Group #8270 — subgroups

Subgroups — #8270

Order 256.

171 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 171 conjugacy classes of subgroups — too many to draw. Only its 91 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	#1744	C₂	1	1	yes	maximal, characteristic
2	#1564	C₂	1	1	yes	maximal, characteristic
3	#1564	C₂	1	1	yes	maximal, characteristic
4	#1452	C₂	1	1	yes	maximal, characteristic
5	#1450	C₂	1	1	yes	maximal, characteristic
6	#1276	C₂	1	1	yes	maximal, characteristic
7	#1276	C₂	1	1	yes	maximal, characteristic
8	#431	C₂²	1	1	yes	characteristic
9	#431	C₂²	1	1	yes	characteristic
10	#431	C₂²	1	1	yes	characteristic
11	#431	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₄³	C₂²	1	1	yes	characteristic
15	#336	C₄	1	2	yes	—
16	#336	C₄	1	2	yes	—
17	#335	C₄	1	2	yes	complemented
18	#335	C₄	1	2	yes	complemented
19	#334	C₄	1	2	yes	complemented
20	#334	C₄	1	2	yes	complemented
21	#322	—	2	1	no	—
22	#322	—	2	1	no	—
23	#322	—	2	1	no	—
24	C₈²	—	2	1	no	—
25	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	2	yes	—
28	C₂×C₄²	—	2	1	no	—
29	C₂×C₄²	—	2	1	no	—
30	C₂×C₄²	C₂×C₄	1	2	yes	—
31	C₂×C₄²	C₂³	1	1	yes	characteristic
32	C₄⋊C₈	C₂×C₄	1	2	yes	—
33	C₄⋊C₈	C₂×C₄	1	2	yes	—
34	C₄⋊C₈	C₂×C₄	1	2	yes	—
35	C₄⋊C₈	C₂×C₄	1	2	yes	—
36	C₂²⋊C₈	—	2	1	no	—
37	C₂²⋊C₈	—	2	1	no	—
38	C₂²⋊C₈	—	2	1	no	—
39	C₂²⋊C₈	—	2	1	no	—
40	C₄×C₈	—	2	1	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈	D₄	1	1	yes	characteristic
49	C₄×C₈	D₄	1	1	yes	characteristic
50	C₄×C₈	Q₈	1	2	yes	—
51	C₂²×C₄	—	2	2	no	—
52	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄	#41	1	1	yes	characteristic
54	C₂²×C₄	#41	1	1	yes	characteristic
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	2	no	—
58	C₂×C₈	—	2	2	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	C₄²	1	1	yes	characteristic
64	C₂×C₈	C₄²	1	1	yes	characteristic
65	C₂×C₈	C₄²	1	1	yes	characteristic
66	C₂×C₈	C₄²	1	1	yes	characteristic
67	C₂×C₈	C₄⋊C₄	1	2	yes	—
68	C₂×C₈	C₄⋊C₄	1	2	yes	—
69	C₄²	—	2	4	no	—
70	C₄²	—	2	2	no	—
71	C₄²	—	2	2	no	—
72	C₄²	—	4	2	no	—
73	C₄²	C₂²×C₄	1	1	yes	characteristic
74	C₄²	C₂²×C₄	1	1	yes	characteristic
75	C₄²	C₂×D₄	1	1	yes	characteristic
76	C₄²	C₂×Q₈	1	1	yes	characteristic
77	C₂³	#117	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄	C₄×D₄	1	1	yes	characteristic
87	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
88	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
89	C₂×C₄	#131	1	2	yes	—
90	C₂×C₄	#131	1	2	yes	—
91	C₈	—	2	1	no	—
92	C₈	—	2	1	no	—
93	C₈	—	2	1	no	—
94	C₈	—	2	1	no	—
95	C₈	—	2	2	no	—
96	C₈	—	2	2	no	—
97	C₈	—	4	2	no	—
98	C₈	—	4	2	no	—
99	C₈	—	4	2	no	—
100	C₈	—	4	2	no	—
101	C₂²	—	2	1	no	—
102	C₂²	—	2	1	no	—
103	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
104	C₂²	#433	1	2	yes	—
105	C₄	—	2	1	no	—
106	C₄	—	2	1	no	—
107	C₄	—	4	4	no	—
108	C₄	#405	1	1	yes	characteristic
109	C₄	#405	1	1	yes	characteristic
110	C₄	#424	1	2	yes	—
111	C₄	#443	1	2	yes	—
112	C₄	#444	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	#1768	1	1	yes	characteristic
115	C₂	#1772	1	1	yes	characteristic
116	C₂	#1773	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1744

C₂

yes

maximal, characteristic

#1564

C₂

yes

maximal, characteristic

#1564

C₂

yes

maximal, characteristic

#1452

C₂

yes

maximal, characteristic

#1450

C₂

yes

maximal, characteristic

#1276

C₂

yes

maximal, characteristic

#1276

C₂

yes

maximal, characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#336

C₄

yes

—

#336

C₄

yes

—

#335

C₄

yes

complemented

#335

C₄

yes

complemented

#334

C₄

yes

complemented

#334

C₄

yes

—

—

—

C₈²

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₄²

yes

characteristic

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₂³

#117

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

#131

yes

—

C₂×C₄

#131

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₄×C₄⋊C₄

yes

characteristic

104

C₂²

#433

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#405

yes

characteristic

109

C₄

#405

yes

characteristic

110

C₄

#424

yes

—

111

C₄

#443

yes

—

112

C₄

#444

yes

—

113

C₂

—

114

C₂

#1768

yes

characteristic

115

C₂

#1772

yes

characteristic

116

C₂

#1773

yes

characteristic

	subgroup	G/H
center	C₄²	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₄×C₄⋊C₄
prime radical	C₂³	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₄²

C₂×D₄

commutator

C₂×C₄

C₂×C₄²

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

C₄×C₄⋊C₄

C₂³

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
17. #335	C₄	semidirect
18. #335	C₄	semidirect
19. #334	C₄	semidirect
20. #334	C₄	semidirect

H ≅ G/N

type

17. #335

C₄

semidirect

18. #335

C₄

semidirect

19. #334

C₄

semidirect

20. #334

C₄

semidirect

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

subgroup	index
#1744	2
#1564	2
#1452	2
#1450	2
#1276	2

subgroup

index

Subgroups — #8270

Conjugacy classes of subgroups

Complements

Maximal subgroups