

—

—

—

—

D₄

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

D₄

yes

—

#500

D₄

yes

—

#500

D₄

yes

—

—

—

—

—

—

—

—

—

—

C₈×Q₈

—

—

—

—

D₄

yes

characteristic

C₂×C₄⋊C₈

—

#405

—

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄⋊C₁₆

—

#346

—

#334

—

#334

D₄

yes

—

#333

—

#332

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₈

yes

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

Q₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

#120

yes

characteristic

106

C₂×C₈

#121

yes

—

107

C₂×C₈

#127

yes

characteristic

108

C₂×C₈

C₂×Q₁₆

yes

—

109

C₂×C₈

#137

yes

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

#120

yes

—

124

C₄²

C₂×D₈

yes

characteristic

125

C₄²

#135

yes

characteristic

126

C₁₆

—

127

C₁₆

—

128

C₂³

#469

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#392

yes

characteristic

138

C₂×C₄

#447

yes

—

139

C₂×C₄

#449

yes

—

140

C₂×C₄

#462

yes

—

141

C₂×C₄

#466

yes

characteristic

142

C₂×C₄

#473

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

#451

yes

—

147

C₈

#494

yes

characteristic

148

C₈

#497

yes

characteristic

149

C₂²

—

150

C₂²

#1694

yes

—

151

C₂²

#1999

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#1674

yes

—

158

C₄

#2016

yes

characteristic

159

C₄

#2020

yes

characteristic

160

C₂

—

161

C₂

#10265

yes

characteristic

162

C₂

#13318

yes

—

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#469
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1999

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#469

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
76. C₂×C₁₆	Q₁₆	semidirect

H ≅ G/N

type

76. C₂×C₁₆

Q₁₆

semidirect

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

subgroup	index
#33728	2
#19210	2
#13267	2
#11936	2
#7156	2

subgroup

index

Subgroups — #144032

Conjugacy classes of subgroups

Complements

Maximal subgroups