Group #513281 — subgroups

Subgroups — #513281

Order 512.

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

normal subgroups only

This group has 300 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	#33947	C₂	1	2	yes	maximal, complemented
2	#13732	C₂	1	2	yes	maximal, complemented
3	#13731	C₂	1	1	yes	maximal, characteristic, complemented
4	#13707	C₂	1	4	yes	maximal, complemented
5	#13705	C₂	1	2	yes	maximal, complemented
6	Q₈×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×D₃₂	C₂	1	1	yes	maximal, characteristic
8	C₄×D₃₂	C₂	1	2	yes	maximal
9	#3380	—	2	2	no	—
10	C₂×D₃₂	C₂²	1	1	yes	characteristic
11	#2253	C₂²	1	4	yes	—
12	#2252	C₂²	1	2	yes	—
13	#2228	—	2	4	no	—
14	#2226	—	2	2	no	—
15	Q₈×C₁₆	C₂²	1	1	yes	characteristic
16	C₄×D₁₆	C₂²	1	2	yes	—
17	C₄×D₁₆	C₂²	1	4	yes	—
18	#1424	C₂²	1	4	yes	—
19	#1423	C₂²	1	2	yes	—
20	#1423	C₂²	1	1	yes	characteristic
21	#1423	C₂²	1	2	yes	—
22	C₄⋊C₃₂	C₂²	1	1	yes	characteristic
23	C₄⋊C₃₂	C₂²	1	2	yes	—
24	#1415	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	#554	—	4	2	no	—
29	C₂×D₁₆	C₂³	1	2	yes	—
30	#500	—	2	2	no	—
31	#499	—	2	4	no	—
32	#478	—	4	4	no	—
33	#476	—	4	2	no	—
34	C₈×Q₈	D₄	1	1	yes	characteristic
35	C₄×D₈	—	2	2	no	—
36	C₄×D₈	—	2	4	no	—
37	D₃₂	Q₈	1	4	yes	complemented
38	C₂×C₃₂	C₂³	1	1	yes	characteristic
39	C₂×C₃₂	C₂³	1	2	yes	—
40	C₂×C₃₂	C₂³	1	1	yes	characteristic
41	#367	—	2	4	no	—
42	#366	—	2	2	no	—
43	#366	C₂³	1	4	yes	—
44	#366	C₂³	1	2	yes	—
45	C₄⋊C₁₆	D₄	1	1	yes	characteristic
46	C₄⋊C₁₆	D₄	1	2	yes	—
47	#357	—	2	4	no	—
48	#357	—	2	2	no	—
49	C₄×C₁₆	C₂³	1	1	yes	characteristic
50	C₄×C₁₆	C₂³	1	2	yes	—
51	C₂×D₈	—	2	2	no	—
52	C₄⋊Q₈	—	4	2	no	—
53	#125	—	4	4	no	—
54	C₂²⋊Q₈	—	8	2	no	—
55	C₂²⋊Q₈	—	8	4	no	—
56	C₄×Q₈	D₈	1	1	yes	characteristic
57	C₄×D₄	—	4	2	no	—
58	C₄×D₄	—	4	4	no	—
59	C₂×C₄⋊C₄	—	8	2	no	—
60	D₁₆	—	2	2	no	—
61	D₁₆	C₂×Q₈	1	4	yes	—
62	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
63	C₂×C₁₆	C₂×D₄	1	2	yes	—
64	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
65	#107	—	2	4	no	—
66	#107	—	2	2	no	—
67	#107	—	4	2	no	—
68	#106	—	4	4	no	—
69	C₄⋊C₈	D₈	1	1	yes	characteristic
70	C₄⋊C₈	D₈	1	2	yes	—
71	#102	—	4	4	no	—
72	#102	—	4	2	no	—
73	C₄×C₈	C₂×D₄	1	2	yes	—
74	C₄×C₈	C₂×D₄	1	1	yes	characteristic
75	C₃₂	—	2	1	no	—
76	C₃₂	—	2	2	no	—
77	C₃₂	C₂×Q₈	1	2	yes	—
78	C₂×Q₈	—	8	2	no	—
79	C₂×Q₈	D₁₆	1	1	yes	characteristic
80	C₂×D₄	—	4	2	no	—
81	C₂²×C₄	—	8	2	no	—
82	C₂²×C₄	—	8	4	no	—
83	D₈	—	2	4	no	—
84	D₈	—	4	2	no	—
85	C₂×C₈	C₂×D₈	1	2	yes	—
86	C₂×C₈	C₂×D₈	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	8	4	no	—
91	C₄⋊C₄	—	8	2	no	—
92	C₄⋊C₄	—	8	4	no	—
93	C₄⋊C₄	D₁₆	1	1	yes	characteristic
94	C₄⋊C₄	D₁₆	1	2	yes	—
95	C₂²⋊C₄	—	8	2	no	—
96	C₂²⋊C₄	—	8	4	no	—
97	C₄²	C₂×D₈	1	2	yes	—
98	C₄²	C₂×D₈	1	1	yes	characteristic
99	C₁₆	—	2	1	no	—
100	C₁₆	—	2	2	no	—
101	C₁₆	C₂²×Q₈	1	1	yes	characteristic
102	C₁₆	2₋¹⁺⁴	1	1	yes	characteristic
103	C₂³	—	8	2	no	—
104	Q₈	—	2	2	no	—
105	Q₈	—	32	2	no	—
106	D₄	—	4	4	no	—
107	D₄	—	8	2	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	—	16	4	no	—
111	C₂×C₄	—	32	4	no	—
112	C₂×C₄	C₂×D₁₆	1	2	yes	—
113	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
114	C₂×C₄	C₂×D₁₆	1	2	yes	—
115	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
117	C₈	—	2	2	no	—
118	C₈	—	2	1	no	—
119	C₈	D₄×Q₈	1	1	yes	characteristic
120	C₈	#578	1	1	yes	characteristic
121	C₂²	—	8	4	no	—
122	C₂²	—	16	2	no	—
123	C₂²	—	16	2	no	—
124	C₂²	C₂²×D₁₆	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	16	2	no	—
129	C₄	—	16	4	no	—
130	C₄	#2262	1	2	yes	—
131	C₄	Q₈×D₈	1	1	yes	characteristic
132	C₄	#3417	1	1	yes	characteristic
133	C₂	—	16	4	no	—
134	C₂	Q₈×D₁₆	1	1	yes	characteristic
135	C₂	C₂×#2262	1	1	yes	characteristic
136	C₂	#33982	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#33947

C₂

yes

maximal, complemented

#13732

C₂

yes

maximal, complemented

#13731

C₂

yes

maximal, characteristic, complemented

#13707

C₂

yes

maximal, complemented

#13705

C₂

yes

maximal, complemented

Q₈×C₃₂

C₂

yes

maximal, characteristic, complemented

C₄×D₃₂

C₂

yes

maximal, characteristic

C₄×D₃₂

C₂

yes

maximal

#3380

—

C₂×D₃₂

C₂²

yes

characteristic

#2253

C₂²

yes

—

#2252

C₂²

yes

—

#2228

—

#2226

—

Q₈×C₁₆

C₂²

yes

characteristic

C₄×D₁₆

C₂²

yes

—

C₄×D₁₆

C₂²

yes

—

#1424

C₂²

yes

—

#1423

C₂²

yes

—

#1423

C₂²

yes

characteristic

#1423

C₂²

yes

—

C₄⋊C₃₂

C₂²

yes

characteristic

C₄⋊C₃₂

C₂²

yes

—

#1415

C₂²

yes

—

#1415

C₂²

yes

—

C₄×C₃₂

C₂²

yes

characteristic

C₄×C₃₂

C₂²

yes

—

#554

—

C₂×D₁₆

C₂³

yes

—

—

—

—

—

C₈×Q₈

D₄

yes

characteristic

C₄×D₈

—

C₄×D₈

—

D₃₂

Q₈

yes

complemented

C₂×C₃₂

C₂³

yes

characteristic

C₂×C₃₂

C₂³

yes

—

C₂×C₃₂

C₂³

yes

—

—

C₂³

yes

—

#366

C₂³

yes

—

C₄⋊C₁₆

D₄

yes

characteristic

C₄⋊C₁₆

D₄

yes

—

#357

—

#357

—

C₄×C₁₆

C₂³

yes

characteristic

C₄×C₁₆

C₂³

yes

—

C₂×D₈

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

D₈

yes

characteristic

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

D₁₆

—

D₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂⁴

yes

—

—

—

—

C₄⋊C₈

D₈

yes

characteristic

C₄⋊C₈

D₈

yes

—

#102

—

#102

—

C₄×C₈

C₂×D₄

yes

—

C₄×C₈

C₂×D₄

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

C₂×Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

D₁₆

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

D₈

—

D₈

—

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂²×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₆

yes

characteristic

C₄⋊C₄

D₁₆

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×D₈

yes

—

C₄²

C₂×D₈

yes

characteristic

C₁₆

—

100

C₁₆

—

101

C₁₆

C₂²×Q₈

yes

characteristic

102

C₁₆

2₋¹⁺⁴

yes

characteristic

103

C₂³

—

104

Q₈

—

105

Q₈

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×D₁₆

yes

—

113

C₂×C₄

C₂×D₁₆

yes

characteristic

114

C₂×C₄

C₂×D₁₆

yes

—

115

C₂×C₄

C₂×D₁₆

yes

characteristic

116

C₂×C₄

C₂²×D₈

yes

characteristic

117

C₈

—

118

C₈

—

119

C₈

D₄×Q₈

yes

characteristic

120

C₈

#578

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂²×D₁₆

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

#2262

yes

—

131

C₄

Q₈×D₈

yes

characteristic

132

C₄

#3417

yes

characteristic

133

C₂

—

134

C₂

Q₈×D₁₆

yes

characteristic

135

C₂

C₂×#2262

yes

characteristic

136

C₂

#33982

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. #33947	C₂	semidirect
2. #13732	C₂	semidirect
3. #13731	C₂	semidirect
4. #13707	C₂	semidirect
5. #13705	C₂	semidirect
6. Q₈×C₃₂	C₂	semidirect
37. D₃₂	Q₈	semidirect

H ≅ G/N

type

1. #33947

C₂

semidirect

2. #13732

C₂

semidirect

3. #13731

C₂

semidirect

4. #13707

C₂

semidirect

5. #13705

C₂

semidirect

6. Q₈×C₃₂

C₂

semidirect

37. D₃₂

Q₈

semidirect

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

subgroup	index
#33947	2
#13732	2
#13731	2
#13707	2
#13705	2
Q₈×C₃₂	2
C₄×D₃₂	2

subgroup

index

Q₈×C₃₂

C₄×D₃₂

Subgroups — #513281

Conjugacy classes of subgroups

Complements

Maximal subgroups