Group #144027 — subgroups

Subgroups — #144027

Order 512.

357 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 100 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×#2252	C₂	1	1	yes	maximal, characteristic
2	C₂×#1676	C₂	1	2	yes	maximal
3	#13263	C₂	1	1	yes	maximal, characteristic
4	C₂×#1365	C₂	1	2	yes	maximal
5	#7154	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	2	yes	—
7	C₂×#477	—	2	2	no	—
8	C₂×C₈×Q₈	C₂²	1	1	yes	characteristic
9	#2252	—	2	4	no	—
10	C₂×#366	—	2	2	no	—
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
13	#1676	—	2	4	no	—
14	#1676	—	2	4	no	—
15	C₂×#335	C₂²	1	2	yes	—
16	C₂×#333	—	2	2	no	—
17	#1374	—	2	2	no	—
18	#1365	—	2	4	no	—
19	C₂×C₄⋊Q₈	—	2	2	no	—
20	C₂×C₄×Q₈	D₄	1	1	yes	characteristic
21	C₂²×C₁₆	—	2	1	no	—
22	C₂²×C₁₆	D₄	1	2	yes	—
23	#500	—	2	2	no	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	D₄	1	4	yes	—
27	#477	—	2	8	no	—
28	#477	—	4	4	no	—
29	C₈×Q₈	—	2	2	no	—
30	C₈×Q₈	—	2	2	no	—
31	C₂×#107	—	2	2	no	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	—	4	2	no	—
34	C₂×C₄⋊C₈	—	2	2	no	—
35	C₂×C₄⋊C₈	—	2	1	no	—
36	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
37	C₂×#103	—	4	2	no	—
38	C₂×C₄×C₈	—	2	1	no	—
39	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
40	#366	—	2	8	no	—
41	C₄⋊C₁₆	—	2	2	no	—
42	C₄×C₁₆	—	2	2	no	—
43	#335	—	2	2	no	—
44	#335	D₄	1	4	yes	—
45	#333	—	2	8	no	—
46	C₂²×Q₈	—	4	2	no	—
47	C₂²×Q₈	D₈	1	1	yes	characteristic
48	C₂²×C₈	—	2	1	no	—
49	C₂²×C₈	—	4	2	no	—
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₈	#41	1	1	yes	characteristic
52	C₄⋊Q₈	—	2	4	no	—
53	C₄⋊Q₈	—	4	2	no	—
54	C₄⋊Q₈	—	4	2	no	—
55	C₄⋊Q₈	—	4	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	C₄×Q₈	—	2	2	no	—
58	C₂×C₄⋊C₄	—	2	1	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
62	C₂×C₄²	—	2	1	no	—
63	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	C₂×C₁₆	—	4	1	no	—
67	C₂×C₁₆	—	4	2	no	—
68	C₂×C₁₆	Q₁₆	1	4	yes	—
69	#107	—	2	4	no	—
70	#107	—	2	4	no	—
71	#107	—	4	8	no	—
72	#107	—	4	2	no	—
73	#107	—	4	2	no	—
74	C₄⋊C₈	—	2	2	no	—
75	C₄⋊C₈	—	2	4	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	#103	—	4	8	no	—
79	C₄×C₈	—	4	1	no	—
80	C₄×C₈	—	4	1	no	—
81	C₄×C₈	C₂×D₄	1	2	yes	—
82	C₄×C₈	C₂×D₄	1	2	yes	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	2	2	no	—
85	C₂×Q₈	—	2	2	no	—
86	C₂×Q₈	—	4	4	no	—
87	C₂×Q₈	—	8	2	no	—
88	C₂×Q₈	—	16	4	no	—
89	C₂×Q₈	Q₃₂	1	2	yes	—
90	C₂²×C₄	—	2	1	no	—
91	C₂²×C₄	—	2	1	no	—
92	C₂²×C₄	—	8	2	no	—
93	C₂²×C₄	#121	1	1	yes	characteristic
94	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
95	C₂²×C₄	#135	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	4	no	—
99	C₂×C₈	—	4	2	no	—
100	C₂×C₈	—	8	2	no	—
101	C₂×C₈	—	8	2	no	—
102	C₂×C₈	#120	1	1	yes	characteristic
103	C₂×C₈	#121	1	2	yes	—
104	C₂×C₈	#127	1	1	yes	characteristic
105	C₂×C₈	C₂×Q₁₆	1	2	yes	—
106	C₂×C₈	#137	1	2	yes	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	4	4	no	—
110	C₄⋊C₄	—	4	4	no	—
111	C₄⋊C₄	—	8	2	no	—
112	C₄⋊C₄	—	8	2	no	—
113	C₄²	—	4	1	no	—
114	C₄²	—	4	1	no	—
115	C₄²	#120	1	2	yes	—
116	C₄²	AGL(1,ℤ/8)	1	2	yes	—
117	C₁₆	—	2	4	no	—
118	C₁₆	—	8	1	no	—
119	C₂³	#466	1	1	yes	characteristic
120	Q₈	—	2	4	no	—
121	Q₈	—	4	2	no	—
122	Q₈	—	16	4	no	—
123	Q₈	—	16	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	16	4	no	—
132	C₂×C₄	#392	1	1	yes	characteristic
133	C₂×C₄	#447	1	2	yes	—
134	C₂×C₄	#449	1	2	yes	—
135	C₂×C₄	#462	1	2	yes	—
136	C₂×C₄	#469	1	1	yes	characteristic
137	C₂×C₄	#473	1	2	yes	—
138	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
139	C₂×C₄	#508	1	1	yes	characteristic
140	C₂×C₄	#510	1	2	yes	—
141	C₈	—	2	2	no	—
142	C₈	—	8	4	no	—
143	C₈	—	8	1	no	—
144	C₈	#451	1	2	yes	—
145	C₈	#494	1	1	yes	characteristic
146	C₈	#497	1	1	yes	characteristic
147	C₂²	#1674	1	2	yes	—
148	C₂²	#1999	1	1	yes	characteristic
149	C₂²	#2216	1	1	yes	characteristic
150	C₂²	#2219	1	1	yes	characteristic
151	C₂²	#2221	1	2	yes	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	8	1	no	—
155	C₄	—	16	4	no	—
156	C₄	#1694	1	2	yes	—
157	C₄	#2016	1	1	yes	characteristic
158	C₄	#2020	1	1	yes	characteristic
159	C₄	#2188	1	2	yes	—
160	C₄	#2189	1	2	yes	—
161	C₂	#10265	1	1	yes	characteristic
162	C₂	#12458	1	2	yes	—
163	C₂	#12461	1	2	yes	—
164	C₂	#13306	1	1	yes	characteristic
165	C₂	#13313	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

C₂×#1676

C₂

yes

maximal

#13263

C₂

yes

maximal, characteristic

C₂×#1365

C₂

yes

maximal

#7154

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

—

C₂×#477

—

C₂×C₈×Q₈

C₂²

yes

characteristic

#2252

—

C₂×#366

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

C₂²

yes

—

C₂×#333

—

#1374

—

#1365

—

C₂×C₄⋊Q₈

—

C₂×C₄×Q₈

D₄

yes

characteristic

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

—

—

—

—

D₄

yes

—

#477

—

#477

—

C₈×Q₈

—

C₈×Q₈

—

C₂×#107

—

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

#366

—

C₄⋊C₁₆

—

C₄×C₁₆

—

#335

—

#335

D₄

yes

—

#333

—

C₂²×Q₈

—

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⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

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

Q₁₆

yes

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

#120

yes

characteristic

103

C₂×C₈

#121

yes

—

104

C₂×C₈

#127

yes

characteristic

105

C₂×C₈

C₂×Q₁₆

yes

—

106

C₂×C₈

#137

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

#120

yes

—

116

C₄²

AGL(1,ℤ/8)

yes

—

117

C₁₆

—

118

C₁₆

—

119

C₂³

#466

yes

characteristic

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#392

yes

characteristic

133

C₂×C₄

#447

yes

—

134

C₂×C₄

#449

yes

—

135

C₂×C₄

#462

yes

—

136

C₂×C₄

#469

yes

characteristic

137

C₂×C₄

#473

yes

—

138

C₂×C₄

C₂×Q₃₂

yes

characteristic

139

C₂×C₄

#508

yes

characteristic

140

C₂×C₄

#510

yes

—

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

#451

yes

—

145

C₈

#494

yes

characteristic

146

C₈

#497

yes

characteristic

147

C₂²

#1674

yes

—

148

C₂²

#1999

yes

characteristic

149

C₂²

#2216

yes

characteristic

150

C₂²

#2219

yes

characteristic

151

C₂²

#2221

yes

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

#1694

yes

—

157

C₄

#2016

yes

characteristic

158

C₄

#2020

yes

characteristic

159

C₄

#2188

yes

—

160

C₄

#2189

yes

—

161

C₂

#10265

yes

characteristic

162

C₂

#12458

yes

—

163

C₂

#12461

yes

—

164

C₂

#13306

yes

characteristic

165

C₂

#13313

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
C₂×#2252	2
C₂×#1676	2
#13263	2
C₂×#1365	2
#7154	2

subgroup

index

Subgroups — #144027

Conjugacy classes of subgroups

Maximal subgroups