Group #8276 — subgroups

Subgroups — #8276

Order 256.

207 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 207 conjugacy classes of subgroups — too many to draw. Only its 115 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	#1767	C₂	1	2	yes	maximal
2	C₄×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	1	yes	maximal, characteristic
4	C₂×#334	C₂	1	1	yes	maximal, characteristic
5	#1277	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×#97	—	2	2	no	—
9	C₂×#97	C₄	1	2	yes	—
10	C₂×#97	C₄	1	2	yes	—
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₄³	C₂²	1	1	yes	characteristic
14	#336	—	2	2	no	—
15	#335	C₄	1	4	yes	complemented
16	#334	C₄	1	4	yes	complemented
17	C₂²×C₈	—	2	2	no	—
18	C₂²×C₈	—	2	2	no	—
19	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²	C₂×C₄	1	2	yes	—
23	C₂×C₄²	D₄	1	2	yes	—
24	C₂×C₄²	Q₈	1	2	yes	—
25	C₂×C₄²	C₂³	1	1	yes	characteristic
26	C₄⋊C₈	—	2	2	no	—
27	C₄⋊C₈	—	2	2	no	—
28	C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₄⋊C₈	C₂×C₄	1	4	yes	—
30	#97	—	2	4	no	—
31	#97	—	2	4	no	—
32	#97	—	4	4	no	—
33	C₄×C₈	—	2	1	no	—
34	C₄×C₈	—	2	1	no	—
35	C₄×C₈	C₂×C₄	1	2	yes	—
36	C₄×C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₄	C₄⋊C₄	1	4	yes	—
38	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₈	—	2	2	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	4	no	—
45	C₂×C₈	—	2	4	no	—
46	C₂×C₈	—	4	2	no	—
47	C₂×C₈	—	4	2	no	—
48	C₂×C₈	—	4	2	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	C₄²	1	2	yes	—
51	C₂×C₈	C₄²	1	2	yes	—
52	C₂×C₈	M₁₆	1	4	yes	—
53	C₂×C₈	M₁₆	1	4	yes	—
54	C₄²	—	2	2	no	—
55	C₄²	—	2	2	no	—
56	C₄²	—	2	2	no	—
57	C₄²	—	2	2	no	—
58	C₄²	—	4	2	no	—
59	C₄²	C₂²×C₄	1	1	yes	characteristic
60	C₄²	C₂²×C₄	1	1	yes	characteristic
61	C₄²	#41	1	1	yes	characteristic
62	C₄²	#41	1	1	yes	characteristic
63	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	4	no	—
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
69	C₂×C₄	#117	1	1	yes	characteristic
70	C₂×C₄	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄	C₄×D₄	1	1	yes	characteristic
72	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
74	C₂×C₄	C₂×M₁₆	1	2	yes	—
75	C₂×C₄	C₂×M₁₆	1	2	yes	—
76	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
77	C₂×C₄	#137	1	2	yes	—
78	C₈	—	2	2	no	—
79	C₈	—	2	4	no	—
80	C₈	—	4	4	no	—
81	C₈	—	4	4	no	—
82	C₈	#97	1	4	yes	complemented
83	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²	#423	1	2	yes	—
85	C₂²	#428	1	2	yes	—
86	C₂²	#430	1	2	yes	—
87	C₄	—	2	1	no	—
88	C₄	—	2	1	no	—
89	C₄	—	4	4	no	—
90	C₄	C₂×#97	1	1	yes	characteristic
91	C₄	C₄×M₁₆	1	1	yes	characteristic
92	C₄	#432	1	2	yes	—
93	C₄	#440	1	2	yes	—
94	C₄	#441	1	1	yes	characteristic
95	C₄	#442	1	1	yes	characteristic
96	C₂	#1569	1	4	yes	—
97	C₂	#1767	1	1	yes	characteristic
98	C₂	#1776	1	1	yes	characteristic
99	C₂	#1778	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1767

C₂

yes

maximal

C₄×#97

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal, characteristic

C₂×#334

C₂

yes

maximal, characteristic

#1277

C₂

yes

maximal

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×#97

—

C₂×#97

C₄

yes

—

C₂×#97

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#336

—

#335

C₄

yes

complemented

#334

C₄

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

Q₈

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₂²×C₄

C₄⋊C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₄²

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

C₄²

yes

C₄²

yes

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

#117

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

C₂×M₁₆

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#97

yes

complemented

C₂²

C₄×C₄⋊C₄

yes

characteristic

C₂²

#423

yes

—

C₂²

#428

yes

—

C₂²

#430

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₂×#97

yes

characteristic

C₄

C₄×M₁₆

yes

characteristic

C₄

#432

yes

—

C₄

#440

yes

—

C₄

yes

C₄

yes

C₂

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂×C₄²

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₂×C₄⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
15. #335	C₄	semidirect
16. #334	C₄	semidirect
82. C₈	#97	semidirect

H ≅ G/N

type

15. #335

C₄

semidirect

16. #334

C₄

semidirect

82. C₈

#97

semidirect

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

subgroup	index
#1767	2
C₄×#97	2
C₂×#335	2
C₂×#334	2
#1277	2

subgroup

index

C₄×#97

Subgroups — #8276

Conjugacy classes of subgroups

Complements

Maximal subgroups