Group #141392 — subgroups

Subgroups — #141392

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	#11972	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	#1365	—	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	8	no	—
51	C₂×M₁₆	—	4	2	no	—
52	C₂²×C₈	D₈	1	2	yes	—
53	C₂²×C₈	SD₁₆	1	2	yes	—
54	C₂²×C₈	#41	1	2	yes	—
55	C₄⋊Q₈	—	2	4	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	C₄⋊Q₈	—	4	2	no	—
58	C₄⋊Q₈	—	4	2	no	—
59	C₂×C₄⋊C₄	—	4	2	no	—
60	C₂×C₄⋊C₄	—	4	2	no	—
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₆	—	2	4	no	—
63	#108	—	8	4	no	—
64	#107	—	2	8	no	—
65	#107	—	4	4	no	—
66	#107	—	4	4	no	—
67	#107	—	4	4	no	—
68	C₄⋊C₈	—	4	2	no	—
69	C₄⋊C₈	—	4	2	no	—
70	C₄×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	D₈	1	2	yes	—
75	C₄×C₈	SD₁₆	1	2	yes	—
76	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₄×C₈	C₂×D₄	1	1	yes	characteristic
78	C₄×C₈	C₂×D₄	1	2	yes	—
79	C₂×Q₈	—	4	4	no	—
80	C₂×Q₈	—	8	8	no	—
81	C₂×Q₈	—	8	2	no	—
82	C₂²×C₄	—	8	2	no	—
83	C₂²×C₄	#124	1	1	yes	characteristic
84	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
85	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
86	Q₁₆	—	8	8	no	—
87	Q₁₆	—	8	8	no	—
88	M₁₆	—	8	4	no	—
89	C₂×C₈	—	2	4	no	—
90	C₂×C₈	—	2	4	no	—
91	C₂×C₈	—	8	2	no	—
92	C₂×C₈	#102	1	2	yes	—
93	C₂×C₈	#102	1	2	yes	—
94	C₂×C₈	#102	1	2	yes	—
95	C₂×C₈	#102	1	2	yes	—
96	C₂×C₈	SD₃₂	1	4	yes	—
97	C₂×C₈	Q₃₂	1	4	yes	—
98	C₂×C₈	C₄×D₄	1	2	yes	—
99	C₂×C₈	#121	1	2	yes	—
100	C₄⋊C₄	—	4	4	no	—
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	8	2	no	—
103	C₄⋊C₄	—	8	2	no	—
104	C₄²	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₄²	#127	1	1	yes	characteristic
106	C₄²	C₂×D₈	1	1	yes	characteristic
107	C₄²	C₂×SD₁₆	1	1	yes	characteristic
108	C₁₆	—	4	4	no	—
109	C₂³	#486	1	1	yes	characteristic
110	Q₈	—	8	8	no	—
111	Q₈	—	16	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	8	4	no	—
115	C₂×C₄	—	16	4	no	—
116	C₂×C₄	#390	1	1	yes	characteristic
117	C₂×C₄	C₂×#102	1	1	yes	characteristic
118	C₂×C₄	C₂×#102	1	1	yes	characteristic
119	C₂×C₄	#486	1	1	yes	characteristic
120	C₂×C₄	#492	1	1	yes	characteristic
121	C₂×C₄	#493	1	1	yes	characteristic
122	C₂×C₄	C₂×SD₃₂	1	2	yes	—
123	C₂×C₄	C₂×Q₃₂	1	2	yes	—
124	C₈	—	2	2	no	—
125	C₈	—	2	2	no	—
126	C₈	—	16	2	no	—
127	C₈	#358	1	4	yes	—
128	C₈	#358	1	4	yes	—
129	C₈	#361	1	4	yes	—
130	C₈	#473	1	4	yes	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#1957	1	1	yes	characteristic
134	C₂²	#2241	1	2	yes	—
135	C₄	—	2	1	no	—
136	C₄	—	2	1	no	—
137	C₄	—	16	4	no	—
138	C₄	#1979	1	2	yes	—
139	C₄	C₂×#358	1	2	yes	—
140	C₄	C₂×#361	1	2	yes	—
141	C₄	#2247	1	1	yes	characteristic
142	C₄	#2248	1	1	yes	characteristic
143	C₂	—	2	2	no	—
144	C₂	#9772	1	1	yes	characteristic
145	C₂	#13203	1	1	yes	characteristic
146	C₂	#13213	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1713

C₂

yes

maximal, characteristic

#11972

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

#1365

—

#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×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

—

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

SD₃₂

yes

—

C₂×C₈

Q₃₂

yes

—

C₂×C₈

C₄×D₄

yes

—

C₂×C₈

#121

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂×C₂²⋊C₄

yes

characteristic

105

C₄²

#127

yes

characteristic

106

C₄²

C₂×D₈

yes

characteristic

107

C₄²

C₂×SD₁₆

yes

characteristic

108

C₁₆

—

109

C₂³

#486

yes

characteristic

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#390

yes

characteristic

117

C₂×C₄

C₂×#102

yes

characteristic

118

C₂×C₄

C₂×#102

yes

characteristic

119

C₂×C₄

#486

yes

characteristic

120

C₂×C₄

#492

yes

characteristic

121

C₂×C₄

#493

yes

characteristic

122

C₂×C₄

C₂×SD₃₂

yes

—

123

C₂×C₄

C₂×Q₃₂

yes

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

#358

yes

—

128

C₈

#358

yes

—

129

C₈

#361

yes

—

130

C₈

#473

yes

—

131

C₂²

—

132

C₂²

—

133

C₂²

#1957

yes

characteristic

134

C₂²

#2241

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#1979

yes

—

139

C₄

C₂×#358

yes

—

140

C₄

C₂×#361

yes

—

141

C₄

#2247

yes

characteristic

142

C₄

#2248

yes

characteristic

143

C₂

—

144

C₂

#9772

yes

characteristic

145

C₂

#13203

yes

characteristic

146

C₂

#13213

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
#11972	2
#11410	2
#8823	2

subgroup

index

Subgroups — #141392

Conjugacy classes of subgroups

Maximal subgroups