Group #141225 — subgroups

Subgroups — #141225

Order 512.

302 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 114 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	#12662	C₂	1	1	yes	maximal, characteristic
3	C₂×#1372	C₂	1	1	yes	maximal, characteristic
4	C₂×#1365	C₂	1	2	yes	maximal
5	#9730	C₂	1	2	yes	maximal
6	C₂×#500	C₂²	1	2	yes	—
7	#2252	—	2	2	no	—
8	#2252	C₄	1	4	yes	complemented
9	C₂×#366	—	2	1	no	—
10	C₂×#366	—	2	1	no	—
11	C₂×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₂×#346	—	2	1	no	—
13	C₂×C₄×C₁₆	C₂²	1	1	yes	characteristic
14	#1987	—	2	2	no	—
15	C₈×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×#335	C₂²	1	2	yes	—
17	C₂×#333	—	2	2	no	—
18	#1374	—	2	1	no	—
19	#1372	—	2	2	no	—
20	#1365	—	2	4	no	—
21	C₂×C₄⋊Q₈	—	2	2	no	—
22	C₂²×C₁₆	—	2	1	no	—
23	C₂²×C₁₆	D₄	1	1	yes	characteristic
24	C₂²×C₁₆	D₄	1	1	yes	characteristic
25	#500	—	2	2	no	—
26	#500	—	2	2	no	—
27	#500	—	2	2	no	—
28	#500	C₂×C₄	1	4	yes	—
29	C₂×#107	—	2	2	no	—
30	C₂×#107	—	2	2	no	—
31	C₂×C₄⋊C₈	—	2	2	no	—
32	C₂×C₄⋊C₈	D₄	1	1	yes	characteristic
33	C₂×C₄×C₈	—	2	1	no	—
34	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
35	C₄×C₄⋊C₄	D₄	1	1	yes	characteristic
36	#366	—	2	4	no	—
37	#366	—	2	4	no	—
38	C₄⋊C₁₆	—	2	1	no	—
39	C₄⋊C₁₆	D₄	1	2	yes	—
40	#346	—	4	2	no	—
41	C₄×C₁₆	C₂×C₄	1	2	yes	—
42	C₄×C₁₆	D₄	1	2	yes	—
43	#340	—	4	4	no	—
44	#336	—	2	1	no	—
45	#335	—	2	2	no	—
46	#335	—	2	2	no	—
47	#333	—	4	4	no	—
48	C₂²×Q₈	—	4	2	no	—
49	C₂²×C₈	—	2	1	no	—
50	C₂²×C₈	—	4	2	no	—
51	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₈	#41	1	1	yes	characteristic
53	C₄⋊Q₈	—	2	4	no	—
54	C₄⋊Q₈	—	4	2	no	—
55	C₄⋊Q₈	—	4	2	no	—
56	C₄⋊Q₈	—	4	2	no	—
57	C₂×C₄⋊C₄	—	4	2	no	—
58	C₂×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄⋊C₄	D₈	1	2	yes	—
60	C₂×C₄²	—	2	1	no	—
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₁₆	—	2	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	C₂×C₁₆	—	2	2	no	—
65	C₂×C₁₆	—	4	1	no	—
66	C₂×C₁₆	—	4	1	no	—
67	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
68	C₂×C₁₆	C₂²⋊C₄	1	2	yes	—
69	C₂×C₁₆	SD₁₆	1	2	yes	—
70	C₂×C₁₆	Q₁₆	1	2	yes	—
71	#107	—	2	4	no	—
72	#107	—	2	4	no	—
73	#107	—	4	2	no	—
74	#107	—	4	2	no	—
75	C₄⋊C₈	—	2	1	no	—
76	C₄⋊C₈	—	4	2	no	—
77	C₄⋊C₈	—	4	2	no	—
78	C₄⋊C₈	D₈	1	2	yes	—
79	C₄×C₈	—	4	1	no	—
80	C₄×C₈	—	4	1	no	—
81	C₄×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₄×C₈	C₂×D₄	1	1	yes	characteristic
83	C₄×C₈	C₂×D₄	1	1	yes	characteristic
84	C₄×C₈	#41	1	1	yes	characteristic
85	#95	—	2	1	no	—
86	C₂×Q₈	—	4	4	no	—
87	C₂×Q₈	—	8	2	no	—
88	C₂×Q₈	—	16	4	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	—	8	2	no	—
91	C₂²×C₄	#121	1	1	yes	characteristic
92	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
93	C₂²×C₄	#135	1	1	yes	characteristic
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	4	1	no	—
98	C₂×C₈	—	4	1	no	—
99	C₂×C₈	—	8	2	no	—
100	C₂×C₈	—	8	4	no	—
101	C₂×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₈	C₄×D₄	1	1	yes	characteristic
103	C₂×C₈	#121	1	1	yes	characteristic
104	C₂×C₈	#124	1	1	yes	characteristic
105	C₂×C₈	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₈	#137	1	1	yes	characteristic
108	C₂×C₈	#137	1	1	yes	characteristic
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	4	4	no	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	8	2	no	—
113	C₄⋊C₄	—	8	2	no	—
114	C₄²	—	4	1	no	—
115	C₄²	—	4	1	no	—
116	C₄²	C₄×D₄	1	1	yes	characteristic
117	C₄²	#127	1	1	yes	characteristic
118	C₄²	C₂×D₈	1	1	yes	characteristic
119	C₄²	#135	1	1	yes	characteristic
120	C₁₆	—	2	2	no	—
121	C₁₆	—	4	2	no	—
122	C₁₆	#103	1	4	yes	—
123	C₂³	#466	1	1	yes	characteristic
124	Q₈	—	16	4	no	—
125	Q₈	—	16	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	8	4	no	—
131	C₂×C₄	—	16	4	no	—
132	C₂×C₄	#390	1	1	yes	characteristic
133	C₂×C₄	C₄×D₈	1	1	yes	characteristic
134	C₂×C₄	C₄×D₈	1	1	yes	characteristic
135	C₂×C₄	#461	1	1	yes	characteristic
136	C₂×C₄	#462	1	1	yes	characteristic
137	C₂×C₄	#466	1	1	yes	characteristic
138	C₂×C₄	#473	1	1	yes	characteristic
139	C₂×C₄	#473	1	1	yes	characteristic
140	C₂×C₄	#493	1	1	yes	characteristic
141	C₂×C₄	#495	1	1	yes	characteristic
142	C₂×C₄	#510	1	4	yes	—
143	C₈	—	2	1	no	—
144	C₈	—	2	1	no	—
145	C₈	—	4	2	no	—
146	C₈	—	16	2	no	—
147	C₈	C₂×#103	1	1	yes	characteristic
148	C₈	#419	1	1	yes	characteristic
149	C₈	#461	1	1	yes	characteristic
150	C₈	#462	1	1	yes	characteristic
151	C₂²	#1673	1	1	yes	characteristic
152	C₂²	#1674	1	1	yes	characteristic
153	C₂²	#1970	1	1	yes	characteristic
154	C₂²	#2219	1	2	yes	—
155	C₂²	#2221	1	2	yes	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	8	2	no	—
159	C₄	—	16	4	no	—
160	C₄	#1673	1	1	yes	characteristic
161	C₄	#1674	1	1	yes	characteristic
162	C₄	#1979	1	1	yes	characteristic
163	C₄	#1981	1	1	yes	characteristic
164	C₄	#2176	1	2	yes	—
165	C₄	#2251	1	2	yes	—
166	C₂	#9731	1	1	yes	characteristic
167	C₂	#12455	1	2	yes	—
168	C₂	#12461	1	2	yes	—
169	C₂	#13240	1	1	yes	characteristic
170	C₂	#13245	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2252

C₂

yes

maximal, characteristic

#12662

C₂

yes

maximal, characteristic

C₂×#1372

C₂

yes

maximal, characteristic

C₂×#1365

C₂

yes

maximal

#9730

C₂

yes

maximal

C₂×#500

C₂²

yes

—

#2252

—

#2252

C₄

yes

complemented

C₂×#366

—

C₂×#366

—

C₂×C₄⋊C₁₆

C₂²

yes

characteristic

C₂×#346

—

C₂×C₄×C₁₆

C₂²

yes

characteristic

#1987

—

C₈×C₄⋊C₄

C₂²

yes

characteristic

C₂×#335

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

—

C₂²×C₁₆

D₄

yes

characteristic

C₂²×C₁₆

D₄

yes

—

—

—

C₂×C₄

yes

—

C₂×#107

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₄

yes

characteristic

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂³

yes

characteristic

C₄×C₄⋊C₄

D₄

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₄

yes

—

#346

—

C₄×C₁₆

C₂×C₄

yes

—

C₄×C₁₆

D₄

yes

—

—

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

#41

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₈

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

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

—

C₄⋊C₈

D₈

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#121

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

#135

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂×C₂²⋊C₄

yes

characteristic

102

C₂×C₈

C₄×D₄

yes

characteristic

103

C₂×C₈

#121

yes

characteristic

104

C₂×C₈

#124

yes

characteristic

105

C₂×C₈

C₂×SD₁₆

yes

characteristic

106

C₂×C₈

C₂×Q₁₆

yes

characteristic

107

C₂×C₈

#137

yes

characteristic

108

C₂×C₈

#137

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄²

—

115

C₄²

—

116

C₄²

C₄×D₄

yes

characteristic

117

C₄²

#127

yes

characteristic

118

C₄²

C₂×D₈

yes

characteristic

119

C₄²

#135

yes

characteristic

120

C₁₆

—

121

C₁₆

—

122

C₁₆

#103

yes

—

123

C₂³

#466

yes

characteristic

124

Q₈

—

125

Q₈

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#390

yes

characteristic

133

C₂×C₄

C₄×D₈

yes

characteristic

134

C₂×C₄

C₄×D₈

yes

characteristic

135

C₂×C₄

#461

yes

characteristic

136

C₂×C₄

#462

yes

characteristic

137

C₂×C₄

#466

yes

characteristic

138

C₂×C₄

#473

yes

characteristic

139

C₂×C₄

#473

yes

characteristic

140

C₂×C₄

#493

yes

characteristic

141

C₂×C₄

#495

yes

characteristic

142

C₂×C₄

#510

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

C₂×#103

yes

characteristic

148

C₈

#419

yes

characteristic

149

C₈

#461

yes

characteristic

150

C₈

#462

yes

characteristic

151

C₂²

#1673

yes

characteristic

152

C₂²

#1674

yes

characteristic

153

C₂²

#1970

yes

characteristic

154

C₂²

#2219

yes

—

155

C₂²

#2221

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

#1673

yes

characteristic

161

C₄

#1674

yes

characteristic

162

C₄

#1979

yes

characteristic

163

C₄

#1981

yes

characteristic

164

C₄

#2176

yes

—

165

C₄

#2251

yes

—

166

C₂

#9731

yes

characteristic

167

C₂

#12455

yes

—

168

C₂

#12461

yes

—

169

C₂

#13240

yes

characteristic

170

C₂

#13245

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#466
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#466

commutator

C₄×C₈

C₂²×C₄

Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G

LECS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
8. #2252	C₄	semidirect

H ≅ G/N

type

8. #2252

C₄

semidirect

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

subgroup	index
C₂×#2252	2
#12662	2
C₂×#1372	2
C₂×#1365	2
#9730	2

subgroup

index

Subgroups — #141225

Conjugacy classes of subgroups

Complements

Maximal subgroups