Group #513172 — subgroups

Subgroups — #513172

Order 512.

444 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 444 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×#2262	C₂	1	2	yes	maximal, complemented
2	#33853	C₂	1	2	yes	maximal, complemented
3	#13725	C₂	1	1	yes	maximal, characteristic, complemented
4	#13694	C₂	1	2	yes	maximal, complemented
5	#13686	C₂	1	2	yes	maximal, complemented
6	#13663	C₂	1	4	yes	maximal, complemented
7	C₄×Q₆₄	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#508	C₂²	1	4	yes	—
10	#3287	—	2	2	no	—
11	#2262	—	2	8	no	—
12	C₂×Q₆₄	C₂²	1	1	yes	characteristic
13	C₂×SD₆₄	C₂²	1	4	yes	—
14	C₂×D₃₂	C₂²	1	2	yes	—
15	C₂×D₃₂	C₂²	1	2	yes	—
16	C₂²×C₃₂	C₂²	1	2	yes	—
17	#2246	C₂²	1	2	yes	—
18	#2215	C₂²	1	4	yes	—
19	#2207	—	2	2	no	—
20	#2186	—	2	4	no	—
21	C₄×Q₃₂	C₂²	1	2	yes	—
22	D₄×C₁₆	C₂²	1	1	yes	characteristic
23	#1423	C₂²	1	1	yes	characteristic
24	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
25	#1416	C₂²	1	2	yes	—
26	#1415	C₂²	1	4	yes	—
27	C₂²⋊C₃₂	C₂²	1	2	yes	—
28	C₄×C₃₂	C₂²	1	1	yes	characteristic
29	C₂×#135	—	2	4	no	—
30	#550	—	4	2	no	—
31	#508	—	2	8	no	—
32	#508	—	4	4	no	—
33	C₂×Q₃₂	C₂³	1	2	yes	—
34	C₂×SD₃₂	—	2	4	no	—
35	C₂×D₁₆	—	2	2	no	—
36	C₂×D₁₆	C₂³	1	4	yes	—
37	C₂²×C₁₆	C₂³	1	2	yes	—
38	#493	—	2	2	no	—
39	#466	—	2	4	no	—
40	#460	—	4	2	no	—
41	#449	—	4	4	no	—
42	C₄×Q₁₆	—	2	2	no	—
43	C₈×D₄	D₄	1	1	yes	characteristic
44	Q₆₄	D₄	1	4	yes	complemented
45	SD₆₄	—	2	8	no	—
46	D₃₂	—	2	4	no	—
47	D₃₂	—	4	2	no	—
48	C₂×C₃₂	—	2	4	no	—
49	C₂×C₃₂	C₂³	1	1	yes	characteristic
50	C₂×C₃₂	C₂³	1	2	yes	—
51	C₂×C₃₂	C₂³	1	1	yes	characteristic
52	#366	C₂³	1	2	yes	—
53	C₄⋊C₁₆	D₄	1	1	yes	characteristic
54	#358	—	2	2	no	—
55	#357	—	2	4	no	—
56	C₂²⋊C₁₆	D₄	1	2	yes	—
57	C₄×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×#41	—	4	4	no	—
59	#135	—	4	8	no	—
60	#135	—	8	4	no	—
61	C₂×Q₁₆	—	2	2	no	—
62	C₂×SD₁₆	—	4	4	no	—
63	C₂×D₈	—	2	4	no	—
64	C₂×D₈	—	4	2	no	—
65	C₂²×C₈	C₂×D₄	1	2	yes	—
66	#127	—	4	2	no	—
67	#127	—	8	2	no	—
68	#121	—	4	4	no	—
69	#121	—	8	4	no	—
70	C₄×Q₈	—	4	2	no	—
71	C₄×D₄	—	8	2	no	—
72	C₄×D₄	D₈	1	1	yes	characteristic
73	Q₃₂	—	2	2	no	—
74	Q₃₂	C₂×D₄	1	4	yes	—
75	SD₃₂	—	4	8	no	—
76	D₁₆	—	2	4	no	—
77	D₁₆	—	2	4	no	—
78	D₁₆	—	8	2	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
82	C₂×C₁₆	C₂×D₄	1	2	yes	—
83	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
84	#107	—	2	2	no	—
85	C₄⋊C₈	D₈	1	1	yes	characteristic
86	#103	—	4	2	no	—
87	#102	—	4	4	no	—
88	C₂²⋊C₈	D₈	1	2	yes	—
89	C₄×C₈	C₂×D₄	1	1	yes	characteristic
90	C₃₂	—	2	2	no	—
91	C₃₂	—	2	1	no	—
92	C₃₂	C₂×D₄	1	2	yes	—
93	#41	—	8	8	no	—
94	#41	—	16	4	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	4	4	no	—
97	C₂×D₄	—	8	4	no	—
98	C₂×D₄	—	8	4	no	—
99	C₂×D₄	—	8	2	no	—
100	C₂×D₄	—	8	2	no	—
101	C₂×D₄	D₁₆	1	1	yes	characteristic
102	C₂²×C₄	—	8	4	no	—
103	C₂²×C₄	C₂×D₈	1	2	yes	—
104	Q₁₆	—	2	4	no	—
105	Q₁₆	—	4	2	no	—
106	SD₁₆	—	8	8	no	—
107	D₈	—	4	4	no	—
108	D₈	—	4	4	no	—
109	D₈	—	16	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	C₂×D₈	1	1	yes	characteristic
113	C₂×C₈	C₂×D₈	1	2	yes	—
114	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
115	C₄⋊C₄	—	4	2	no	—
116	C₄⋊C₄	—	8	2	no	—
117	C₄⋊C₄	D₁₆	1	1	yes	characteristic
118	C₂²⋊C₄	—	8	4	no	—
119	C₂²⋊C₄	D₁₆	1	2	yes	—
120	C₄²	—	8	2	no	—
121	C₄²	C₂×D₈	1	1	yes	characteristic
122	C₁₆	—	2	2	no	—
123	C₁₆	—	2	1	no	—
124	C₁₆	C₂²×D₄	1	1	yes	characteristic
125	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂³	—	8	4	no	—
127	C₂³	C₂×D₁₆	1	2	yes	—
128	Q₈	—	4	4	no	—
129	Q₈	—	8	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	8	4	no	—
132	D₄	—	8	4	no	—
133	D₄	—	16	8	no	—
134	D₄	—	16	4	no	—
135	D₄	—	32	2	no	—
136	D₄	—	32	4	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	8	2	no	—
140	C₂×C₄	—	8	2	no	—
141	C₂×C₄	—	16	8	no	—
142	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
143	C₂×C₄	C₂×D₁₆	1	2	yes	—
144	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
145	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
146	C₈	—	2	1	no	—
147	C₈	—	2	2	no	—
148	C₈	D₄²	1	1	yes	characteristic
149	C₈	#576	1	1	yes	characteristic
150	C₂²	—	2	2	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	16	4	no	—
154	C₂²	—	16	4	no	—
155	C₂²	—	16	4	no	—
156	C₂²	C₂²×D₁₆	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	1	no	—
159	C₄	—	8	4	no	—
160	C₄	—	16	2	no	—
161	C₄	#2262	1	2	yes	complemented
162	C₄	D₄×D₈	1	1	yes	characteristic
163	C₄	#3415	1	1	yes	characteristic
164	C₂	—	4	2	no	—
165	C₂	—	32	4	no	—
166	C₂	D₄×D₁₆	1	1	yes	characteristic
167	C₂	C₂×#2262	1	1	yes	characteristic
168	C₂	#33982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#2262

C₂

yes

maximal, complemented

#33853

C₂

yes

maximal, complemented

#13725

C₂

yes

maximal, characteristic, complemented

#13694

C₂

yes

maximal, complemented

#13686

C₂

yes

maximal, complemented

#13663

C₂

yes

maximal, complemented

C₄×Q₆₄

C₂

yes

maximal, characteristic, complemented

D₄×C₃₂

C₂

yes

maximal, characteristic, complemented

C₂×#508

C₂²

yes

—

#3287

—

#2262

—

C₂×Q₆₄

C₂²

yes

characteristic

C₂×SD₆₄

C₂²

yes

—

C₂×D₃₂

C₂²

yes

—

C₂×D₃₂

C₂²

yes

—

C₂²×C₃₂

C₂²

yes

—

#2246

C₂²

yes

—

#2215

C₂²

yes

—

#2207

—

#2186

—

C₄×Q₃₂

C₂²

yes

—

D₄×C₁₆

C₂²

yes

characteristic

#1423

C₂²

yes

characteristic

C₄⋊C₃₂

C₂²

yes

characteristic

#1416

C₂²

yes

—

#1415

C₂²

yes

—

C₂²⋊C₃₂

C₂²

yes

—

C₄×C₃₂

C₂²

yes

—

—

—

—

C₂×Q₃₂

C₂³

yes

—

C₂×SD₃₂

—

C₂×D₁₆

—

C₂×D₁₆

C₂³

yes

—

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

C₄×Q₁₆

—

C₈×D₄

D₄

yes

characteristic

Q₆₄

D₄

yes

complemented

SD₆₄

—

D₃₂

—

D₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂³

yes

characteristic

C₂×C₃₂

C₂³

yes

—

C₂×C₃₂

C₂³

yes

characteristic

#366

C₂³

yes

—

C₄⋊C₁₆

D₄

yes

characteristic

#358

—

#357

—

C₂²⋊C₁₆

D₄

yes

—

C₄×C₁₆

C₂³

yes

characteristic

C₂×#41

—

#135

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

C₂×D₄

yes

—

—

—

—

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₈

yes

characteristic

Q₃₂

—

Q₃₂

C₂×D₄

yes

—

SD₃₂

—

D₁₆

—

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

characteristic

#107

—

C₄⋊C₈

D₈

yes

characteristic

#103

—

#102

—

C₂²⋊C₈

D₈

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

C₂×D₄

yes

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

D₁₆

yes

characteristic

102

C₂²×C₄

—

103

C₂²×C₄

C₂×D₈

yes

—

104

Q₁₆

—

105

Q₁₆

—

106

SD₁₆

—

107

D₈

—

108

D₈

—

109

D₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

C₂×D₈

yes

characteristic

113

C₂×C₈

C₂×D₈

yes

—

114

C₂×C₈

C₂²×D₄

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

D₁₆

yes

characteristic

118

C₂²⋊C₄

—

119

C₂²⋊C₄

D₁₆

yes

—

120

C₄²

—

121

C₄²

C₂×D₈

yes

characteristic

122

C₁₆

—

123

C₁₆

—

124

C₁₆

C₂²×D₄

yes

characteristic

125

C₁₆

H(5,𝔽₂)

yes

characteristic

126

C₂³

—

127

C₂³

C₂×D₁₆

yes

—

128

Q₈

—

129

Q₈

—

130

D₄

—

131

D₄

—

132

D₄

—

133

D₄

—

134

D₄

—

135

D₄

—

136

D₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×D₁₆

yes

characteristic

143

C₂×C₄

C₂×D₁₆

yes

—

144

C₂×C₄

C₂×D₁₆

yes

characteristic

145

C₂×C₄

C₂²×D₈

yes

characteristic

146

C₈

—

147

C₈

—

148

C₈

D₄²

yes

characteristic

149

C₈

#576

yes

characteristic

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

C₂²×D₁₆

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

#2262

yes

complemented

162

C₄

D₄×D₈

yes

characteristic

163

C₄

#3415

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

D₄×D₁₆

yes

characteristic

167

C₂

C₂×#2262

yes

characteristic

168

C₂

#33982

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂⁴

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. C₂×#2262	C₂	semidirect
2. #33853	C₂	semidirect
3. #13725	C₂	semidirect
4. #13694	C₂	semidirect
5. #13686	C₂	semidirect
6. #13663	C₂	semidirect
7. C₄×Q₆₄	C₂	semidirect
8. D₄×C₃₂	C₂	semidirect
44. Q₆₄	D₄	semidirect
161. C₄	#2262	semidirect

H ≅ G/N

type

1. C₂×#2262

C₂

semidirect

2. #33853

C₂

semidirect

3. #13725

C₂

semidirect

4. #13694

C₂

semidirect

5. #13686

C₂

semidirect

6. #13663

C₂

semidirect

7. C₄×Q₆₄

C₂

semidirect

8. D₄×C₃₂

C₂

semidirect

44. Q₆₄

D₄

semidirect

161. C₄

#2262

semidirect

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

subgroup	index
C₂×#2262	2
#33853	2
#13725	2
#13694	2
#13686	2
#13663	2
C₄×Q₆₄	2
D₄×C₃₂	2

subgroup

index

C₄×Q₆₄

D₄×C₃₂

Subgroups — #513172

Conjugacy classes of subgroups

Complements

Maximal subgroups