Group #513064 — subgroups

Subgroups — #513064

Order 512.

304 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 162 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	#33729	C₂	1	2	yes	maximal
2	#13734	C₂	1	4	yes	maximal, complemented
3	#13733	C₂	1	2	yes	maximal, complemented
4	#13732	C₂	1	2	yes	maximal, complemented
5	#13634	C₂	1	2	yes	maximal
6	#13633	C₂	1	1	yes	maximal, characteristic
7	C₂×#1424	C₂	1	1	yes	maximal, characteristic
8	#13549	C₂	1	1	yes	maximal, characteristic
9	#3164	—	2	2	no	—
10	C₂×M₆₄	C₂²	1	1	yes	characteristic
11	C₂²×C₃₂	C₂²	1	1	yes	characteristic
12	#2255	—	2	4	no	—
13	#2253	C₂²	1	4	yes	—
14	#2252	C₂²	1	4	yes	—
15	#2159	—	2	2	no	—
16	#2158	C₂²	1	2	yes	—
17	C₂×#366	C₂²	1	2	yes	—
18	#2108	C₂²	1	1	yes	characteristic
19	#1424	C₂²	1	2	yes	—
20	#1424	C₂²	1	2	yes	—
21	#1424	C₂²	1	1	yes	characteristic
22	#1424	C₂²	1	1	yes	characteristic
23	#1424	C₂²	1	4	yes	—
24	#1423	C₂²	1	2	yes	—
25	#1423	C₂²	1	4	yes	—
26	#1397	C₂²	1	2	yes	—
27	C₄×C₃₂	C₂²	1	2	yes	—
28	#544	—	4	2	no	—
29	C₂×M₃₂	D₄	1	1	yes	characteristic
30	C₂²×C₁₆	C₂³	1	1	yes	characteristic
31	#501	—	4	4	no	—
32	#500	—	2	4	no	—
33	#499	—	2	4	no	—
34	#428	—	4	2	no	—
35	#427	—	2	2	no	—
36	C₂×#107	—	2	2	no	—
37	#405	D₄	1	1	yes	characteristic
38	M₆₄	Q₈	1	4	yes	complemented
39	C₂×C₃₂	Q₈	1	4	yes	complemented
40	C₂×C₃₂	C₂³	1	1	yes	characteristic
41	C₂×C₃₂	C₂³	1	2	yes	—
42	C₂×C₃₂	C₂³	1	1	yes	characteristic
43	#367	—	2	2	no	—
44	#367	—	2	4	no	—
45	#366	—	2	2	no	—
46	#366	—	2	4	no	—
47	#366	C₂³	1	4	yes	—
48	#366	C₂³	1	2	yes	—
49	#366	C₂³	1	2	yes	—
50	#346	D₄	1	2	yes	—
51	C₄×C₁₆	C₂³	1	2	yes	—
52	C₂×M₁₆	D₈	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
54	C₄⋊Q₈	—	4	4	no	—
55	C₄⋊Q₈	—	8	2	no	—
56	#125	—	4	4	no	—
57	#125	—	8	2	no	—
58	C₂²⋊Q₈	—	8	4	no	—
59	#117	—	4	2	no	—
60	#117	D₈	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	4	2	no	—
62	M₃₂	Q₁₆	1	4	yes	—
63	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
64	C₂×C₁₆	C₂×D₄	1	2	yes	—
65	C₂×C₁₆	C₂×Q₈	1	2	yes	—
66	C₂×C₁₆	C₂×Q₈	1	2	yes	—
67	C₂×C₁₆	C₂⁴	1	1	yes	characteristic
68	#107	—	2	4	no	—
69	#107	—	2	2	no	—
70	#107	—	2	2	no	—
71	#107	—	4	2	no	—
72	#107	—	4	4	no	—
73	#106	—	4	2	no	—
74	#106	—	4	4	no	—
75	#97	D₈	1	2	yes	—
76	C₄×C₈	C₂×D₄	1	2	yes	—
77	C₃₂	C₂×Q₈	1	2	yes	—
78	C₃₂	C₂×Q₈	1	2	yes	—
79	C₃₂	C₂×Q₈	1	4	yes	—
80	C₂×Q₈	—	8	4	no	—
81	C₂²×C₄	—	8	2	no	—
82	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
83	M₁₆	Q₃₂	1	4	yes	—
84	C₂×C₈	C₄⋊Q₈	1	2	yes	—
85	C₂×C₈	C₂×D₈	1	1	yes	characteristic
86	C₂×C₈	C₂×D₈	1	2	yes	—
87	C₂×C₈	C₂×Q₁₆	1	2	yes	—
88	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	8	2	no	—
93	C₄⋊C₄	—	8	4	no	—
94	C₄⋊C₄	—	8	4	no	—
95	C₄⋊C₄	—	8	4	no	—
96	C₄⋊C₄	D₁₆	1	2	yes	—
97	C₂²⋊C₄	—	8	2	no	—
98	C₂²⋊C₄	D₁₆	1	2	yes	—
99	C₄²	—	8	2	no	—
100	C₄²	C₂×D₈	1	2	yes	—
101	C₁₆	C₄⋊Q₈	1	2	yes	—
102	C₁₆	C₂×Q₁₆	1	4	yes	—
103	C₁₆	C₂²×Q₈	1	1	yes	characteristic
104	C₁₆	C₂²×Q₈	1	1	yes	characteristic
105	C₂³	C₂×D₁₆	1	1	yes	characteristic
106	Q₈	—	32	4	no	—
107	C₂×C₄	—	8	2	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	8	4	no	—
110	C₂×C₄	—	32	2	no	—
111	C₂×C₄	#500	1	2	yes	—
112	C₂×C₄	C₂×D₁₆	1	4	yes	—
113	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
114	C₂×C₄	C₂×Q₃₂	1	2	yes	—
115	C₂×C₄	C₂²×D₈	1	1	yes	characteristic
116	C₈	#500	1	2	yes	—
117	C₈	C₂×Q₃₂	1	4	yes	—
118	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
119	C₈	C₂²×Q₁₆	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#2252	1	2	yes	—
122	C₂²	C₂²×D₁₆	1	1	yes	characteristic
123	C₄	—	2	4	no	—
124	C₄	—	16	2	no	—
125	C₄	—	16	4	no	—
126	C₄	—	16	2	no	—
127	C₄	#2252	1	2	yes	—
128	C₄	C₂×#500	1	1	yes	characteristic
129	C₄	C₂²×Q₃₂	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×#2252	1	1	yes	characteristic
132	C₂	#33983	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#33729

C₂

yes

maximal

#13734

C₂

yes

maximal, complemented

#13733

C₂

yes

maximal, complemented

#13732

C₂

yes

maximal, complemented

#13634

C₂

yes

maximal

#13633

C₂

yes

maximal, characteristic

C₂×#1424

C₂

yes

maximal, characteristic

#13549

C₂

yes

maximal, characteristic

#3164

—

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

#2255

—

#2253

C₂²

yes

—

#2252

C₂²

yes

—

#2159

—

#2158

C₂²

yes

—

C₂×#366

C₂²

yes

—

#2108

C₂²

yes

characteristic

#1424

C₂²

yes

—

#1424

C₂²

yes

—

#1424

C₂²

yes

characteristic

#1424

C₂²

yes

characteristic

#1424

C₂²

yes

—

#1423

C₂²

yes

—

#1423

C₂²

yes

—

#1397

C₂²

yes

—

C₄×C₃₂

C₂²

yes

—

#544

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

—

—

—

D₄

yes

characteristic

M₆₄

Q₈

yes

complemented

C₂×C₃₂

Q₈

yes

complemented

C₂×C₃₂

C₂³

yes

characteristic

C₂×C₃₂

C₂³

yes

—

C₂×C₃₂

C₂³

yes

—

—

—

—

C₂³

yes

—

#366

C₂³

yes

—

#366

C₂³

yes

—

#346

D₄

yes

—

C₄×C₁₆

C₂³

yes

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

#117

—

#117

D₈

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂×Q₈

yes

—

C₂×C₁₆

C₂⁴

yes

—

—

—

—

—

—

—

D₈

yes

—

C₄×C₈

C₂×D₄

yes

—

C₃₂

C₂×Q₈

yes

—

C₃₂

C₂×Q₈

yes

—

C₃₂

C₂×Q₈

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

Q₃₂

yes

—

C₂×C₈

C₄⋊Q₈

yes

—

C₂×C₈

C₂×D₈

yes

characteristic

C₂×C₈

C₂×D₈

yes

—

C₂×C₈

C₂×Q₁₆

yes

—

C₂×C₈

C₂²×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₆

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

D₁₆

yes

—

C₄²

—

100

C₄²

C₂×D₈

yes

—

101

C₁₆

C₄⋊Q₈

yes

—

102

C₁₆

C₂×Q₁₆

yes

—

103

C₁₆

C₂²×Q₈

yes

characteristic

104

C₁₆

C₂²×Q₈

yes

characteristic

105

C₂³

C₂×D₁₆

yes

characteristic

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#500

yes

—

112

C₂×C₄

C₂×D₁₆

yes

—

113

C₂×C₄

C₂×D₁₆

yes

characteristic

114

C₂×C₄

C₂×Q₃₂

yes

—

115

C₂×C₄

C₂²×D₈

yes

characteristic

116

C₈

#500

yes

—

117

C₈

C₂×Q₃₂

yes

—

118

C₈

C₂×C₄⋊Q₈

yes

characteristic

119

C₈

C₂²×Q₁₆

yes

characteristic

120

C₂²

—

121

C₂²

#2252

yes

—

122

C₂²

C₂²×D₁₆

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#2252

yes

—

128

C₄

C₂×#500

yes

characteristic

129

C₄

C₂²×Q₃₂

yes

characteristic

130

C₂

—

131

C₂

C₂×#2252

yes

characteristic

132

C₂

#33983

yes

—

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ 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

C₂³

C₂×D₁₆

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #13734	C₂	semidirect
3. #13733	C₂	semidirect
4. #13732	C₂	semidirect
38. M₆₄	Q₈	semidirect
39. C₂×C₃₂	Q₈	semidirect

H ≅ G/N

type

2. #13734

C₂

semidirect

3. #13733

C₂

semidirect

4. #13732

C₂

semidirect

38. M₆₄

Q₈

semidirect

39. C₂×C₃₂

Q₈

semidirect

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

subgroup	index
#33729	2
#13734	2
#13733	2
#13732	2
#13634	2
#13633	2
C₂×#1424	2
#13549	2

subgroup

index

Subgroups — #513064

Conjugacy classes of subgroups

Complements

Maximal subgroups