Group #513294 — subgroups

Subgroups — #513294

Order 512.

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

normal subgroups only

This group has 328 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	#33960	C₂	1	2	yes	maximal, complemented
2	#13728	C₂	1	2	yes	maximal
3	#13726	C₂	1	1	yes	maximal, characteristic, complemented
4	#13690	C₂	1	4	yes	maximal, complemented
5	#13687	C₂	1	2	yes	maximal, complemented
6	Q₈×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Q₆₄	C₂	1	2	yes	maximal, complemented
8	C₄×D₃₂	C₂	1	1	yes	maximal, characteristic
9	#3393	—	2	2	no	—
10	C₂×Q₆₄	C₂²	1	2	yes	—
11	C₂×Q₆₄	C₂²	1	2	yes	—
12	C₂×SD₆₄	C₂²	1	4	yes	—
13	C₂×D₃₂	C₂²	1	1	yes	characteristic
14	#2249	C₂²	1	4	yes	—
15	#2247	C₂²	1	2	yes	—
16	#2211	—	2	4	no	—
17	#2208	—	2	2	no	—
18	Q₈×C₁₆	C₂²	1	1	yes	characteristic
19	C₄×Q₃₂	C₂²	1	4	yes	—
20	C₄×D₁₆	C₂²	1	2	yes	—
21	#1423	C₂²	1	1	yes	characteristic
22	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
23	C₄⋊C₃₂	C₂²	1	2	yes	—
24	#1416	C₂²	1	4	yes	—
25	#1415	C₂²	1	2	yes	—
26	C₄×C₃₂	C₂²	1	1	yes	characteristic
27	C₄×C₃₂	C₂²	1	2	yes	—
28	#556	—	4	2	no	—
29	C₂×Q₃₂	—	2	2	no	—
30	C₂×Q₃₂	C₂³	1	4	yes	—
31	C₂×SD₃₂	—	2	4	no	—
32	C₂×D₁₆	C₂³	1	2	yes	—
33	#495	—	2	4	no	—
34	#494	—	2	2	no	—
35	#464	—	4	4	no	—
36	#461	—	4	2	no	—
37	C₈×Q₈	D₄	1	1	yes	characteristic
38	C₄×Q₁₆	—	2	4	no	—
39	C₄×D₈	—	2	2	no	—
40	Q₆₄	—	2	4	no	—
41	Q₆₄	—	4	2	no	—
42	SD₆₄	—	4	4	no	—
43	D₃₂	—	2	2	no	—
44	C₂×C₃₂	C₂³	1	1	yes	characteristic
45	C₂×C₃₂	C₂³	1	2	yes	—
46	C₂×C₃₂	C₂³	1	1	yes	characteristic
47	#366	C₂³	1	2	yes	—
48	C₄⋊C₁₆	D₄	1	1	yes	characteristic
49	C₄⋊C₁₆	D₄	1	2	yes	—
50	#358	—	2	4	no	—
51	#357	—	2	2	no	—
52	C₄×C₁₆	C₂³	1	1	yes	characteristic
53	C₄×C₁₆	C₂³	1	2	yes	—
54	C₂×Q₁₆	—	2	4	no	—
55	C₂×Q₁₆	—	4	2	no	—
56	C₂×SD₁₆	—	4	4	no	—
57	C₂×D₈	—	2	2	no	—
58	C₄⋊Q₈	—	4	2	no	—
59	#126	—	8	4	no	—
60	#124	—	4	4	no	—
61	C₂²⋊Q₈	—	8	2	no	—
62	C₄×Q₈	—	4	4	no	—
63	C₄×Q₈	D₈	1	1	yes	characteristic
64	C₄×D₄	—	4	2	no	—
65	#117	—	8	2	no	—
66	Q₃₂	—	2	4	no	—
67	Q₃₂	—	2	4	no	—
68	Q₃₂	—	8	2	no	—
69	SD₃₂	—	8	4	no	—
70	D₁₆	—	2	2	no	—
71	D₁₆	—	2	2	no	—
72	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
73	C₂×C₁₆	C₂×D₄	1	2	yes	—
74	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
75	#107	—	2	2	no	—
76	C₄⋊C₈	D₈	1	1	yes	characteristic
77	C₄⋊C₈	D₈	1	2	yes	—
78	#103	—	4	4	no	—
79	#102	—	4	2	no	—
80	C₄×C₈	C₂×D₄	1	1	yes	characteristic
81	C₄×C₈	C₂×D₄	1	2	yes	—
82	C₃₂	—	2	2	no	—
83	C₃₂	—	2	1	no	—
84	C₃₂	#41	1	2	yes	—
85	C₂×Q₈	—	4	4	no	—
86	C₂×Q₈	D₁₆	1	1	yes	characteristic
87	C₂×D₄	—	4	2	no	—
88	C₂²×C₄	—	8	2	no	—
89	Q₁₆	—	4	4	no	—
90	Q₁₆	—	4	4	no	—
91	Q₁₆	—	16	2	no	—
92	SD₁₆	—	16	4	no	—
93	D₈	—	4	2	no	—
94	D₈	—	4	2	no	—
95	C₂×C₈	C₂×D₈	1	1	yes	characteristic
96	C₂×C₈	C₂×D₈	1	2	yes	—
97	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	8	2	no	—
100	C₄⋊C₄	—	8	4	no	—
101	C₄⋊C₄	—	8	2	no	—
102	C₄⋊C₄	D₁₆	1	1	yes	characteristic
103	C₄⋊C₄	D₁₆	1	2	yes	—
104	C₂²⋊C₄	—	8	4	no	—
105	C₂²⋊C₄	—	8	2	no	—
106	C₂²⋊C₄	—	8	4	no	—
107	C₄²	—	8	4	no	—
108	C₄²	C₂×D₈	1	1	yes	characteristic
109	C₄²	C₂×D₈	1	2	yes	—
110	C₁₆	—	2	2	no	—
111	C₁₆	—	2	1	no	—
112	C₁₆	C₂×#41	1	1	yes	characteristic
113	C₁₆	H(5,𝔽₂)	1	1	yes	characteristic
114	C₂³	—	8	2	no	—
115	Q₈	—	2	2	no	—
116	Q₈	—	8	4	no	—
117	Q₈	—	8	4	no	—
118	D₄	—	8	2	no	—
119	D₄	—	8	2	no	—
120	C₂×C₄	—	8	2	no	—
121	C₂×C₄	—	8	4	no	—
122	C₂×C₄	—	16	4	no	—
123	C₂×C₄	C₂×D₁₆	1	2	yes	—
124	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₆	1	2	yes	—
126	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
127	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	2	1	no	—
130	C₈	#550	1	1	yes	characteristic
131	C₈	#576	1	1	yes	characteristic
132	C₂²	—	16	2	no	—
133	C₂²	—	16	2	no	—
134	C₂²	—	16	2	no	—
135	C₂²	C₂²×D₁₆	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	16	4	no	—
140	C₄	—	16	2	no	—
141	C₄	#2262	1	2	yes	—
142	C₄	#3391	1	1	yes	characteristic
143	C₄	#3415	1	1	yes	characteristic
144	C₂	—	32	2	no	—
145	C₂	#33958	1	1	yes	characteristic
146	C₂	C₂×#2262	1	1	yes	characteristic
147	C₂	#33984	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33960

C₂

yes

maximal, complemented

#13728

C₂

yes

maximal

#13726

C₂

yes

maximal, characteristic, complemented

#13690

C₂

yes

maximal, complemented

#13687

C₂

yes

maximal, complemented

Q₈×C₃₂

C₂

yes

maximal, characteristic, complemented

C₄×Q₆₄

C₂

yes

maximal, complemented

C₄×D₃₂

C₂

yes

maximal, characteristic

#3393

—

C₂×Q₆₄

C₂²

yes

—

C₂×Q₆₄

C₂²

yes

—

C₂×SD₆₄

C₂²

yes

—

C₂×D₃₂

C₂²

yes

characteristic

#2249

C₂²

yes

—

#2247

C₂²

yes

—

#2211

—

#2208

—

Q₈×C₁₆

C₂²

yes

characteristic

C₄×Q₃₂

C₂²

yes

—

C₄×D₁₆

C₂²

yes

—

#1423

C₂²

yes

characteristic

C₄⋊C₃₂

C₂²

yes

characteristic

C₄⋊C₃₂

C₂²

yes

—

#1416

C₂²

yes

—

#1415

C₂²

yes

—

C₄×C₃₂

C₂²

yes

characteristic

C₄×C₃₂

C₂²

yes

—

#556

—

C₂×Q₃₂

—

C₂×Q₃₂

C₂³

yes

—

C₂×SD₃₂

—

C₂×D₁₆

C₂³

yes

—

—

—

—

—

C₈×Q₈

D₄

yes

characteristic

C₄×Q₁₆

—

C₄×D₈

—

Q₆₄

—

Q₆₄

—

SD₆₄

—

D₃₂

—

C₂×C₃₂

C₂³

yes

characteristic

C₂×C₃₂

C₂³

yes

—

C₂×C₃₂

C₂³

yes

characteristic

#366

C₂³

yes

—

C₄⋊C₁₆

D₄

yes

characteristic

C₄⋊C₁₆

D₄

yes

—

#358

—

#357

—

C₄×C₁₆

C₂³

yes

characteristic

C₄×C₁₆

C₂³

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

#126

—

#124

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

D₈

yes

characteristic

C₄×D₄

—

#117

—

Q₃₂

—

Q₃₂

—

Q₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

characteristic

#107

—

C₄⋊C₈

D₈

yes

characteristic

C₄⋊C₈

D₈

yes

—

#103

—

#102

—

C₄×C₈

C₂×D₄

yes

characteristic

C₄×C₈

C₂×D₄

yes

—

C₃₂

—

C₃₂

—

C₃₂

#41

yes

—

C₂×Q₈

—

C₂×Q₈

D₁₆

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂²×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

D₁₆

yes

characteristic

103

C₄⋊C₄

D₁₆

yes

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

C₂×D₈

yes

characteristic

109

C₄²

C₂×D₈

yes

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

C₂×#41

yes

characteristic

113

C₁₆

H(5,𝔽₂)

yes

characteristic

114

C₂³

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

D₄

—

119

D₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×D₁₆

yes

—

124

C₂×C₄

C₂×D₁₆

yes

characteristic

125

C₂×C₄

C₂×D₁₆

yes

—

126

C₂×C₄

C₂×D₁₆

yes

characteristic

127

C₂×C₄

C₂²×D₈

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

#550

yes

characteristic

131

C₈

#576

yes

characteristic

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₂²×D₁₆

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#2262

yes

—

142

C₄

#3391

yes

characteristic

143

C₄

#3415

yes

characteristic

144

C₂

—

145

C₂

#33958

yes

characteristic

146

C₂

C₂×#2262

yes

characteristic

147

C₂

#33984

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×D₃₂	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂²×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂⁴

prime radical

C₂×D₃₂

C₂²

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆ ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
1. #33960	C₂	semidirect
3. #13726	C₂	semidirect
4. #13690	C₂	semidirect
5. #13687	C₂	semidirect
6. Q₈×C₃₂	C₂	semidirect
7. C₄×Q₆₄	C₂	semidirect

H ≅ G/N

type

1. #33960

C₂

semidirect

3. #13726

C₂

semidirect

4. #13690

C₂

semidirect

5. #13687

C₂

semidirect

6. Q₈×C₃₂

C₂

semidirect

7. C₄×Q₆₄

C₂

semidirect

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

subgroup	index
#33960	2
#13728	2
#13726	2
#13690	2
#13687	2
Q₈×C₃₂	2
C₄×Q₆₄	2
C₄×D₃₂	2

subgroup

index

Q₈×C₃₂

C₄×Q₆₄

C₄×D₃₂

Subgroups — #513294

Conjugacy classes of subgroups

Complements

Maximal subgroups