Group #146267 — subgroups

Subgroups — #146267

Order 512.

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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	#13288	C₂	1	1	yes	maximal, characteristic
2	#13285	C₂	1	1	yes	maximal, characteristic
3	#12904	C₂	1	1	yes	maximal, characteristic
4	#12889	C₂	1	1	yes	maximal, characteristic
5	#9755	C₂	1	1	yes	maximal, characteristic
6	#8793	C₂	1	1	yes	maximal, characteristic
7	#7141	C₂	1	1	yes	maximal, characteristic
8	#3160	C₂²	1	1	yes	characteristic
9	C₂×#365	—	2	2	no	—
10	C₄×M₃₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
12	#1848	C₂²	1	1	yes	characteristic
13	#1846	C₂²	1	1	yes	characteristic
14	#1846	C₂²	1	1	yes	characteristic
15	#1518	—	2	2	no	—
16	#1450	C₂²	1	1	yes	characteristic
17	#1386	—	2	2	no	—
18	#1382	—	2	2	no	—
19	#1380	—	2	2	no	—
20	#1286	—	2	2	no	—
21	#533	—	2	1	no	—
22	C₂×M₃₂	D₄	1	2	yes	—
23	C₂²×C₁₆	D₄	1	2	yes	—
24	#500	Q₈	1	2	yes	—
25	#499	—	2	1	no	—
26	#499	—	2	1	no	—
27	#498	—	2	1	no	—
28	#431	D₄	1	2	yes	—
29	C₂×#108	—	2	1	no	—
30	C₂×#108	—	2	1	no	—
31	#427	—	2	1	no	—
32	#427	—	2	1	no	—
33	C₂×#107	—	2	1	no	—
34	C₂×#107	—	2	1	no	—
35	C₂×#106	—	2	1	no	—
36	C₂×#106	—	2	1	no	—
37	#423	—	2	1	no	—
38	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
39	#382	—	2	1	no	—
40	#382	—	2	1	no	—
41	#365	—	4	4	no	—
42	#346	—	2	1	no	—
43	C₄×C₁₆	—	2	1	no	—
44	C₄×C₁₆	—	2	1	no	—
45	C₄×C₁₆	—	2	1	no	—
46	#328	—	4	4	no	—
47	#322	—	2	1	no	—
48	C₈²	—	2	1	no	—
49	C₂×M₁₆	—	4	1	no	—
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₄⋊Q₈	—	2	2	no	—
53	#125	—	4	1	no	—
54	C₂²⋊Q₈	—	4	2	no	—
55	C₄×Q₈	—	4	2	no	—
56	#117	—	4	1	no	—
57	C₂×C₄⋊C₄	—	4	1	no	—
58	C₂×C₄⋊C₄	—	4	1	no	—
59	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
60	M₃₂	—	2	2	no	—
61	M₃₂	SD₁₆	1	4	yes	—
62	C₂×C₁₆	—	2	2	no	—
63	C₂×C₁₆	—	2	2	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	2	2	no	—
66	#108	—	4	2	no	—
67	#108	—	4	2	no	—
68	#107	—	2	2	no	—
69	#107	—	2	2	no	—
70	#107	—	4	2	no	—
71	#107	—	4	2	no	—
72	#106	—	4	1	no	—
73	#106	—	4	1	no	—
74	#106	—	4	2	no	—
75	#106	—	4	2	no	—
76	C₄⋊C₈	—	2	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₂²⋊C₈	—	2	2	no	—
79	C₄×C₈	—	2	2	no	—
80	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
81	C₄×C₈	#41	1	1	yes	characteristic
82	C₄×C₈	#41	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	#95	—	4	2	no	—
85	#95	—	4	2	no	—
86	C₂×Q₈	—	4	2	no	—
87	C₂²×C₄	—	8	1	no	—
88	C₂²×C₄	—	8	1	no	—
89	C₂²×C₄	#127	1	1	yes	characteristic
90	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
91	M₁₆	—	8	1	no	—
92	M₁₆	—	8	1	no	—
93	C₂×C₈	—	4	2	no	—
94	C₂×C₈	—	8	1	no	—
95	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂×C₈	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂×C₈	#123	1	1	yes	characteristic
98	C₂×C₈	#123	1	1	yes	characteristic
99	C₂×C₈	C₂×SD₁₆	1	2	yes	—
100	C₂×C₈	#135	1	2	yes	—
101	C₂×C₈	#137	1	2	yes	—
102	C₂×C₈	#137	1	2	yes	—
103	C₄⋊C₄	—	4	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	8	2	no	—
106	C₄⋊C₄	—	8	1	no	—
107	C₄⋊C₄	—	8	1	no	—
108	C₄⋊C₄	—	8	1	no	—
109	C₄⋊C₄	—	8	1	no	—
110	C₂²⋊C₄	—	8	1	no	—
111	C₄²	—	8	1	no	—
112	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
113	C₄²	#123	1	1	yes	characteristic
114	C₄²	#135	1	2	yes	—
115	C₁₆	—	2	4	no	—
116	C₁₆	—	4	2	no	—
117	C₂³	#496	1	1	yes	characteristic
118	Q₈	—	16	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	16	2	no	—
122	C₂×C₄	—	16	2	no	—
123	C₂×C₄	—	16	1	no	—
124	C₂×C₄	—	16	1	no	—
125	C₂×C₄	#399	1	1	yes	characteristic
126	C₂×C₄	#478	1	2	yes	—
127	C₂×C₄	#482	1	2	yes	—
128	C₂×C₄	#492	1	1	yes	characteristic
129	C₂×C₄	#495	1	1	yes	characteristic
130	C₂×C₄	#495	1	1	yes	characteristic
131	C₂×C₄	#497	1	1	yes	characteristic
132	C₂×C₄	#497	1	1	yes	characteristic
133	C₈	—	8	2	no	—
134	C₈	—	16	1	no	—
135	C₈	#475	1	2	yes	—
136	C₈	#479	1	2	yes	—
137	C₈	#483	1	2	yes	—
138	C₈	#485	1	2	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	#1714	1	1	yes	characteristic
141	C₂²	#1720	1	1	yes	characteristic
142	C₂²	#2092	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	2	no	—
145	C₄	—	16	2	no	—
146	C₄	—	32	1	no	—
147	C₄	—	32	1	no	—
148	C₄	#1708	1	1	yes	characteristic
149	C₄	#1718	1	1	yes	characteristic
150	C₄	#2096	1	1	yes	characteristic
151	C₄	#2097	1	1	yes	characteristic
152	C₂	—	2	1	no	—
153	C₂	—	2	1	no	—
154	C₂	#10681	1	1	yes	characteristic
155	C₂	#13525	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13288

C₂

yes

maximal, characteristic

#13285

C₂

yes

maximal, characteristic

#12904

C₂

yes

maximal, characteristic

#12889

C₂

yes

maximal, characteristic

#9755

C₂

yes

maximal, characteristic

#8793

C₂

yes

maximal, characteristic

#7141

C₂

yes

maximal, characteristic

#3160

C₂²

yes

characteristic

C₂×#365

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

characteristic

#1846

C₂²

yes

characteristic

#1846

C₂²

yes

characteristic

#1518

—

#1450

C₂²

yes

—

—

—

—

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

#500

Q₈

yes

—

—

—

—

D₄

yes

—

—

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

#328

—

#322

—

C₈²

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

SD₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

AGL(1,ℤ/8)

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊Q₈

yes

characteristic

C₂×C₈

C₂²⋊Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×SD₁₆

yes

—

100

C₂×C₈

#135

yes

—

101

C₂×C₈

#137

yes

—

102

C₂×C₈

#137

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

C₂²⋊Q₈

yes

characteristic

113

C₄²

#123

yes

characteristic

114

C₄²

#135

yes

—

115

C₁₆

—

116

C₁₆

—

117

C₂³

#496

yes

characteristic

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#399

yes

characteristic

126

C₂×C₄

#478

yes

—

127

C₂×C₄

#482

yes

—

128

C₂×C₄

#492

yes

characteristic

129

C₂×C₄

#495

yes

characteristic

130

C₂×C₄

#495

yes

characteristic

131

C₂×C₄

#497

yes

characteristic

132

C₂×C₄

#497

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

#475

yes

—

136

C₈

#479

yes

—

137

C₈

#483

yes

—

138

C₈

#485

yes

—

139

C₂²

—

140

C₂²

#1714

yes

characteristic

141

C₂²

#1720

yes

characteristic

142

C₂²

#2092

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

#1708

yes

characteristic

149

C₄

#1718

yes

characteristic

150

C₄

#2096

yes

characteristic

151

C₄

#2097

yes

characteristic

152

C₂

—

153

C₂

—

154

C₂

#10681

yes

characteristic

155

C₂

#13525

yes

—

	subgroup	G/H
center, socle	C₂²	#2092
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#496
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2092

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#496

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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
#13288	2
#13285	2
#12904	2
#12889	2
#9755	2
#8793	2
#7141	2

subgroup

index

Subgroups — #146267

Conjugacy classes of subgroups

Maximal subgroups