Group #512515 — subgroups

Subgroups — #512515

Order 512.

518 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 218 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	#33313	C₂	1	1	yes	maximal, characteristic
2	#33311	C₂	1	1	yes	maximal, characteristic
3	#13283	C₂	1	4	yes	maximal
4	#13281	C₂	1	4	yes	maximal
5	#12890	C₂	1	4	yes	maximal
6	C₄×#346	C₂	1	1	yes	maximal, characteristic
7	C₂×#500	C₂²	1	2	yes	—
8	C₂×#499	—	2	1	no	—
9	C₂×#499	C₂²	1	2	yes	—
10	C₂×#498	—	2	1	no	—
11	#2862	—	2	1	no	—
12	#2859	—	2	1	no	—
13	C₂×#346	C₂²	1	2	yes	—
14	C₂×#346	C₂²	1	2	yes	—
15	C₂×C₄×C₁₆	C₂²	1	2	yes	—
16	#1846	—	2	1	no	—
17	#1846	—	2	1	no	—
18	#1846	—	2	1	no	—
19	#1846	C₂²	1	2	yes	—
20	#1846	C₂²	1	2	yes	—
21	#1846	C₂²	1	2	yes	—
22	#1845	C₂²	1	2	yes	—
23	#1844	—	2	1	no	—
24	C₄²×C₈	C₂²	1	1	yes	characteristic
25	#1380	C₂²	1	16	yes	—
26	C₂×C₄⋊Q₈	—	2	2	no	—
27	C₂×#125	—	2	2	no	—
28	C₂²×C₁₆	C₂³	1	4	yes	—
29	#500	—	2	8	no	—
30	#499	—	2	8	no	—
31	#499	—	4	4	no	—
32	#498	—	4	4	no	—
33	C₂×#107	—	2	2	no	—
34	C₂×#107	—	2	2	no	—
35	C₂×#107	C₂³	1	4	yes	—
36	C₂×#107	C₂³	1	4	yes	—
37	C₂×#106	—	2	2	no	—
38	C₂×#106	—	2	2	no	—
39	C₂×#106	—	2	2	no	—
40	C₂×#106	—	2	2	no	—
41	C₂×C₄×C₈	D₄	1	1	yes	characteristic
42	C₂×C₄×C₈	D₄	1	1	yes	characteristic
43	C₂×C₄×C₈	D₄	1	1	yes	characteristic
44	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₈	C₂³	1	1	yes	characteristic
47	#391	—	4	4	no	—
48	#389	—	4	4	no	—
49	#384	—	2	2	no	—
50	#382	—	2	2	no	—
51	#382	—	2	2	no	—
52	#382	—	2	2	no	—
53	C₄³	D₄	1	1	yes	characteristic
54	#346	—	2	4	no	—
55	#346	Q₈	1	8	yes	complemented
56	C₄×C₁₆	—	2	4	no	—
57	C₂²×Q₈	—	4	2	no	—
58	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₈	C₂⁴	1	1	yes	characteristic
62	C₄⋊Q₈	—	4	4	no	—
63	C₄⋊Q₈	—	4	4	no	—
64	#125	—	4	4	no	—
65	#125	—	4	4	no	—
66	C₂×C₄⋊C₄	—	2	4	no	—
67	C₂×C₄⋊C₄	—	2	4	no	—
68	C₂×C₄⋊C₄	—	4	4	no	—
69	C₂×C₄⋊C₄	—	4	2	no	—
70	C₂×C₄⋊C₄	—	4	4	no	—
71	C₂×C₄²	D₈	1	4	yes	—
72	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
73	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
74	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
75	C₂×C₁₆	—	2	4	no	—
76	C₂×C₁₆	—	2	4	no	—
77	C₂×C₁₆	C₂×Q₈	1	8	yes	—
78	#107	—	2	8	no	—
79	#107	—	2	8	no	—
80	#107	—	4	4	no	—
81	#107	—	4	4	no	—
82	#106	—	4	4	no	—
83	#106	—	4	4	no	—
84	#106	—	4	4	no	—
85	#106	—	4	4	no	—
86	C₄×C₈	SD₁₆	1	4	yes	—
87	C₄×C₈	SD₁₆	1	4	yes	—
88	C₄×C₈	Q₁₆	1	4	yes	—
89	C₄×C₈	C₂×Q₈	1	4	yes	—
90	C₄×C₈	#41	1	4	yes	—
91	C₄×C₈	#41	1	4	yes	—
92	#95	—	4	4	no	—
93	#95	—	4	4	no	—
94	#95	—	4	4	no	—
95	C₂×Q₈	—	8	8	no	—
96	C₂×Q₈	—	8	4	no	—
97	C₂²×C₄	—	4	4	no	—
98	C₂²×C₄	—	4	4	no	—
99	C₂²×C₄	C₂×D₈	1	2	yes	—
100	C₂²×C₄	C₂×D₈	1	2	yes	—
101	C₂²×C₄	C₂×D₈	1	2	yes	—
102	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₈	#124	1	2	yes	—
104	C₂×C₈	#124	1	2	yes	—
105	C₂×C₈	C₄⋊Q₈	1	2	yes	—
106	C₂×C₈	C₂×SD₁₆	1	2	yes	—
107	C₂×C₈	C₂×SD₁₆	1	2	yes	—
108	C₂×C₈	C₂×SD₁₆	1	2	yes	—
109	C₂×C₈	C₂×SD₁₆	1	2	yes	—
110	C₂×C₈	C₂×Q₁₆	1	2	yes	—
111	C₂×C₈	C₂×Q₁₆	1	2	yes	—
112	C₂×C₈	C₂²×Q₈	1	2	yes	—
113	C₂×C₈	C₂×#41	1	2	yes	—
114	C₂×C₈	C₂×#41	1	2	yes	—
115	C₄⋊C₄	—	4	4	no	—
116	C₄⋊C₄	—	4	4	no	—
117	C₄⋊C₄	—	4	4	no	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄⋊C₄	—	8	8	no	—
120	C₄⋊C₄	—	8	8	no	—
121	C₄⋊C₄	—	8	4	no	—
122	C₄²	—	4	4	no	—
123	C₄²	#124	1	2	yes	—
124	C₄²	#124	1	2	yes	—
125	C₄²	C₄⋊Q₈	1	2	yes	—
126	C₄²	C₂×SD₁₆	1	2	yes	—
127	C₄²	C₂×SD₁₆	1	2	yes	—
128	C₄²	C₂×Q₁₆	1	2	yes	—
129	C₁₆	—	2	8	no	—
130	C₂³	C₂²×D₈	1	1	yes	characteristic
131	Q₈	—	16	8	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	—	8	4	no	—
137	C₂×C₄	—	8	8	no	—
138	C₂×C₄	—	8	8	no	—
139	C₂×C₄	#486	1	4	yes	—
140	C₂×C₄	#486	1	4	yes	—
141	C₂×C₄	#500	1	4	yes	—
142	C₂×C₄	C₂×#124	1	1	yes	characteristic
143	C₂×C₄	C₂×#124	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
146	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
147	C₂×C₄	C₂²×Q₁₆	1	1	yes	characteristic
148	C₈	#487	1	4	yes	—
149	C₈	#487	1	4	yes	—
150	C₈	#498	1	4	yes	—
151	C₈	#533	1	4	yes	—
152	C₂²	C₂×#486	1	1	yes	characteristic
153	C₂²	C₂×#486	1	1	yes	characteristic
154	C₂²	C₂×#500	1	1	yes	characteristic
155	C₂²	#3414	1	4	yes	—
156	C₄	—	2	4	no	—
157	C₄	—	2	4	no	—
158	C₄	—	2	4	no	—
159	C₄	—	16	4	no	—
160	C₄	—	16	4	no	—
161	C₄	#2862	1	1	yes	characteristic
162	C₄	C₂×#487	1	1	yes	characteristic
163	C₄	C₂×#487	1	1	yes	characteristic
164	C₄	C₂×#498	1	1	yes	characteristic
165	C₂	#33289	1	1	yes	characteristic
166	C₂	#33701	1	2	yes	—
167	C₂	#33706	1	2	yes	—
168	C₂	#33732	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33313

C₂

yes

maximal, characteristic

#33311

C₂

yes

maximal, characteristic

#13283

C₂

yes

maximal

#13281

C₂

yes

maximal

#12890

C₂

yes

maximal

C₄×#346

C₂

yes

maximal, characteristic

C₂×#500

C₂²

yes

—

C₂×#499

—

C₂×#499

C₂²

yes

—

—

—

—

C₂²

yes

—

C₂×#346

C₂²

yes

—

C₂×C₄×C₁₆

C₂²

yes

—

—

—

—

C₂²

yes

—

#1846

C₂²

yes

—

#1846

C₂²

yes

—

#1845

C₂²

yes

—

#1844

—

C₄²×C₈

C₂²

yes

characteristic

#1380

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂×#125

—

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

—

C₂³

yes

—

C₂×#107

C₂³

yes

—

—

—

—

—

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₂×C₄×C₈

C₂³

yes

characteristic

C₂×C₄×C₈

C₂³

yes

—

—

—

—

—

—

C₄³

D₄

yes

characteristic

#346

—

#346

Q₈

yes

complemented

C₄×C₁₆

—

C₂²×Q₈

—

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₂²×C₈

C₂⁴

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

D₈

yes

—

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Q₈

yes

—

—

—

—

—

—

—

—

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

—

C₄×C₈

C₂×Q₈

yes

—

C₄×C₈

#41

yes

—

C₄×C₈

#41

yes

—

#95

—

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

—

100

C₂²×C₄

C₂×D₈

yes

—

101

C₂²×C₄

C₂×D₈

yes

—

102

C₂²×C₄

C₂²×D₄

yes

characteristic

103

C₂×C₈

#124

yes

—

104

C₂×C₈

#124

yes

—

105

C₂×C₈

C₄⋊Q₈

yes

—

106

C₂×C₈

C₂×SD₁₆

yes

—

107

C₂×C₈

C₂×SD₁₆

yes

—

108

C₂×C₈

C₂×SD₁₆

yes

—

109

C₂×C₈

C₂×SD₁₆

yes

—

110

C₂×C₈

C₂×Q₁₆

yes

—

111

C₂×C₈

C₂×Q₁₆

yes

—

112

C₂×C₈

C₂²×Q₈

yes

—

113

C₂×C₈

C₂×#41

yes

—

114

C₂×C₈

C₂×#41

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

—

123

C₄²

#124

yes

—

124

C₄²

#124

yes

—

125

C₄²

C₄⋊Q₈

yes

—

126

C₄²

C₂×SD₁₆

yes

—

127

C₄²

C₂×SD₁₆

yes

—

128

C₄²

C₂×Q₁₆

yes

—

129

C₁₆

—

130

C₂³

C₂²×D₈

yes

characteristic

131

Q₈

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

#486

yes

—

140

C₂×C₄

#486

yes

—

141

C₂×C₄

#500

yes

—

142

C₂×C₄

C₂×#124

yes

characteristic

143

C₂×C₄

C₂×#124

yes

characteristic

144

C₂×C₄

C₂×C₄⋊Q₈

yes

characteristic

145

C₂×C₄

C₂²×SD₁₆

yes

characteristic

146

C₂×C₄

C₂²×SD₁₆

yes

characteristic

147

C₂×C₄

C₂²×Q₁₆

yes

characteristic

148

C₈

#487

yes

—

149

C₈

#487

yes

—

150

C₈

#498

yes

—

151

C₈

#533

yes

—

152

C₂²

C₂×#486

yes

characteristic

153

C₂²

C₂×#486

yes

characteristic

154

C₂²

C₂×#500

yes

characteristic

155

C₂²

#3414

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#2862

yes

characteristic

162

C₄

C₂×#487

yes

characteristic

163

C₄

C₂×#487

yes

characteristic

164

C₄

C₂×#498

yes

characteristic

165

C₂

#33289

yes

characteristic

166

C₂

#33701

yes

—

167

C₂

#33706

yes

—

168

C₂

#33732

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₈

commutator, Frattini, cosocle

C₂²×C₈

C₂⁴

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
55. #346	Q₈	semidirect

H ≅ G/N

type

55. #346

Q₈

semidirect

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

subgroup	index
#33313	2
#33311	2
#13283	2
#13281	2
#12890	2
C₄×#346	2

subgroup

index

Subgroups — #512515

Conjugacy classes of subgroups

Complements

Maximal subgroups