Group #8143 — subgroups

Subgroups — C₈×C₂²⋊C₈ #8143

Order 256.

363 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 199 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	C₂²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₈²	C₂	1	2	yes	maximal, complemented
4	#1273	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈	C₄	1	2	yes	—
6	C₂²×C₄²	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊C₈	C₄	1	4	yes	—
8	C₂×C₄×C₈	—	2	4	no	—
9	C₂×C₄×C₈	C₄	1	2	yes	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#336	C₄	1	4	yes	—
14	C₈²	—	2	4	no	—
15	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄	C₂×C₄	1	2	yes	—
17	C₂²×C₈	—	2	4	no	—
18	C₂²×C₈	—	2	4	no	—
19	C₂²×C₈	C₈	1	8	yes	complemented
20	C₂²×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₈	C₂×C₄	1	8	yes	—
22	C₂²×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄²	—	2	2	no	—
24	C₂×C₄²	—	2	2	no	—
25	C₂×C₄²	C₂×C₄	1	2	yes	—
26	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	C₂²⋊C₈	C₈	1	16	yes	direct product factor
29	C₄×C₈	—	2	4	no	—
30	C₄×C₈	—	2	4	no	—
31	C₄×C₈	—	2	2	no	—
32	C₄×C₈	—	2	4	no	—
33	C₄×C₈	D₄	1	4	yes	—
34	C₂⁴	C₄²	1	1	yes	characteristic
35	C₂²×C₄	—	2	1	no	—
36	C₂²×C₄	—	2	1	no	—
37	C₂²×C₄	—	2	4	no	—
38	C₂²×C₄	—	2	2	no	—
39	C₂²×C₄	—	2	4	no	—
40	C₂²×C₄	C₄²	1	1	yes	characteristic
41	C₂²×C₄	C₄²	1	2	yes	—
42	C₂²×C₄	C₂×C₈	1	8	yes	—
43	C₂²×C₄	C₂×C₈	1	4	yes	—
44	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄	C₂²×C₄	1	2	yes	—
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	—	2	8	no	—
48	C₂×C₈	—	2	2	no	—
49	C₂×C₈	—	2	8	no	—
50	C₂×C₈	—	2	16	no	—
51	C₂×C₈	C₂²⋊C₄	1	4	yes	—
52	C₂×C₈	C₂×C₈	1	4	yes	—
53	C₂×C₈	C₂×C₈	1	4	yes	—
54	C₂×C₈	C₂×C₈	1	16	yes	complemented
55	C₂×C₈	M₁₆	1	4	yes	—
56	C₄²	—	2	1	no	—
57	C₄²	—	2	1	no	—
58	C₄²	—	2	4	no	—
59	C₄²	C₂×D₄	1	2	yes	—
60	C₄²	#41	1	2	yes	—
61	C₂³	—	2	1	no	—
62	C₂³	—	2	1	no	—
63	C₂³	—	2	2	no	—
64	C₂³	C₄×C₈	1	4	yes	—
65	C₂³	C₄×C₈	1	2	yes	—
66	C₂³	C₂×C₄²	1	1	yes	characteristic
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	2	4	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	2	16	no	—
74	C₂×C₄	C₄×C₈	1	2	yes	—
75	C₂×C₄	C₄×C₈	1	4	yes	—
76	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₄	#117	1	1	yes	characteristic
78	C₂×C₄	C₄×D₄	1	4	yes	—
79	C₂×C₄	C₂²×C₈	1	2	yes	—
80	C₂×C₄	C₂²×C₈	1	4	yes	—
81	C₂×C₄	C₂×M₁₆	1	2	yes	—
82	C₂×C₄	#131	1	4	yes	—
83	C₈	—	2	4	no	—
84	C₈	—	2	16	no	—
85	C₈	C₂²⋊C₈	1	8	yes	direct product factor
86	C₂²	—	2	4	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	2	2	no	—
89	C₂²	—	2	4	no	—
90	C₂²	C₈²	1	4	yes	complemented
91	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
92	C₂²	C₂×C₄×C₈	1	2	yes	—
93	C₂²	C₂×C₄×C₈	1	1	yes	characteristic
94	C₂²	C₄×M₁₆	1	1	yes	characteristic
95	C₂²	#405	1	2	yes	—
96	C₄	—	2	2	no	—
97	C₄	—	2	4	no	—
98	C₄	C₂×C₂²⋊C₈	1	2	yes	—
99	C₄	#431	1	2	yes	—
100	C₄	C₈×D₄	1	8	yes	—
101	C₂	—	2	4	no	—
102	C₂	C₂×C₈²	1	1	yes	characteristic
103	C₂	C₈×M₁₆	1	2	yes	—
104	C₂	#1450	1	1	yes	characteristic
105	C₂	C₄×C₂²⋊C₈	1	1	yes	characteristic
106	C₂	C₈×C₂²⋊C₄	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×C₈

C₂

yes

maximal, characteristic

C₄×C₂²⋊C₈

C₂

yes

maximal

C₂×C₈²

C₂

yes

maximal, complemented

#1273

C₂

yes

maximal, complemented

C₂³×C₈

C₄

yes

—

C₂²×C₄²

C₂²

yes

characteristic

C₂×C₂²⋊C₈

C₄

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#336

C₄

yes

—

C₈²

—

C₂³×C₄

C₂×C₄

yes

characteristic

C₂³×C₄

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₈

yes

complemented

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₂²⋊C₈

C₈

yes

direct product factor

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

—

C₂⁴

C₄²

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄²

yes

characteristic

C₂²×C₄

C₄²

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

yes

—

C₂×C₈

yes

complemented

C₂×C₈

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₄

yes

—

C₄²

#41

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₄×C₈

yes

—

C₂³

C₄×C₈

yes

—

C₂³

C₂×C₄²

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

C₄×C₈

yes

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

#131

yes

—

C₈

—

C₈

—

C₈

C₂²⋊C₈

yes

direct product factor

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₈²

yes

complemented

C₂²

C₄×C₂²⋊C₄

yes

characteristic

C₂²

C₂×C₄×C₈

yes

—

C₂²

C₂×C₄×C₈

yes

characteristic

C₂²

C₄×M₁₆

yes

characteristic

C₂²

#405

yes

—

C₄

—

C₄

—

C₄

C₂×C₂²⋊C₈

yes

—

C₄

#431

yes

—

100

C₄

C₈×D₄

yes

—

101

C₂

—

102

C₂

C₂×C₈²

yes

characteristic

103

C₂

C₈×M₁₆

yes

—

104

C₂

#1450

yes

characteristic

105

C₂

C₄×C₂²⋊C₈

yes

characteristic

106

C₂

C₈×C₂²⋊C₄

yes

—

	subgroup	G/H
center	C₂×C₄×C₈	C₂²
commutator	C₂	C₂×C₈²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂³	C₂×C₄²
prime radical	C₂⁴	C₄²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄×C₈

C₂²

commutator

C₂

C₂×C₈²

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂³

C₂×C₄²

prime radical

C₂⁴

C₄²

LCS

G ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. C₂×C₈²	C₂	semidirect
4. #1273	C₂	semidirect
19. C₂²×C₈	C₈	semidirect
28. C₂²⋊C₈	C₈	direct (N × H)
54. C₂×C₈	C₂×C₈	semidirect
85. C₈	C₂²⋊C₈	direct (N × H)
90. C₂²	C₈²	semidirect

H ≅ G/N

type

3. C₂×C₈²

C₂

semidirect

4. #1273

C₂

semidirect

19. C₂²×C₈

C₈

semidirect

28. C₂²⋊C₈

C₈

direct (N × H)

54. C₂×C₈

C₂×C₈

semidirect

85. C₈

C₂²⋊C₈

direct (N × H)

90. C₂²

C₈²

semidirect

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

subgroup	index
C₂²×C₄×C₈	2
C₄×C₂²⋊C₈	2
C₂×C₈²	2
#1273	2

subgroup

index

C₂²×C₄×C₈

C₄×C₂²⋊C₈

C₂×C₈²

#1273

Subgroups — C₈×C₂²⋊C₈ #8143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×C22⋊C8 #8143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₂²⋊C₈ #8143