Group #8828 — subgroups

Subgroups — #8828

Order 256.

161 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 87 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	#1848	C₂	1	1	yes	maximal, characteristic
2	#1837	C₂	1	1	yes	maximal, characteristic
3	#1567	C₂	1	1	yes	maximal, characteristic
4	#1567	C₂	1	1	yes	maximal, characteristic
5	#1566	C₂	1	1	yes	maximal, characteristic
6	#1566	C₂	1	1	yes	maximal, characteristic
7	C₂×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×#108	—	2	2	no	—
9	#423	C₂²	1	1	yes	characteristic
10	#423	C₂²	1	1	yes	characteristic
11	#423	C₂²	1	1	yes	characteristic
12	#423	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#341	—	2	2	no	—
17	#335	—	2	2	no	—
18	#335	—	2	2	no	—
19	#334	—	2	2	no	—
20	#334	—	2	2	no	—
21	#322	C₄	1	2	yes	—
22	#322	C₄	1	2	yes	—
23	C₂×M₁₆	—	2	1	no	—
24	C₂×M₁₆	—	2	1	no	—
25	C₂×M₁₆	—	2	1	no	—
26	C₂×M₁₆	—	2	1	no	—
27	C₂²×C₈	D₄	1	2	yes	—
28	C₂²×C₈	Q₈	1	2	yes	—
29	C₂²×C₈	Q₈	1	2	yes	—
30	C₂×C₄²	C₂³	1	1	yes	characteristic
31	#108	—	4	4	no	—
32	C₄⋊C₈	—	2	1	no	—
33	C₄⋊C₈	—	2	1	no	—
34	C₄⋊C₈	—	2	1	no	—
35	C₄⋊C₈	—	2	1	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄×C₈	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈	D₄	1	2	yes	—
45	C₄×C₈	D₄	1	2	yes	—
46	C₄×C₈	Q₈	1	2	yes	—
47	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
50	M₁₆	—	4	2	no	—
51	M₁₆	—	4	2	no	—
52	M₁₆	—	4	2	no	—
53	M₁₆	—	4	2	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	4	no	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	—	4	1	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	—	4	1	no	—
61	C₂×C₈	—	4	1	no	—
62	C₂×C₈	C₄⋊C₄	1	2	yes	—
63	C₂×C₈	C₄⋊C₄	1	4	yes	—
64	C₂×C₈	C₄⋊C₄	1	4	yes	—
65	C₂×C₈	C₄⋊C₄	1	2	yes	—
66	C₄²	C₂²×C₄	1	1	yes	characteristic
67	C₄²	C₂×D₄	1	1	yes	characteristic
68	C₄²	C₂×D₄	1	1	yes	characteristic
69	C₄²	C₂×Q₈	1	1	yes	characteristic
70	C₂³	C₄⋊Q₈	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₂×C₄	#127	1	1	yes	characteristic
77	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
78	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
79	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
80	C₂×C₄	#137	1	2	yes	—
81	C₈	—	2	4	no	—
82	C₈	—	2	2	no	—
83	C₈	—	2	4	no	—
84	C₈	—	8	1	no	—
85	C₈	—	8	1	no	—
86	C₈	—	8	1	no	—
87	C₈	—	8	1	no	—
88	C₈	#108	1	4	yes	complemented
89	C₂²	—	2	1	no	—
90	C₂²	—	2	1	no	—
91	C₂²	#384	1	1	yes	characteristic
92	C₂²	#501	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	#428	1	2	yes	—
96	C₄	C₂×#108	1	2	yes	—
97	C₄	#430	1	2	yes	—
98	C₄	#496	1	1	yes	characteristic
99	C₄	#497	1	1	yes	characteristic
100	C₂	—	2	2	no	—
101	C₂	#1847	1	1	yes	characteristic
102	C₂	#1848	1	1	yes	characteristic
103	C₂	#1849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1848

C₂

yes

maximal, characteristic

#1837

C₂

yes

maximal, characteristic

#1567

C₂

yes

maximal, characteristic

#1567

C₂

yes

maximal, characteristic

#1566

C₂

yes

maximal, characteristic

#1566

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

C₂×#108

—

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

#423

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

—

—

C₄

yes

—

#322

C₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

Q₈

yes

—

C₂²×C₈

Q₈

yes

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₂³

C₄⋊Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

#127

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#108

yes

complemented

C₂²

—

C₂²

—

C₂²

#384

yes

characteristic

C₂²

#501

yes

—

C₄

—

C₄

—

C₄

#428

yes

—

C₄

C₂×#108

yes

—

C₄

#430

yes

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

#1847

yes

characteristic

102

C₂

#1848

yes

characteristic

103

C₂

#1849

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#384
prime radical	C₂³	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#127

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#384

prime radical

C₂³

C₄⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
88. C₈	#108	semidirect

H ≅ G/N

type

88. C₈

#108

semidirect

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

subgroup	index
#1848	2
#1837	2
#1567	2
#1566	2
C₂×#322	2

subgroup

index

Subgroups — #8828

Conjugacy classes of subgroups

Complements

Maximal subgroups