Group #152857 — subgroups

Subgroups — #152857

Order 512.

176 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 86 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×#1425	C₂	1	1	yes	maximal, characteristic
2	C₂×#1423	C₂	1	1	yes	maximal, characteristic
3	#13549	C₂	1	1	yes	maximal, characteristic
4	#13171	C₂	1	2	yes	maximal
5	#7478	C₂	1	2	yes	maximal
6	C₂×M₆₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃₂	C₂²	1	1	yes	characteristic
8	C₂×#368	C₂²	1	2	yes	—
9	C₂×#366	C₂²	1	2	yes	—
10	#2108	C₂²	1	1	yes	characteristic
11	#1952	—	2	2	no	—
12	#1425	C₄	1	4	yes	complemented
13	#1423	—	2	1	no	—
14	#1423	—	2	1	no	—
15	#1397	—	2	1	no	—
16	C₄×C₃₂	—	2	1	no	—
17	#1387	—	2	2	no	—
18	C₂×M₃₂	D₄	1	1	yes	characteristic
19	C₂²×C₁₆	C₂³	1	1	yes	characteristic
20	C₂×#108	—	2	2	no	—
21	#428	—	4	2	no	—
22	C₂×#107	—	2	2	no	—
23	#405	D₄	1	1	yes	characteristic
24	M₆₄	—	2	1	no	—
25	M₆₄	Q₈	1	2	yes	—
26	C₂×C₃₂	—	2	1	no	—
27	C₂×C₃₂	C₂×C₄	1	2	yes	—
28	C₂×C₃₂	D₄	1	1	yes	characteristic
29	C₂×C₃₂	D₄	1	1	yes	characteristic
30	C₂×C₃₂	Q₈	1	2	yes	—
31	#368	—	2	2	no	—
32	#368	C₂×C₄	1	4	yes	—
33	#366	—	2	2	no	—
34	#366	—	2	2	no	—
35	#346	—	2	1	no	—
36	C₄×C₁₆	—	2	1	no	—
37	#343	—	4	2	no	—
38	C₂×M₁₆	—	4	2	no	—
39	C₂×M₁₆	D₈	1	1	yes	characteristic
40	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
41	#117	D₈	1	1	yes	characteristic
42	C₂×C₄⋊C₄	—	4	2	no	—
43	M₃₂	—	2	1	no	—
44	M₃₂	Q₁₆	1	2	yes	—
45	C₂×C₁₆	—	2	1	no	—
46	C₂×C₁₆	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₁₆	C₂×D₄	1	1	yes	characteristic
48	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₁₆	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₁₆	#41	1	1	yes	characteristic
51	C₂×C₁₆	#41	1	1	yes	characteristic
52	#108	—	2	4	no	—
53	#108	—	4	2	no	—
54	#107	—	4	2	no	—
55	#107	—	4	2	no	—
56	#107	—	8	2	no	—
57	#106	—	8	2	no	—
58	#97	—	2	1	no	—
59	C₄×C₈	—	2	1	no	—
60	C₃₂	—	2	2	no	—
61	C₃₂	C₄⋊C₄	1	2	yes	—
62	C₃₂	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄	—	8	2	no	—
64	C₂²×C₄	C₂×D₈	1	1	yes	characteristic
65	M₁₆	—	2	1	no	—
66	M₁₆	—	4	4	no	—
67	M₁₆	—	8	2	no	—
68	M₁₆	Q₃₂	1	2	yes	—
69	C₂×C₈	—	2	1	no	—
70	C₂×C₈	—	8	2	no	—
71	C₂×C₈	C₄×D₄	1	1	yes	characteristic
72	C₂×C₈	#121	1	1	yes	characteristic
73	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₈	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₈	#135	1	1	yes	characteristic
76	C₂×C₈	#135	1	1	yes	characteristic
77	C₄⋊C₄	—	8	2	no	—
78	C₄⋊C₄	—	8	2	no	—
79	C₄⋊C₄	D₁₆	1	1	yes	characteristic
80	C₄⋊C₄	D₁₆	1	1	yes	characteristic
81	C₂²⋊C₄	—	2	1	no	—
82	C₄²	—	2	1	no	—
83	C₁₆	—	2	2	no	—
84	C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₁₆	C₄×Q₈	1	1	yes	characteristic
86	C₁₆	C₂²⋊Q₈	1	1	yes	characteristic
87	C₁₆	#125	1	1	yes	characteristic
88	C₂³	#508	1	1	yes	characteristic
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	16	2	no	—
92	C₂×C₄	—	32	2	no	—
93	C₂×C₄	C₄×D₈	1	1	yes	characteristic
94	C₂×C₄	#466	1	1	yes	characteristic
95	C₂×C₄	#500	1	1	yes	characteristic
96	C₂×C₄	C₂×D₁₆	1	1	yes	characteristic
97	C₂×C₄	C₂×Q₃₂	1	1	yes	characteristic
98	C₂×C₄	#508	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	8	4	no	—
101	C₈	#389	1	1	yes	characteristic
102	C₈	C₄×Q₁₆	1	1	yes	characteristic
103	C₈	#467	1	1	yes	characteristic
104	C₈	#499	1	1	yes	characteristic
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	C₄×Q₃₂	1	1	yes	characteristic
108	C₂²	#2215	1	1	yes	characteristic
109	C₂²	#2253	1	1	yes	characteristic
110	C₄	—	4	1	no	—
111	C₄	—	4	1	no	—
112	C₄	—	32	2	no	—
113	C₄	#1942	1	1	yes	characteristic
114	C₄	C₄×D₁₆	1	1	yes	characteristic
115	C₄	#2216	1	1	yes	characteristic
116	C₄	#2252	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	#13165	1	1	yes	characteristic
120	C₂	#13700	1	1	yes	characteristic
121	C₂	#13701	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#1425

C₂

yes

maximal, characteristic

C₂×#1423

C₂

yes

maximal, characteristic

#13549

C₂

yes

maximal, characteristic

#13171

C₂

yes

maximal

#7478

C₂

yes

maximal

C₂×M₆₄

C₂²

yes

characteristic

C₂²×C₃₂

C₂²

yes

characteristic

C₂×#368

C₂²

yes

—

C₂×#366

C₂²

yes

—

#2108

C₂²

yes

characteristic

#1952

—

#1425

C₄

yes

—

—

—

C₄×C₃₂

—

#1387

—

C₂×M₃₂

D₄

yes

characteristic

C₂²×C₁₆

C₂³

yes

—

—

—

D₄

yes

characteristic

M₆₄

—

M₆₄

Q₈

yes

—

C₂×C₃₂

—

C₂×C₃₂

C₂×C₄

yes

—

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

D₄

yes

characteristic

C₂×C₃₂

Q₈

yes

—

#368

—

#368

C₂×C₄

yes

—

#366

—

#366

—

#346

—

C₄×C₁₆

—

#343

—

C₂×M₁₆

—

C₂×M₁₆

D₈

yes

characteristic

C₂²×C₈

C₂×D₄

yes

characteristic

#117

D₈

yes

characteristic

C₂×C₄⋊C₄

—

M₃₂

—

M₃₂

Q₁₆

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×C₄

yes

characteristic

C₂×C₁₆

C₂×D₄

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

characteristic

C₂×C₁₆

C₂×Q₈

yes

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₃₂

—

C₃₂

C₄⋊C₄

yes

—

C₃₂

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

Q₃₂

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₄

yes

characteristic

C₂×C₈

#121

yes

characteristic

C₂×C₈

C₄⋊Q₈

yes

characteristic

C₂×C₈

C₂×Q₁₆

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₆

yes

characteristic

C₄⋊C₄

D₁₆

yes

characteristic

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

C₂×C₄⋊C₄

yes

characteristic

C₁₆

C₄×Q₈

yes

characteristic

C₁₆

C₂²⋊Q₈

yes

C₁₆

yes

C₂³

yes

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×D₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×D₁₆

yes

characteristic

C₂×C₄

C₂×Q₃₂

yes

characteristic

C₂×C₄

#508

yes

characteristic

C₈

—

100

C₈

—

101

C₈

#389

yes

characteristic

102

C₈

C₄×Q₁₆

yes

characteristic

103

C₈

#467

yes

characteristic

104

C₈

#499

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

C₄×Q₃₂

yes

characteristic

108

C₂²

#2215

yes

characteristic

109

C₂²

#2253

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#1942

yes

characteristic

114

C₄

C₄×D₁₆

yes

characteristic

115

C₄

#2216

yes

characteristic

116

C₄

#2252

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

#13165

yes

characteristic

120

C₂

#13700

yes

characteristic

121

C₂

#13701

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#2215
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#508
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle

C₂²

#2215

commutator

C₂×C₁₆

C₂²×C₄

Frattini, cosocle

C₂²×C₁₆

C₂³

prime radical

C₂³

#508

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G

LECS

G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
12. #1425	C₄	semidirect

H ≅ G/N

type

12. #1425

C₄

semidirect

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

subgroup	index
C₂×#1425	2
C₂×#1423	2
#13549	2
#13171	2
#7478	2

subgroup

index

Subgroups — #152857

Conjugacy classes of subgroups

Complements

Maximal subgroups