Group #141663 — subgroups

Subgroups — #141663

Order 512.

304 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 122 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	#13290	C₂	1	2	yes	maximal
3	#12909	C₂	1	1	yes	maximal, characteristic
4	#11399	C₂	1	1	yes	maximal, characteristic
5	#9771	C₂	1	2	yes	maximal
6	C₂×#500	C₂²	1	2	yes	—
7	#2252	—	2	1	no	—
8	#2252	—	2	1	no	—
9	#2252	C₄	1	4	yes	—
10	C₂×#368	—	2	2	no	—
11	C₂×#366	—	2	2	no	—
12	C₄×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1848	C₂²	1	2	yes	—
15	#1549	—	2	2	no	—
16	#1548	—	2	2	no	—
17	#1450	C₂²	1	1	yes	characteristic
18	#1387	—	2	4	no	—
19	#1311	—	2	1	no	—
20	C₈×C₁₆	—	2	1	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂×M₃₂	D₄	1	2	yes	—
23	C₂²×C₁₆	D₄	1	2	yes	—
24	#500	—	2	2	no	—
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	C₂×C₄	1	4	yes	—
28	#431	D₄	1	1	yes	characteristic
29	#431	D₄	1	1	yes	characteristic
30	C₂×#108	—	2	2	no	—
31	C₂×#108	—	2	2	no	—
32	C₂×#107	—	2	2	no	—
33	C₂×#107	—	2	2	no	—
34	#423	—	2	2	no	—
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	#368	—	4	4	no	—
37	#366	—	2	4	no	—
38	#366	—	2	4	no	—
39	#346	—	2	1	no	—
40	C₄×C₁₆	—	2	1	no	—
41	C₄×C₁₆	C₂×C₄	1	2	yes	—
42	C₄×C₁₆	D₄	1	1	yes	characteristic
43	C₄×C₁₆	D₄	1	1	yes	characteristic
44	#333	—	4	4	no	—
45	#332	—	4	4	no	—
46	#322	—	2	1	no	—
47	C₈²	—	2	1	no	—
48	C₂²×Q₈	—	4	2	no	—
49	C₂×M₁₆	—	4	2	no	—
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₈	C₂×D₄	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	C₂×C₄⋊C₄	—	4	2	no	—
57	C₂×C₄⋊C₄	—	4	2	no	—
58	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
59	M₃₂	—	2	2	no	—
60	M₃₂	Q₁₆	1	4	yes	—
61	C₂×C₁₆	—	2	2	no	—
62	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
63	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
64	C₂×C₁₆	SD₁₆	1	4	yes	—
65	C₂×C₁₆	Q₁₆	1	4	yes	—
66	#108	—	4	4	no	—
67	#108	—	4	4	no	—
68	#107	—	2	4	no	—
69	#107	—	2	4	no	—
70	#107	—	4	2	no	—
71	#107	—	4	2	no	—
72	C₄⋊C₈	—	4	2	no	—
73	C₄⋊C₈	—	4	2	no	—
74	C₄⋊C₈	D₈	1	2	yes	—
75	C₄⋊C₈	D₈	1	2	yes	—
76	C₂²⋊C₈	—	2	1	no	—
77	C₂²⋊C₈	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₄×C₈	—	2	1	no	—
80	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
81	C₄×C₈	C₂×D₄	1	1	yes	characteristic
82	C₄×C₈	#41	1	1	yes	characteristic
83	C₄×C₈	#41	1	1	yes	characteristic
84	C₂×Q₈	—	4	4	no	—
85	C₂×Q₈	—	8	2	no	—
86	C₂×Q₈	—	16	4	no	—
87	C₂²×C₄	—	8	2	no	—
88	C₂²×C₄	#127	1	1	yes	characteristic
89	C₂²×C₄	#135	1	1	yes	characteristic
90	C₂²×C₄	#135	1	1	yes	characteristic
91	M₁₆	—	8	2	no	—
92	M₁₆	—	8	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	8	2	no	—
96	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈	C₄×D₄	1	1	yes	characteristic
98	C₂×C₈	#121	1	1	yes	characteristic
99	C₂×C₈	#124	1	1	yes	characteristic
100	C₂×C₈	C₂×SD₁₆	1	2	yes	—
101	C₂×C₈	C₂×Q₁₆	1	2	yes	—
102	C₂×C₈	C₂×Q₁₆	1	2	yes	—
103	C₂×C₈	#135	1	2	yes	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	8	2	no	—
107	C₄⋊C₄	—	8	2	no	—
108	C₄²	C₄×D₄	1	1	yes	characteristic
109	C₄²	#121	1	1	yes	characteristic
110	C₄²	C₂×D₈	1	1	yes	characteristic
111	C₄²	C₂×D₈	1	1	yes	characteristic
112	C₁₆	—	2	4	no	—
113	C₁₆	#103	1	4	yes	—
114	C₁₆	#103	1	4	yes	—
115	C₂³	#495	1	1	yes	characteristic
116	Q₈	—	16	4	no	—
117	Q₈	—	16	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	8	4	no	—
121	C₂×C₄	—	16	4	no	—
122	C₂×C₄	#390	1	1	yes	characteristic
123	C₂×C₄	C₄×D₈	1	1	yes	characteristic
124	C₂×C₄	C₄×D₈	1	1	yes	characteristic
125	C₂×C₄	#466	1	1	yes	characteristic
126	C₂×C₄	#466	1	1	yes	characteristic
127	C₂×C₄	#492	1	1	yes	characteristic
128	C₂×C₄	#493	1	1	yes	characteristic
129	C₂×C₄	#494	1	1	yes	characteristic
130	C₂×C₄	#494	1	1	yes	characteristic
131	C₂×C₄	#495	1	1	yes	characteristic
132	C₈	—	4	2	no	—
133	C₈	—	4	2	no	—
134	C₈	—	16	2	no	—
135	C₈	C₂×#103	1	2	yes	—
136	C₈	C₄×Q₁₆	1	2	yes	—
137	C₈	#467	1	2	yes	—
138	C₈	#487	1	2	yes	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#1707	1	1	yes	characteristic
142	C₂²	#1710	1	1	yes	characteristic
143	C₂²	#1970	1	1	yes	characteristic
144	C₄	—	2	1	no	—
145	C₄	—	2	1	no	—
146	C₄	—	16	4	no	—
147	C₄	#1706	1	1	yes	characteristic
148	C₄	#1713	1	1	yes	characteristic
149	C₄	#1956	1	1	yes	characteristic
150	C₄	#1969	1	1	yes	characteristic
151	C₄	#2221	1	2	yes	—
152	C₄	#2222	1	2	yes	—
153	C₂	—	2	1	no	—
154	C₂	—	2	1	no	—
155	C₂	#9823	1	1	yes	characteristic
156	C₂	#13244	1	1	yes	characteristic
157	C₂	#13245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

#13290

C₂

yes

maximal

#12909

C₂

yes

maximal, characteristic

#11399

C₂

yes

maximal, characteristic

#9771

C₂

yes

maximal

C₂×#500

C₂²

yes

—

#2252

—

#2252

—

#2252

C₄

yes

—

C₂×#368

—

C₂×#366

—

C₄×M₃₂

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1848

C₂²

yes

—

#1549

—

#1548

—

#1450

C₂²

yes

characteristic

#1387

—

#1311

—

C₈×C₁₆

—

C₂×C₄⋊Q₈

—

C₂×M₃₂

D₄

yes

—

C₂²×C₁₆

D₄

yes

—

—

—

—

C₂×C₄

yes

—

#431

D₄

yes

characteristic

#431

D₄

yes

—

—

—

—

—

C₂×C₄×C₈

C₂³

yes

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

characteristic

C₄×C₁₆

D₄

yes

—

—

—

C₈²

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

M₃₂

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

C₂²⋊C₄

yes

—

C₂×C₁₆

SD₁₆

yes

—

C₂×C₁₆

Q₁₆

yes

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

D₈

yes

—

C₄⋊C₈

D₈

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

C₄×C₈

yes

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

C₄×D₄

yes

C₂×C₈

yes

C₂×C₈

yes

100

C₂×C₈

C₂×SD₁₆

yes

—

101

C₂×C₈

C₂×Q₁₆

yes

—

102

C₂×C₈

C₂×Q₁₆

yes

—

103

C₂×C₈

#135

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₄×D₄

yes

characteristic

109

C₄²

#121

yes

characteristic

110

C₄²

C₂×D₈

yes

characteristic

111

C₄²

C₂×D₈

yes

characteristic

112

C₁₆

—

113

C₁₆

#103

yes

—

114

C₁₆

#103

yes

—

115

C₂³

#495

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

#390

yes

characteristic

123

C₂×C₄

C₄×D₈

yes

characteristic

124

C₂×C₄

C₄×D₈

yes

characteristic

125

C₂×C₄

#466

yes

characteristic

126

C₂×C₄

#466

yes

characteristic

127

C₂×C₄

#492

yes

characteristic

128

C₂×C₄

#493

yes

characteristic

129

C₂×C₄

#494

yes

characteristic

130

C₂×C₄

#494

yes

characteristic

131

C₂×C₄

#495

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

C₂×#103

yes

—

136

C₈

C₄×Q₁₆

yes

—

137

C₈

#467

yes

—

138

C₈

#487

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

#1707

yes

characteristic

142

C₂²

#1710

yes

characteristic

143

C₂²

#1970

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#1706

yes

characteristic

148

C₄

#1713

yes

characteristic

149

C₄

#1956

yes

characteristic

150

C₄

#1969

yes

characteristic

151

C₄

#2221

yes

—

152

C₄

#2222

yes

—

153

C₂

—

154

C₂

—

155

C₂

#9823

yes

characteristic

156

C₂

#13244

yes

characteristic

157

C₂

#13245

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1970
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#495
LCS	G ▷ 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

#1970

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#495

LCS

G ▷ 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
C₂×#2252	2
#13290	2
#12909	2
#11399	2
#9771	2

subgroup

index

Subgroups — #141663

Conjugacy classes of subgroups

Maximal subgroups