Group #141003 — subgroups

Subgroups — #141003

Order 512.

334 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 126 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	#33311	C₂	1	1	yes	maximal, characteristic
2	#12900	C₂	1	1	yes	maximal, characteristic
3	C₄×#364	C₂	1	1	yes	maximal, characteristic
4	C₂×#1366	C₂	1	4	yes	maximal
5	C₂×#499	C₄	1	2	yes	—
6	C₂×#499	C₂²	1	2	yes	—
7	#2859	C₄	1	2	yes	—
8	C₂×#364	—	2	1	no	—
9	C₂×#364	C₂²	1	2	yes	—
10	C₂×#364	C₂²	1	2	yes	—
11	#2150	—	2	1	no	—
12	C₄×M₃₂	—	2	1	no	—
13	#1846	—	2	1	no	—
14	#1846	—	2	1	no	—
15	#1845	—	2	1	no	—
16	#1844	—	2	1	no	—
17	C₄²×C₈	C₂²	1	1	yes	characteristic
18	#1376	—	2	1	no	—
19	#1366	—	2	8	no	—
20	C₂×#125	C₂×C₄	1	4	yes	—
21	C₂×M₃₂	—	2	2	no	—
22	C₂×M₃₂	—	2	2	no	—
23	#499	—	2	4	no	—
24	#499	—	2	4	no	—
25	#499	—	4	4	no	—
26	C₂×#107	—	2	2	no	—
27	C₂×#107	—	2	2	no	—
28	C₂×#106	—	2	2	no	—
29	C₂×#106	—	2	2	no	—
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	D₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	D₄	1	1	yes	characteristic
34	C₂×C₄×C₈	D₄	1	1	yes	characteristic
35	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
36	#389	—	4	4	no	—
37	#384	—	2	2	no	—
38	#382	—	2	2	no	—
39	C₄³	C₂×C₄	1	1	yes	characteristic
40	#364	—	2	4	no	—
41	#364	—	2	2	no	—
42	#364	—	4	2	no	—
43	#364	D₄	1	4	yes	—
44	C₄⋊C₁₆	—	4	2	no	—
45	#346	—	4	1	no	—
46	C₄×C₁₆	—	4	1	no	—
47	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₈	C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
51	#125	—	2	8	no	—
52	#125	—	4	4	no	—
53	C₂×C₄⋊C₄	—	2	4	no	—
54	C₂×C₄⋊C₄	—	4	4	no	—
55	C₂×C₄⋊C₄	—	4	4	no	—
56	C₂×C₄²	—	2	2	no	—
57	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄²	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
60	M₃₂	—	2	4	no	—
61	M₃₂	—	4	2	no	—
62	M₃₂	—	4	2	no	—
63	M₃₂	—	4	2	no	—
64	C₂×C₁₆	—	4	2	no	—
65	C₂×C₁₆	—	4	2	no	—
66	#107	—	4	4	no	—
67	#107	—	4	2	no	—
68	#107	—	4	2	no	—
69	#106	—	4	4	no	—
70	#106	—	4	4	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	SD₁₆	1	1	yes	characteristic
74	C₄×C₈	SD₁₆	1	1	yes	characteristic
75	C₄×C₈	SD₁₆	1	2	yes	—
76	C₄×C₈	SD₁₆	1	2	yes	—
77	C₄×C₈	Q₁₆	1	2	yes	—
78	C₄×C₈	Q₁₆	1	2	yes	—
79	C₄×C₈	Q₁₆	1	1	yes	characteristic
80	C₄×C₈	Q₁₆	1	1	yes	characteristic
81	C₄×C₈	C₂×D₄	1	2	yes	—
82	C₄×C₈	#41	1	2	yes	—
83	#95	—	4	4	no	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	—	2	1	no	—
86	C₂²×C₄	—	4	4	no	—
87	C₂²×C₄	#100	1	2	yes	—
88	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	#100	1	2	yes	—
92	C₂×C₈	#103	1	2	yes	—
93	C₂×C₈	#103	1	2	yes	—
94	C₂×C₈	#103	1	1	yes	characteristic
95	C₂×C₈	#103	1	1	yes	characteristic
96	C₂×C₈	#121	1	2	yes	—
97	C₂×C₈	#124	1	1	yes	characteristic
98	C₂×C₈	#127	1	1	yes	characteristic
99	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
100	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
105	C₂×C₈	#137	1	2	yes	—
106	C₄⋊C₄	—	4	4	no	—
107	C₄⋊C₄	—	4	4	no	—
108	C₄⋊C₄	—	8	8	no	—
109	C₄⋊C₄	—	8	8	no	—
110	C₄²	—	2	1	no	—
111	C₄²	—	2	1	no	—
112	C₄²	—	4	4	no	—
113	C₄²	#103	1	2	yes	—
114	C₄²	#103	1	1	yes	characteristic
115	C₄²	#103	1	2	yes	—
116	C₄²	#103	1	1	yes	characteristic
117	C₄²	C₄×D₄	1	2	yes	—
118	C₁₆	—	4	4	no	—
119	C₁₆	—	8	2	no	—
120	C₂³	C₂×#100	1	1	yes	characteristic
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	—	8	8	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	#333	1	2	yes	—
129	C₂×C₄	#333	1	2	yes	—
130	C₂×C₄	#390	1	1	yes	characteristic
131	C₂×C₄	C₂×#100	1	1	yes	characteristic
132	C₂×C₄	C₂×#103	1	1	yes	characteristic
133	C₂×C₄	C₂×#103	1	1	yes	characteristic
134	C₂×C₄	C₂×#103	1	1	yes	characteristic
135	C₂×C₄	#419	1	1	yes	characteristic
136	C₈	—	2	2	no	—
137	C₈	#333	1	2	yes	—
138	C₈	#333	1	2	yes	—
139	C₈	#461	1	2	yes	—
140	C₈	#462	1	2	yes	—
141	C₈	#487	1	1	yes	characteristic
142	C₈	#487	1	1	yes	characteristic
143	C₈	#492	1	1	yes	characteristic
144	C₈	#494	1	1	yes	characteristic
145	C₂²	—	2	2	no	—
146	C₂²	C₂×#333	1	1	yes	characteristic
147	C₂²	#1548	1	1	yes	characteristic
148	C₂²	#1965	1	1	yes	characteristic
149	C₄	—	2	2	no	—
150	C₄	—	4	2	no	—
151	C₄	—	4	2	no	—
152	C₄	—	16	4	no	—
153	C₄	#1366	1	4	yes	complemented
154	C₄	C₂×#333	1	1	yes	characteristic
155	C₄	#1548	1	1	yes	characteristic
156	C₄	#1956	1	1	yes	characteristic
157	C₄	#1981	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#9691	1	1	yes	characteristic
161	C₂	C₂×#1366	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33311

C₂

yes

maximal, characteristic

#12900

C₂

yes

maximal, characteristic

C₄×#364

C₂

yes

maximal, characteristic

C₂×#1366

C₂

yes

maximal

C₂×#499

C₄

yes

—

C₂×#499

C₂²

yes

—

#2859

C₄

yes

—

C₂×#364

—

C₂×#364

C₂²

yes

—

C₂×#364

C₂²

yes

—

#2150

—

C₄×M₃₂

—

—

—

—

—

C₄²×C₈

C₂²

yes

—

—

C₂×C₄

yes

—

C₂×M₃₂

—

C₂×M₃₂

—

—

—

—

—

—

—

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

C₄³

C₂×C₄

yes

—

—

—

D₄

yes

—

C₄⋊C₁₆

—

#346

—

C₄×C₁₆

—

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²⋊C₄

yes

characteristic

C₂×C₄²

C₂²×C₄

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

characteristic

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

Q₁₆

yes

characteristic

C₄×C₈

Q₁₆

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

#41

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#100

yes

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#100

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

#103

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

C₂×SD₁₆

yes

characteristic

100

C₂×C₈

C₂×SD₁₆

yes

characteristic

101

C₂×C₈

C₂×SD₁₆

yes

characteristic

102

C₂×C₈

C₂×Q₁₆

yes

characteristic

103

C₂×C₈

C₂×Q₁₆

yes

characteristic

104

C₂×C₈

C₂×Q₁₆

yes

characteristic

105

C₂×C₈

#137

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

#103

yes

—

114

C₄²

#103

yes

characteristic

115

C₄²

#103

yes

—

116

C₄²

#103

yes

characteristic

117

C₄²

C₄×D₄

yes

—

118

C₁₆

—

119

C₁₆

—

120

C₂³

C₂×#100

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

#333

yes

—

129

C₂×C₄

#333

yes

—

130

C₂×C₄

#390

yes

characteristic

131

C₂×C₄

C₂×#100

yes

characteristic

132

C₂×C₄

C₂×#103

yes

characteristic

133

C₂×C₄

C₂×#103

yes

characteristic

134

C₂×C₄

C₂×#103

yes

characteristic

135

C₂×C₄

#419

yes

characteristic

136

C₈

—

137

C₈

#333

yes

—

138

C₈

#333

yes

—

139

C₈

#461

yes

—

140

C₈

#462

yes

—

141

C₈

#487

yes

characteristic

142

C₈

#487

yes

characteristic

143

C₈

#492

yes

characteristic

144

C₈

#494

yes

characteristic

145

C₂²

—

146

C₂²

C₂×#333

yes

characteristic

147

C₂²

#1548

yes

characteristic

148

C₂²

#1965

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#1366

yes

complemented

154

C₄

C₂×#333

yes

characteristic

155

C₄

#1548

yes

characteristic

156

C₄

#1956

yes

characteristic

157

C₄

#1981

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

#9691

yes

characteristic

161

C₂

C₂×#1366

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#333
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#333

commutator

C₂×C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

prime radical

C₂³

C₂×#100

LCS

G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
153. C₄	#1366	semidirect

H ≅ G/N

type

153. C₄

#1366

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#33311	2
#12900	2
C₄×#364	2
C₂×#1366	2

subgroup

index

Subgroups — #141003

Conjugacy classes of subgroups

Complements

Maximal subgroups