Group #146155 — subgroups

Subgroups — #146155

Order 512.

235 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 235 conjugacy classes of subgroups — too many to draw. Only its 96 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	#13159	C₂	1	1	yes	maximal, characteristic
2	#12905	C₂	1	1	yes	maximal, characteristic
3	#9615	C₂	1	2	yes	maximal
4	#7274	C₂	1	2	yes	maximal
5	#7154	C₂	1	1	yes	maximal, characteristic
6	C₂×#366	—	2	1	no	—
7	C₂×#366	—	2	1	no	—
8	C₂×C₄⋊C₁₆	—	2	1	no	—
9	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#1948	—	2	2	no	—
12	#1845	C₂²	1	2	yes	—
13	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×#335	C₂²	1	2	yes	—
15	#1374	—	2	1	no	—
16	#1296	—	2	2	no	—
17	C₂²×C₁₆	—	2	1	no	—
18	C₂²×C₁₆	D₄	1	1	yes	characteristic
19	C₂²×C₁₆	D₄	1	1	yes	characteristic
20	C₂×#107	—	2	2	no	—
21	C₂×#107	—	2	2	no	—
22	C₂×#106	—	4	2	no	—
23	C₂×C₄⋊C₈	—	2	2	no	—
24	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
25	C₂×C₄×C₈	—	2	1	no	—
26	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
27	#384	—	2	2	no	—
28	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
29	#366	—	4	2	no	—
30	#366	—	4	2	no	—
31	C₄⋊C₁₆	—	2	2	no	—
32	C₄⋊C₁₆	—	4	2	no	—
33	C₄×C₁₆	—	2	2	no	—
34	#340	—	4	2	no	—
35	#336	—	2	1	no	—
36	#335	—	2	2	no	—
37	#335	Q₈	1	4	yes	—
38	C₂²×C₈	—	2	1	no	—
39	C₂²×C₈	—	4	2	no	—
40	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₈	#41	1	1	yes	characteristic
42	C₂×C₄⋊C₄	—	4	2	no	—
43	C₂×C₄⋊C₄	—	4	2	no	—
44	C₂×C₄⋊C₄	—	4	2	no	—
45	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
46	C₂×C₄⋊C₄	D₈	1	1	yes	characteristic
47	C₂×C₄²	—	2	1	no	—
48	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
49	C₂×C₁₆	—	2	1	no	—
50	C₂×C₁₆	—	2	1	no	—
51	C₂×C₁₆	—	2	1	no	—
52	C₂×C₁₆	—	2	1	no	—
53	C₂×C₁₆	—	4	1	no	—
54	C₂×C₁₆	—	4	1	no	—
55	C₂×C₁₆	—	4	1	no	—
56	C₂×C₁₆	SD₁₆	1	2	yes	—
57	C₂×C₁₆	SD₁₆	1	2	yes	—
58	#107	—	4	2	no	—
59	#107	—	4	2	no	—
60	#107	—	8	2	no	—
61	#106	—	8	4	no	—
62	C₄⋊C₈	—	2	4	no	—
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	4	2	no	—
65	C₄×C₈	—	4	1	no	—
66	C₄×C₈	—	4	1	no	—
67	C₄×C₈	C₂×Q₈	1	2	yes	—
68	C₄×C₈	#41	1	2	yes	—
69	#95	—	2	1	no	—
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	—	2	1	no	—
72	C₂²×C₄	—	8	2	no	—
73	C₂²×C₄	#121	1	1	yes	characteristic
74	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
75	C₂²×C₄	#135	1	1	yes	characteristic
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	4	1	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	4	4	no	—
82	C₂×C₈	—	8	2	no	—
83	C₂×C₈	—	8	2	no	—
84	C₂×C₈	#123	1	1	yes	characteristic
85	C₂×C₈	#125	1	2	yes	—
86	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
87	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₈	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×C₈	#137	1	1	yes	characteristic
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	8	2	no	—
94	C₄⋊C₄	—	8	2	no	—
95	C₄⋊C₄	—	8	2	no	—
96	C₄⋊C₄	—	8	2	no	—
97	C₄⋊C₄	—	16	2	no	—
98	C₄⋊C₄	Q₃₂	1	2	yes	—
99	C₄²	—	4	1	no	—
100	C₄²	—	4	1	no	—
101	C₄²	C₂²⋊Q₈	1	2	yes	—
102	C₄²	#137	1	2	yes	—
103	C₁₆	—	2	2	no	—
104	C₁₆	—	2	2	no	—
105	C₁₆	—	8	1	no	—
106	C₂³	#466	1	1	yes	characteristic
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	16	2	no	—
116	C₂×C₄	—	16	2	no	—
117	C₂×C₄	—	16	2	no	—
118	C₂×C₄	#398	1	1	yes	characteristic
119	C₂×C₄	#460	1	1	yes	characteristic
120	C₂×C₄	#461	1	1	yes	characteristic
121	C₂×C₄	#470	1	1	yes	characteristic
122	C₂×C₄	#472	1	1	yes	characteristic
123	C₂×C₄	#473	1	1	yes	characteristic
124	C₂×C₄	#474	1	2	yes	—
125	C₂×C₄	#479	1	2	yes	—
126	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
127	C₂×C₄	#508	1	1	yes	characteristic
128	C₂×C₄	#509	1	2	yes	—
129	C₈	—	2	2	no	—
130	C₈	—	8	1	no	—
131	C₈	—	8	4	no	—
132	C₈	#481	1	1	yes	characteristic
133	C₈	#483	1	1	yes	characteristic
134	C₈	#498	1	1	yes	characteristic
135	C₈	#501	1	1	yes	characteristic
136	C₂²	#1668	1	1	yes	characteristic
137	C₂²	#1673	1	1	yes	characteristic
138	C₂²	#2086	1	1	yes	characteristic
139	C₂²	#2216	1	1	yes	characteristic
140	C₂²	#2218	1	1	yes	characteristic
141	C₂²	#2221	1	1	yes	characteristic
142	C₂²	#2222	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	2	no	—
145	C₄	—	8	1	no	—
146	C₄	—	32	2	no	—
147	C₄	#1690	1	1	yes	characteristic
148	C₄	#1695	1	1	yes	characteristic
149	C₄	#2084	1	1	yes	characteristic
150	C₄	#2089	1	1	yes	characteristic
151	C₄	#2227	1	2	yes	—
152	C₄	#2228	1	2	yes	—
153	C₂	#10651	1	1	yes	characteristic
154	C₂	#12440	1	1	yes	characteristic
155	C₂	#12442	1	1	yes	characteristic
156	C₂	#12452	1	1	yes	characteristic
157	C₂	#12454	1	1	yes	characteristic
158	C₂	#13515	1	1	yes	characteristic
159	C₂	#13517	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13159

C₂

yes

maximal, characteristic

#12905

C₂

yes

maximal, characteristic

#9615

C₂

yes

maximal

#7274

C₂

yes

maximal

#7154

C₂

yes

maximal, characteristic

C₂×#366

—

C₂×#366

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1948

—

#1845

C₂²

yes

—

C₈×C₄⋊C₄

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

#1374

—

#1296

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

C₄×C₄⋊C₄

D₄

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

—

—

—

Q₈

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₈

yes

characteristic

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

SD₁₆

yes

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

#95

—

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

#123

yes

characteristic

C₂×C₈

#125

yes

—

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

characteristic

C₂×C₈

C₂×SD₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Q₃₂

yes

—

C₄²

—

100

C₄²

—

101

C₄²

C₂²⋊Q₈

yes

—

102

C₄²

#137

yes

—

103

C₁₆

—

104

C₁₆

—

105

C₁₆

—

106

C₂³

#466

yes

characteristic

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

#398

yes

characteristic

119

C₂×C₄

#460

yes

characteristic

120

C₂×C₄

#461

yes

characteristic

121

C₂×C₄

#470

yes

characteristic

122

C₂×C₄

#472

yes

characteristic

123

C₂×C₄

#473

yes

characteristic

124

C₂×C₄

#474

yes

—

125

C₂×C₄

#479

yes

—

126

C₂×C₄

C₂×Q₃₂

yes

characteristic

127

C₂×C₄

#508

yes

characteristic

128

C₂×C₄

#509

yes

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

#481

yes

characteristic

133

C₈

#483

yes

characteristic

134

C₈

#498

yes

characteristic

135

C₈

#501

yes

characteristic

136

C₂²

#1668

yes

characteristic

137

C₂²

#1673

yes

characteristic

138

C₂²

#2086

yes

characteristic

139

C₂²

#2216

yes

characteristic

140

C₂²

#2218

yes

characteristic

141

C₂²

#2221

yes

characteristic

142

C₂²

#2222

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#1690

yes

characteristic

148

C₄

#1695

yes

characteristic

149

C₄

#2084

yes

characteristic

150

C₄

#2089

yes

characteristic

151

C₄

#2227

yes

—

152

C₄

#2228

yes

—

153

C₂

#10651

yes

characteristic

154

C₂

#12440

yes

characteristic

155

C₂

#12442

yes

characteristic

156

C₂

#12452

yes

characteristic

157

C₂

#12454

yes

characteristic

158

C₂

#13515

yes

characteristic

159

C₂

#13517

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 ◁ C₈×C₄⋊C₄ ◁ 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 ◁ C₈×C₄⋊C₄ ◁ 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
#13159	2
#12905	2
#9615	2
#7274	2
#7154	2

subgroup

index

Subgroups — #146155

Conjugacy classes of subgroups

Maximal subgroups