Group #141398 — subgroups

Subgroups — #141398

Order 512.

305 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 102 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	#19604	C₂	1	1	yes	maximal, characteristic
2	#11968	C₂	1	4	yes	maximal
3	#11410	C₂	1	1	yes	maximal, characteristic
4	#8822	C₂	1	1	yes	maximal, characteristic
5	#3170	—	2	1	no	—
6	#3170	—	2	1	no	—
7	#3160	C₂²	1	2	yes	—
8	#2162	C₂²	1	2	yes	—
9	#1848	—	2	1	no	—
10	#1713	C₄	1	1	yes	characteristic
11	#1713	C₄	1	1	yes	characteristic
12	#1710	C₄	1	2	yes	—
13	#1707	—	2	1	no	—
14	#1707	—	2	1	no	—
15	#1566	—	2	1	no	—
16	#1566	C₂²	1	2	yes	—
17	C₂×C₈²	C₂²	1	1	yes	characteristic
18	#1363	—	2	8	no	—
19	#1312	—	2	1	no	—
20	C₈×C₁₆	—	2	1	no	—
21	#533	—	2	2	no	—
22	#500	—	2	1	no	—
23	#500	—	2	1	no	—
24	#500	C₂×C₄	1	2	yes	—
25	#500	C₂×C₄	1	2	yes	—
26	#499	—	2	4	no	—
27	#498	—	2	2	no	—
28	#494	—	2	1	no	—
29	#494	—	2	1	no	—
30	#487	—	2	2	no	—
31	#487	—	2	2	no	—
32	#467	—	4	2	no	—
33	#467	—	4	2	no	—
34	C₄×Q₁₆	—	4	2	no	—
35	C₄×Q₁₆	—	4	2	no	—
36	C₂×#108	—	4	2	no	—
37	#427	—	2	4	no	—
38	#423	—	2	2	no	—
39	C₂×C₄×C₈	D₄	1	1	yes	characteristic
40	C₂×C₄×C₈	D₄	1	1	yes	characteristic
41	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊C₁₆	D₄	1	4	yes	—
43	C₂²⋊C₁₆	—	2	2	no	—
44	C₄×C₁₆	—	2	2	no	—
45	#334	—	2	4	no	—
46	#334	—	4	2	no	—
47	C₈²	C₂×C₄	1	1	yes	characteristic
48	C₈²	C₂×C₄	1	1	yes	characteristic
49	C₈²	D₄	1	1	yes	characteristic
50	C₈²	D₄	1	1	yes	characteristic
51	C₂×Q₁₆	—	4	2	no	—
52	C₂×Q₁₆	—	4	2	no	—
53	C₂×M₁₆	—	4	2	no	—
54	C₂²×C₈	D₈	1	2	yes	—
55	C₂²×C₈	SD₁₆	1	2	yes	—
56	C₂²×C₈	#41	1	2	yes	—
57	C₄⋊Q₈	—	2	2	no	—
58	C₄⋊Q₈	—	2	2	no	—
59	#125	—	4	2	no	—
60	C₂²⋊Q₈	—	4	2	no	—
61	C₂²⋊Q₈	—	4	2	no	—
62	C₄×Q₈	—	4	2	no	—
63	C₄×Q₈	—	4	2	no	—
64	#117	—	4	2	no	—
65	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
66	C₂×C₁₆	—	2	4	no	—
67	#108	—	8	4	no	—
68	#107	—	2	4	no	—
69	#107	—	2	4	no	—
70	#107	—	4	2	no	—
71	#107	—	4	2	no	—
72	#106	—	4	4	no	—
73	C₄⋊C₈	—	4	2	no	—
74	C₄⋊C₈	—	4	2	no	—
75	#103	—	4	4	no	—
76	#103	—	4	4	no	—
77	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₄×C₈	D₈	1	2	yes	—
82	C₄×C₈	SD₁₆	1	2	yes	—
83	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₄×C₈	C₂×D₄	1	1	yes	characteristic
85	C₄×C₈	C₂×D₄	1	2	yes	—
86	C₂×Q₈	—	4	2	no	—
87	C₂×Q₈	—	4	2	no	—
88	C₂²×C₄	#124	1	1	yes	characteristic
89	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
90	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
91	Q₁₆	—	8	4	no	—
92	Q₁₆	—	8	4	no	—
93	M₁₆	—	8	4	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	2	4	no	—
97	C₂×C₈	—	8	2	no	—
98	C₂×C₈	#102	1	2	yes	—
99	C₂×C₈	#102	1	2	yes	—
100	C₂×C₈	#102	1	2	yes	—
101	C₂×C₈	#102	1	2	yes	—
102	C₂×C₈	C₄×D₄	1	2	yes	—
103	C₂×C₈	#121	1	2	yes	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	8	2	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₂²⋊C₄	—	8	2	no	—
111	C₄²	—	8	2	no	—
112	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₄²	#127	1	1	yes	characteristic
114	C₄²	C₂×D₈	1	1	yes	characteristic
115	C₄²	C₂×SD₁₆	1	1	yes	characteristic
116	C₁₆	—	4	4	no	—
117	C₂³	#486	1	1	yes	characteristic
118	Q₈	—	8	4	no	—
119	Q₈	—	8	4	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	8	2	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	#390	1	1	yes	characteristic
126	C₂×C₄	C₂×#102	1	1	yes	characteristic
127	C₂×C₄	C₂×#102	1	1	yes	characteristic
128	C₂×C₄	#486	1	1	yes	characteristic
129	C₂×C₄	#492	1	1	yes	characteristic
130	C₂×C₄	#493	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	2	2	no	—
133	C₈	—	2	2	no	—
134	C₈	—	16	2	no	—
135	C₈	#359	1	4	yes	—
136	C₈	#360	1	4	yes	—
137	C₈	#473	1	4	yes	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	—	2	1	no	—
141	C₂²	#1957	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	16	2	no	—
146	C₄	—	16	2	no	—
147	C₄	#1979	1	2	yes	—
148	C₄	C₂×#359	1	2	yes	—
149	C₄	C₂×#360	1	2	yes	—
150	C₂	—	4	1	no	—
151	C₂	#9772	1	1	yes	characteristic
152	C₂	#13208	1	1	yes	characteristic
153	C₂	#13210	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19604

C₂

yes

maximal, characteristic

#11968

C₂

yes

maximal

#11410

C₂

yes

maximal, characteristic

#8822

C₂

yes

maximal, characteristic

#3170

—

#3170

—

#3160

C₂²

yes

—

#2162

C₂²

yes

—

#1848

—

#1713

C₄

yes

characteristic

#1713

C₄

yes

characteristic

#1710

C₄

yes

—

—

—

—

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1363

—

#1312

—

C₈×C₁₆

—

—

—

—

C₂×C₄

yes

—

#500

C₂×C₄

yes

—

—

—

—

—

—

—

—

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×#108

—

#427

—

#423

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₄⋊C₁₆

D₄

yes

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

#334

—

#334

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

#41

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#103

—

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

C₂²⋊C₄

yes

characteristic

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#124

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×SD₁₆

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#102

yes

—

C₂×C₈

#102

yes

—

100

C₂×C₈

#102

yes

—

101

C₂×C₈

#102

yes

—

102

C₂×C₈

C₄×D₄

yes

—

103

C₂×C₈

#121

yes

—

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

C₂×C₂²⋊C₄

yes

characteristic

113

C₄²

#127

yes

characteristic

114

C₄²

C₂×D₈

yes

characteristic

115

C₄²

C₂×SD₁₆

yes

characteristic

116

C₁₆

—

117

C₂³

#486

yes

characteristic

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#390

yes

characteristic

126

C₂×C₄

C₂×#102

yes

characteristic

127

C₂×C₄

C₂×#102

yes

characteristic

128

C₂×C₄

#486

yes

characteristic

129

C₂×C₄

#492

yes

characteristic

130

C₂×C₄

#493

yes

characteristic

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

#359

yes

—

136

C₈

#360

yes

—

137

C₈

#473

yes

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

#1957

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#1979

yes

—

148

C₄

C₂×#359

yes

—

149

C₄

C₂×#360

yes

—

150

C₂

—

151

C₂

#9772

yes

characteristic

152

C₂

#13208

yes

characteristic

153

C₂

#13210

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#1957
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#1957

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

#486

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ 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
#19604	2
#11968	2
#11410	2
#8822	2

subgroup

index

Subgroups — #141398

Conjugacy classes of subgroups

Maximal subgroups