Group #8280 — subgroups

Subgroups — C₈×#106 #8280

Order 256.

223 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 223 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₄×#106	C₂	1	1	yes	maximal, characteristic
2	C₈×C₄⋊C₄	C₂	1	2	yes	maximal
3	C₂×#334	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈²	C₂	1	1	yes	maximal, characteristic
5	#1279	C₂	1	2	yes	maximal
6	C₂×#106	C₄	1	2	yes	—
7	C₂×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄×C₈	—	2	2	no	—
9	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄×C₄⋊C₄	C₂²	1	2	yes	—
13	#340	C₄	1	2	yes	—
14	#336	—	2	2	no	—
15	#334	C₄	1	4	yes	complemented
16	C₈²	C₄	1	4	yes	complemented
17	C₂²×C₈	—	2	1	no	—
18	C₂²×C₈	—	2	2	no	—
19	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈	D₄	1	1	yes	characteristic
22	C₂²×C₈	D₄	1	1	yes	characteristic
23	C₂×C₄⋊C₄	C₂×C₄	1	4	yes	—
24	C₂×C₄²	—	2	2	no	—
25	C₂×C₄²	C₂³	1	1	yes	characteristic
26	#106	C₈	1	8	yes	direct product factor
27	C₄⋊C₈	—	2	2	no	—
28	C₄⋊C₈	C₂×C₄	1	4	yes	—
29	C₄×C₈	—	2	1	no	—
30	C₄×C₈	—	4	4	no	—
31	C₄×C₈	C₂×C₄	1	2	yes	—
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	Q₈	1	2	yes	—
36	#95	—	2	2	no	—
37	C₂²×C₄	—	2	4	no	—
38	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄	#41	1	1	yes	characteristic
41	C₂×C₈	—	2	4	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	—	2	4	no	—
46	C₂×C₈	—	4	2	no	—
47	C₂×C₈	—	4	2	no	—
48	C₂×C₈	C₄²	1	2	yes	—
49	C₂×C₈	C₄²	1	2	yes	—
50	C₂×C₈	C₄⋊C₄	1	2	yes	—
51	C₂×C₈	C₄⋊C₄	1	2	yes	—
52	C₂×C₈	C₂×C₈	1	4	yes	—
53	C₂×C₈	SD₁₆	1	4	yes	—
54	C₄⋊C₄	—	2	4	no	—
55	C₄⋊C₄	C₂×C₈	1	8	yes	—
56	C₄²	—	4	2	no	—
57	C₄²	—	4	2	no	—
58	C₄²	C₂²×C₄	1	1	yes	characteristic
59	C₄²	C₂²×C₄	1	1	yes	characteristic
60	C₄²	C₂×Q₈	1	1	yes	characteristic
61	C₄²	#41	1	1	yes	characteristic
62	C₂³	C₄×D₄	1	1	yes	characteristic
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	2	8	no	—
65	C₂×C₄	—	4	4	no	—
66	C₂×C₄	—	4	4	no	—
67	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
68	C₂×C₄	C₂×C₄⋊C₄	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×Q₈	1	1	yes	characteristic
72	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
73	C₂×C₄	C₂²×C₈	1	2	yes	—
74	C₂×C₄	#131	1	2	yes	—
75	C₂×C₄	C₂×SD₁₆	1	2	yes	—
76	C₂×C₄	#135	1	2	yes	—
77	C₈	—	2	4	no	—
78	C₈	—	2	2	no	—
79	C₈	—	2	2	no	—
80	C₈	—	4	4	no	—
81	C₈	C₄×C₈	1	4	yes	complemented
82	C₈	#106	1	4	yes	direct product factor
83	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
84	C₂²	C₈×D₄	1	2	yes	—
85	C₂²	C₄×SD₁₆	1	2	yes	—
86	C₂²	#443	1	2	yes	—
87	C₄	—	2	1	no	—
88	C₄	—	2	1	no	—
89	C₄	—	4	8	no	—
90	C₄	C₂×C₄×C₈	1	1	yes	characteristic
91	C₄	#405	1	1	yes	characteristic
92	C₄	C₂×#106	1	1	yes	characteristic
93	C₄	#427	1	1	yes	characteristic
94	C₄	C₄×SD₁₆	1	2	yes	—
95	C₄	C₈×Q₈	1	2	yes	—
96	C₂	C₈×SD₁₆	1	4	yes	—
97	C₂	C₈×C₄⋊C₄	1	1	yes	characteristic
98	C₂	#1772	1	1	yes	characteristic
99	C₂	C₄×#106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#106

C₂

yes

maximal, characteristic

C₈×C₄⋊C₄

C₂

yes

maximal

C₂×#334

C₂

yes

maximal, characteristic

C₂×C₈²

C₂

yes

maximal, characteristic

#1279

C₂

yes

maximal

C₂×#106

C₄

yes

—

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

—

#340

C₄

yes

—

#336

—

#334

C₄

yes

complemented

C₈²

C₄

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#106

C₈

yes

direct product factor

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

Q₈

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

SD₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₈

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

#135

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₄×C₈

yes

complemented

C₈

#106

yes

direct product factor

C₂²

C₄×C₄⋊C₄

yes

characteristic

C₂²

C₈×D₄

yes

—

C₂²

C₄×SD₁₆

yes

—

C₂²

#443

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₂×C₄×C₈

yes

C₄

yes

C₄

yes

C₄

yes

C₄

C₄×SD₁₆

yes

—

C₄

C₈×Q₈

yes

—

C₂

C₈×SD₁₆

yes

—

C₂

C₈×C₄⋊C₄

yes

C₂

yes

C₂

yes

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

subgroup

G/H

center

C₂²×C₈

D₄

commutator

C₄

C₂×C₄×C₈

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₄×D₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
15. #334	C₄	semidirect
16. C₈²	C₄	semidirect
26. #106	C₈	direct (N × H)
81. C₈	C₄×C₈	semidirect
82. C₈	#106	direct (N × H)

H ≅ G/N

type

15. #334

C₄

semidirect

16. C₈²

C₄

semidirect

26. #106

C₈

direct (N × H)

81. C₈

C₄×C₈

semidirect

82. C₈

#106

direct (N × H)

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

subgroup	index
C₄×#106	2
C₈×C₄⋊C₄	2
C₂×#334	2
C₂×C₈²	2
#1279	2

subgroup

index

C₄×#106

C₈×C₄⋊C₄

C₂×#334

C₂×C₈²

#1279

Subgroups — C₈×#106 #8280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×#106 #8280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#106 #8280