Group #146249 — subgroups

Subgroups — #146249

Order 512.

299 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 116 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	#13282	C₂	1	2	yes	maximal
2	#12888	C₂	1	2	yes	maximal
3	#9823	C₂	1	1	yes	maximal, characteristic
4	#8818	C₂	1	1	yes	maximal, characteristic
5	#7136	C₂	1	1	yes	maximal, characteristic
6	C₂×#500	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₁₆	C₂²	1	2	yes	—
8	#1990	—	2	1	no	—
9	#1956	—	2	1	no	—
10	#1845	—	2	1	no	—
11	#1845	—	2	1	no	—
12	#1845	C₂²	1	1	yes	characteristic
13	#1845	C₂²	1	2	yes	—
14	C₂×C₈²	C₂²	1	1	yes	characteristic
15	#1380	—	2	4	no	—
16	#1380	—	2	4	no	—
17	C₂×C₄⋊Q₈	—	2	1	no	—
18	C₂²×C₁₆	D₄	1	4	yes	—
19	#500	—	2	2	no	—
20	#500	—	2	1	no	—
21	#500	Q₈	1	2	yes	—
22	C₂×#107	—	2	2	no	—
23	C₂×#107	—	2	2	no	—
24	C₂×#107	—	2	2	no	—
25	C₂×#107	—	2	2	no	—
26	C₂×#107	—	4	2	no	—
27	C₂×#107	—	4	2	no	—
28	C₂×#103	—	4	2	no	—
29	C₂×C₄×C₈	D₄	1	1	yes	characteristic
30	C₂×C₄×C₈	D₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
32	#384	—	2	2	no	—
33	#384	—	2	1	no	—
34	C₄×C₁₆	—	2	2	no	—
35	C₄×C₁₆	—	2	2	no	—
36	#340	—	4	2	no	—
37	C₈²	—	2	2	no	—
38	C₂²×Q₈	—	4	1	no	—
39	C₂²×C₈	D₈	1	2	yes	—
40	C₂²×C₈	D₈	1	2	yes	—
41	C₂²×C₈	C₂×D₄	1	2	yes	—
42	C₄⋊Q₈	—	2	2	no	—
43	C₄⋊Q₈	—	4	1	no	—
44	C₄⋊Q₈	—	4	1	no	—
45	C₄⋊Q₈	—	4	1	no	—
46	C₂×C₄⋊C₄	—	4	1	no	—
47	C₂×C₄⋊C₄	—	4	1	no	—
48	C₂×C₄⋊C₄	—	4	2	no	—
49	C₂×C₄⋊C₄	—	4	1	no	—
50	C₂×C₄⋊C₄	—	4	1	no	—
51	C₂×C₄⋊C₄	—	4	1	no	—
52	C₂×C₄⋊C₄	—	4	2	no	—
53	C₂×C₄⋊C₄	—	4	2	no	—
54	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
55	C₂×C₁₆	—	2	4	no	—
56	C₂×C₁₆	—	2	4	no	—
57	C₂×C₁₆	—	2	4	no	—
58	#107	—	2	4	no	—
59	#107	—	4	2	no	—
60	#107	—	4	4	no	—
61	#107	—	4	4	no	—
62	#107	—	4	4	no	—
63	#107	—	8	4	no	—
64	#107	—	8	4	no	—
65	#103	—	8	4	no	—
66	C₄×C₈	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₄×C₈	—	2	1	no	—
69	C₄×C₈	—	2	1	no	—
70	C₄×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₄×C₈	#41	1	1	yes	characteristic
72	C₄×C₈	#41	1	2	yes	—
73	C₂×Q₈	—	4	2	no	—
74	C₂×Q₈	—	8	1	no	—
75	C₂×Q₈	—	8	2	no	—
76	C₂×Q₈	—	8	2	no	—
77	C₂²×C₄	—	8	1	no	—
78	C₂²×C₄	—	8	2	no	—
79	C₂²×C₄	—	8	1	no	—
80	C₂²×C₄	#127	1	1	yes	characteristic
81	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	D₁₆	1	4	yes	—
88	C₂×C₈	SD₃₂	1	4	yes	—
89	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₈	#123	1	2	yes	—
91	C₂×C₈	#135	1	4	yes	—
92	C₂×C₈	#137	1	4	yes	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	8	1	no	—
96	C₄⋊C₄	—	8	1	no	—
97	C₄⋊C₄	—	8	2	no	—
98	C₄⋊C₄	—	8	2	no	—
99	C₄⋊C₄	—	8	2	no	—
100	C₄⋊C₄	—	8	2	no	—
101	C₄⋊C₄	—	8	4	no	—
102	C₄⋊C₄	—	8	4	no	—
103	C₄⋊C₄	—	8	2	no	—
104	C₄²	C₂²⋊Q₈	1	1	yes	characteristic
105	C₄²	#123	1	1	yes	characteristic
106	C₄²	#137	1	1	yes	characteristic
107	C₄²	#137	1	1	yes	characteristic
108	C₁₆	—	4	4	no	—
109	C₂³	#493	1	1	yes	characteristic
110	Q₈	—	8	4	no	—
111	Q₈	—	16	2	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	16	1	no	—
114	C₂×C₄	—	16	1	no	—
115	C₂×C₄	—	16	2	no	—
116	C₂×C₄	—	16	1	no	—
117	C₂×C₄	—	16	2	no	—
118	C₂×C₄	—	16	1	no	—
119	C₂×C₄	—	16	2	no	—
120	C₂×C₄	—	16	1	no	—
121	C₂×C₄	#399	1	1	yes	characteristic
122	C₂×C₄	#474	1	1	yes	characteristic
123	C₂×C₄	#474	1	1	yes	characteristic
124	C₂×C₄	#480	1	1	yes	characteristic
125	C₂×C₄	#480	1	1	yes	characteristic
126	C₂×C₄	#495	1	2	yes	—
127	C₂×C₄	#497	1	2	yes	—
128	C₂×C₄	#497	1	1	yes	characteristic
129	C₂×C₄	C₂×D₁₆	1	2	yes	—
130	C₂×C₄	C₂×SD₃₂	1	2	yes	—
131	C₂×C₄	#510	1	2	yes	—
132	C₂×C₄	#510	1	2	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	2	4	no	—
135	C₈	#479	1	4	yes	—
136	C₈	#485	1	4	yes	—
137	C₂²	#1709	1	1	yes	characteristic
138	C₂²	#1722	1	1	yes	characteristic
139	C₂²	#2093	1	1	yes	characteristic
140	C₂²	#2246	1	1	yes	characteristic
141	C₂²	#2248	1	1	yes	characteristic
142	C₂²	#2251	1	1	yes	characteristic
143	C₂²	#2251	1	1	yes	characteristic
144	C₄	—	16	2	no	—
145	C₄	—	32	2	no	—
146	C₄	—	32	1	no	—
147	C₄	#1718	1	2	yes	—
148	C₄	#2097	1	2	yes	—
149	C₄	#2224	1	2	yes	—
150	C₄	#2225	1	2	yes	—
151	C₄	#2232	1	2	yes	—
152	C₄	#2235	1	2	yes	—
153	C₂	#10676	1	1	yes	characteristic
154	C₂	#12509	1	1	yes	characteristic
155	C₂	#12514	1	1	yes	characteristic
156	C₂	#12531	1	1	yes	characteristic
157	C₂	#12535	1	1	yes	characteristic
158	C₂	#13526	1	1	yes	characteristic
159	C₂	#13531	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13282

C₂

yes

maximal

#12888

C₂

yes

maximal

#9823

C₂

yes

maximal, characteristic

#8818

C₂

yes

maximal, characteristic

#7136

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

characteristic

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

#1845

C₂²

yes

—

C₂×C₈²

C₂²

yes

characteristic

#1380

—

#1380

—

C₂×C₄⋊Q₈

—

C₂²×C₁₆

D₄

yes

—

#500

—

#500

—

#500

Q₈

yes

—

—

—

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

#384

—

#384

—

C₄×C₁₆

—

C₄×C₁₆

—

#340

—

C₈²

—

C₂²×Q₈

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

D₈

yes

—

C₂²×C₈

C₂×D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

—

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×Q₈

yes

C₄×C₈

yes

C₄×C₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#127

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂²×C₄

C₂×D₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₆

yes

—

C₂×C₈

SD₃₂

yes

—

C₂×C₈

C₂²⋊Q₈

yes

—

C₂×C₈

#123

yes

—

C₂×C₈

#135

yes

—

C₂×C₈

#137

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂²⋊Q₈

yes

characteristic

105

C₄²

#123

yes

characteristic

106

C₄²

#137

yes

characteristic

107

C₄²

#137

yes

characteristic

108

C₁₆

—

109

C₂³

#493

yes

characteristic

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

#399

yes

characteristic

122

C₂×C₄

#474

yes

characteristic

123

C₂×C₄

#474

yes

characteristic

124

C₂×C₄

#480

yes

characteristic

125

C₂×C₄

#480

yes

characteristic

126

C₂×C₄

#495

yes

—

127

C₂×C₄

#497

yes

—

128

C₂×C₄

#497

yes

characteristic

129

C₂×C₄

C₂×D₁₆

yes

—

130

C₂×C₄

C₂×SD₃₂

yes

—

131

C₂×C₄

#510

yes

—

132

C₂×C₄

#510

yes

—

133

C₈

—

134

C₈

—

135

C₈

#479

yes

—

136

C₈

#485

yes

—

137

C₂²

#1709

yes

characteristic

138

C₂²

#1722

yes

characteristic

139

C₂²

#2093

yes

characteristic

140

C₂²

#2246

yes

characteristic

141

C₂²

#2248

yes

characteristic

142

C₂²

#2251

yes

characteristic

143

C₂²

#2251

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

#1718

yes

—

148

C₄

#2097

yes

—

149

C₄

#2224

yes

—

150

C₄

#2225

yes

—

151

C₄

#2232

yes

—

152

C₄

#2235

yes

—

153

C₂

#10676

yes

characteristic

154

C₂

#12509

yes

characteristic

155

C₂

#12514

yes

characteristic

156

C₂

#12531

yes

characteristic

157

C₂

#12535

yes

characteristic

158

C₂

#13526

yes

characteristic

159

C₂

#13531

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#493

commutator, Frattini, cosocle

C₂×C₄×C₈

C₂³

LCS

G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ 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
#13282	2
#12888	2
#9823	2
#8818	2
#7136	2

subgroup

index

Subgroups — #146249

Conjugacy classes of subgroups

Maximal subgroups