Group #141391 — subgroups

Subgroups — #141391

Order 512.

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

normal subgroups only

This group has 348 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×#1713	C₂	1	1	yes	maximal, characteristic
2	#11971	C₂	1	4	yes	maximal
3	#11410	C₂	1	1	yes	maximal, characteristic
4	#8823	C₂	1	1	yes	maximal, characteristic
5	C₂×#500	—	2	1	no	—
6	C₂×#500	C₂²	1	2	yes	—
7	C₂×#494	—	2	1	no	—
8	#2162	C₂²	1	2	yes	—
9	#1848	—	2	1	no	—
10	#1713	—	2	1	no	—
11	#1713	—	2	1	no	—
12	#1713	C₄	1	2	yes	—
13	#1713	C₄	1	2	yes	—
14	#1567	—	2	1	no	—
15	#1567	C₂²	1	2	yes	—
16	C₂×C₈²	C₂²	1	1	yes	characteristic
17	#1364	—	2	8	no	—
18	#1312	—	2	1	no	—
19	C₈×C₁₆	—	2	1	no	—
20	C₂²×Q₁₆	—	4	2	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	#500	—	2	2	no	—
23	#500	—	2	2	no	—
24	#500	—	2	4	no	—
25	#500	—	2	2	no	—
26	#500	—	4	2	no	—
27	#500	C₂×C₄	1	4	yes	—
28	#494	—	2	2	no	—
29	#494	—	2	2	no	—
30	#494	—	4	2	no	—
31	C₂×#108	—	4	2	no	—
32	C₂×#107	—	2	4	no	—
33	C₂×#107	—	4	2	no	—
34	#423	—	2	2	no	—
35	C₂×C₄×C₈	D₄	1	1	yes	characteristic
36	C₂×C₄×C₈	D₄	1	1	yes	characteristic
37	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
38	C₄⋊C₁₆	D₄	1	4	yes	—
39	C₂²⋊C₁₆	—	2	2	no	—
40	C₄×C₁₆	—	2	2	no	—
41	#335	—	2	4	no	—
42	#335	—	4	2	no	—
43	C₈²	C₂×C₄	1	1	yes	characteristic
44	C₈²	C₂×C₄	1	1	yes	characteristic
45	C₈²	D₄	1	1	yes	characteristic
46	C₈²	D₄	1	1	yes	characteristic
47	C₂²×Q₈	—	4	2	no	—
48	C₂×Q₁₆	—	4	4	no	—
49	C₂×Q₁₆	—	4	4	no	—
50	C₂×Q₁₆	—	8	4	no	—
51	C₂×Q₁₆	—	8	4	no	—
52	C₂×M₁₆	—	4	2	no	—
53	C₂²×C₈	D₈	1	2	yes	—
54	C₂²×C₈	SD₁₆	1	2	yes	—
55	C₂²×C₈	#41	1	2	yes	—
56	C₄⋊Q₈	—	2	4	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₄⋊Q₈	—	4	2	no	—
59	C₄⋊Q₈	—	4	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄⋊C₄	—	4	2	no	—
62	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
63	C₂×C₁₆	—	2	4	no	—
64	#108	—	8	4	no	—
65	#107	—	2	8	no	—
66	#107	—	4	4	no	—
67	#107	—	4	4	no	—
68	#107	—	4	4	no	—
69	C₄⋊C₈	—	4	2	no	—
70	C₄⋊C₈	—	4	2	no	—
71	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
75	C₄×C₈	D₈	1	2	yes	—
76	C₄×C₈	SD₁₆	1	2	yes	—
77	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
78	C₄×C₈	C₂×D₄	1	1	yes	characteristic
79	C₄×C₈	C₂×D₄	1	2	yes	—
80	C₂×Q₈	—	4	4	no	—
81	C₂×Q₈	—	8	2	no	—
82	C₂×Q₈	—	8	4	no	—
83	C₂×Q₈	—	8	4	no	—
84	C₂²×C₄	—	8	2	no	—
85	C₂²×C₄	#124	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
87	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
88	Q₁₆	—	8	8	no	—
89	Q₁₆	—	8	8	no	—
90	M₁₆	—	8	4	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	8	2	no	—
94	C₂×C₈	#102	1	2	yes	—
95	C₂×C₈	#102	1	2	yes	—
96	C₂×C₈	#102	1	2	yes	—
97	C₂×C₈	#102	1	2	yes	—
98	C₂×C₈	D₁₆	1	4	yes	—
99	C₂×C₈	SD₃₂	1	4	yes	—
100	C₂×C₈	C₄×D₄	1	2	yes	—
101	C₂×C₈	#121	1	2	yes	—
102	C₄⋊C₄	—	4	4	no	—
103	C₄⋊C₄	—	4	4	no	—
104	C₄⋊C₄	—	8	2	no	—
105	C₄⋊C₄	—	8	2	no	—
106	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₄²	#127	1	1	yes	characteristic
108	C₄²	C₂×D₈	1	1	yes	characteristic
109	C₄²	C₂×SD₁₆	1	1	yes	characteristic
110	C₁₆	—	4	4	no	—
111	C₂³	#486	1	1	yes	characteristic
112	Q₈	—	8	8	no	—
113	Q₈	—	16	2	no	—
114	Q₈	—	16	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	8	4	no	—
118	C₂×C₄	—	16	4	no	—
119	C₂×C₄	#390	1	1	yes	characteristic
120	C₂×C₄	C₂×#102	1	1	yes	characteristic
121	C₂×C₄	C₂×#102	1	1	yes	characteristic
122	C₂×C₄	#486	1	1	yes	characteristic
123	C₂×C₄	#492	1	1	yes	characteristic
124	C₂×C₄	#493	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₆	1	2	yes	—
126	C₂×C₄	C₂×SD₃₂	1	2	yes	—
127	C₈	—	2	2	no	—
128	C₈	—	2	2	no	—
129	C₈	—	16	2	no	—
130	C₈	#357	1	4	yes	—
131	C₈	#357	1	4	yes	—
132	C₈	#362	1	4	yes	—
133	C₈	#473	1	4	yes	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	#1957	1	1	yes	characteristic
137	C₂²	#2240	1	2	yes	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	16	4	no	—
141	C₄	#1979	1	2	yes	—
142	C₄	C₂×#357	1	2	yes	—
143	C₄	C₂×#362	1	2	yes	—
144	C₄	#2246	1	1	yes	characteristic
145	C₄	#2248	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#9772	1	1	yes	characteristic
148	C₂	#13202	1	1	yes	characteristic
149	C₂	#13212	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1713

C₂

yes

maximal, characteristic

#11971

C₂

yes

maximal

#11410

C₂

yes

maximal, characteristic

#8823

C₂

yes

maximal, characteristic

C₂×#500

—

C₂×#500

C₂²

yes

—

C₂×#494

—

#2162

C₂²

yes

—

—

—

—

C₄

yes

—

#1713

C₄

yes

—

#1567

—

#1567

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1364

—

#1312

—

C₈×C₁₆

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

—

—

C₂×C₄

yes

—

—

—

—

—

—

—

—

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

—

#335

—

#335

—

C₈²

C₂×C₄

yes

characteristic

C₈²

C₂×C₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₈²

D₄

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

SD₁₆

yes

—

C₂²×C₈

#41

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

—

—

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

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

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

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

#102

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

#102

yes

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

SD₃₂

yes

—

100

C₂×C₈

C₄×D₄

yes

—

101

C₂×C₈

#121

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄²

C₂×C₂²⋊C₄

yes

characteristic

107

C₄²

#127

yes

characteristic

108

C₄²

C₂×D₈

yes

characteristic

109

C₄²

C₂×SD₁₆

yes

characteristic

110

C₁₆

—

111

C₂³

#486

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#390

yes

characteristic

120

C₂×C₄

C₂×#102

yes

characteristic

121

C₂×C₄

C₂×#102

yes

characteristic

122

C₂×C₄

#486

yes

characteristic

123

C₂×C₄

#492

yes

characteristic

124

C₂×C₄

#493

yes

characteristic

125

C₂×C₄

C₂×D₁₆

yes

—

126

C₂×C₄

C₂×SD₃₂

yes

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

#357

yes

—

131

C₈

#357

yes

—

132

C₈

#362

yes

—

133

C₈

#473

yes

—

134

C₂²

—

135

C₂²

—

136

C₂²

#1957

yes

characteristic

137

C₂²

#2240

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#1979

yes

—

142

C₄

C₂×#357

yes

—

143

C₄

C₂×#362

yes

—

144

C₄

#2246

yes

characteristic

145

C₄

#2248

yes

characteristic

146

C₂

—

147

C₂

#9772

yes

characteristic

148

C₂

#13202

yes

characteristic

149

C₂

#13212

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
C₂×#1713	2
#11971	2
#11410	2
#8823	2

subgroup

index

Subgroups — #141391

Conjugacy classes of subgroups

Maximal subgroups