Group #8808 — subgroups

Subgroups — #8808

Order 256.

209 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 123 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	#1842	C₂	1	1	yes	maximal, characteristic
2	#1831	C₂	1	1	yes	maximal, characteristic
3	C₄×#97	C₂	1	1	yes	maximal, characteristic
4	C₂×#335	C₂	1	2	yes	maximal
5	C₂×#334	C₂	1	2	yes	maximal
6	C₂×C₄⋊C₈	—	2	2	no	—
7	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×#97	C₄	1	2	yes	—
11	C₂×#97	C₄	1	1	yes	characteristic
12	C₂×#97	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	2	yes	—
14	C₄³	C₂²	1	1	yes	characteristic
15	#336	—	2	2	no	—
16	#335	—	2	4	no	—
17	#334	—	2	4	no	—
18	C₂²×C₈	—	2	1	no	—
19	C₂²×C₈	—	2	1	no	—
20	C₂²×C₈	—	2	2	no	—
21	C₂²×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²	D₄	1	2	yes	—
26	C₂×C₄²	Q₈	1	2	yes	—
27	C₂×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊C₈	—	2	2	no	—
29	C₄⋊C₈	—	2	2	no	—
30	C₄⋊C₈	—	2	2	no	—
31	C₄⋊C₈	—	2	2	no	—
32	C₄⋊C₈	—	4	4	no	—
33	#97	—	2	4	no	—
34	#97	—	2	2	no	—
35	#97	C₈	1	4	yes	complemented
36	C₄×C₈	D₄	1	4	yes	—
37	C₄×C₈	Q₈	1	4	yes	—
38	C₂²×C₄	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₈	—	2	2	no	—
44	C₂×C₈	—	2	2	no	—
45	C₂×C₈	—	4	2	no	—
46	C₂×C₈	—	4	1	no	—
47	C₂×C₈	—	4	1	no	—
48	C₂×C₈	—	4	4	no	—
49	C₂×C₈	—	4	2	no	—
50	C₂×C₈	—	4	2	no	—
51	C₂×C₈	C₄⋊C₄	1	4	yes	—
52	C₂×C₈	C₄⋊C₄	1	4	yes	—
53	C₂×C₈	C₂×C₈	1	4	yes	—
54	C₂×C₈	M₁₆	1	4	yes	—
55	C₄²	—	2	2	no	—
56	C₄²	—	2	2	no	—
57	C₄²	—	2	2	no	—
58	C₄²	—	4	2	no	—
59	C₄²	C₂×C₈	1	2	yes	—
60	C₄²	M₁₆	1	2	yes	—
61	C₄²	C₂×D₄	1	2	yes	—
62	C₄²	C₂×Q₈	1	2	yes	—
63	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	2	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	2	no	—
70	C₂×C₄	C₄⋊C₈	1	4	yes	—
71	C₂×C₄	C₂×C₄⋊C₄	1	2	yes	—
72	C₂×C₄	#127	1	1	yes	characteristic
73	C₂×C₄	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄	C₂×M₁₆	1	2	yes	—
77	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
79	C₂×C₄	#137	1	2	yes	—
80	C₈	—	2	4	no	—
81	C₈	—	8	2	no	—
82	C₈	—	8	1	no	—
83	C₈	—	8	1	no	—
84	C₈	C₄⋊C₈	1	8	yes	complemented
85	C₂²	#384	1	1	yes	characteristic
86	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
87	C₂²	#423	1	1	yes	characteristic
88	C₂²	#428	1	2	yes	—
89	C₂²	#430	1	2	yes	—
90	C₄	—	2	4	no	—
91	C₄	—	2	2	no	—
92	C₄	—	4	2	no	—
93	C₄	C₂×C₄⋊C₈	1	2	yes	—
94	C₄	#423	1	2	yes	—
95	C₄	#496	1	1	yes	characteristic
96	C₄	#497	1	1	yes	characteristic
97	C₄	#501	1	2	yes	—
98	C₂	#1565	1	2	yes	—
99	C₂	#1569	1	2	yes	—
100	C₂	#1842	1	1	yes	characteristic
101	C₂	#1847	1	1	yes	characteristic
102	C₂	#1849	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1842

C₂

yes

maximal, characteristic

#1831

C₂

yes

maximal, characteristic

C₄×#97

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal

C₂×#334

C₂

yes

maximal

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

C₄

yes

—

C₂×#97

C₄

yes

characteristic

C₂×#97

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₄³

C₂²

yes

—

—

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

—

C₂×C₄²

Q₈

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

C₈

yes

complemented

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

yes

—

C₂²×C₄

C₄⋊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

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₈

yes

—

C₄²

M₁₆

yes

—

C₄²

C₂×D₄

yes

—

C₄²

C₂×Q₈

yes

—

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

C₂×C₄⋊C₄

yes

—

C₂×C₄

#127

yes

characteristic

C₂×C₄

C₄⋊Q₈

yes

—

C₂×C₄

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₄⋊C₈

yes

complemented

C₂²

#384

yes

characteristic

C₂²

C₂×C₄⋊C₈

yes

C₂²

yes

C₂²

yes

—

C₂²

#430

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₂×C₄⋊C₈

yes

—

C₄

#423

yes

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

#1565

yes

—

C₂

#1569

yes

—

100

C₂

#1842

yes

characteristic

101

C₂

#1847

yes

characteristic

102

C₂

#1849

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₈
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

C₂²×C₈

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
35. #97	C₈	semidirect
84. C₈	C₄⋊C₈	semidirect

H ≅ G/N

type

35. #97

C₈

semidirect

84. C₈

C₄⋊C₈

semidirect

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

subgroup	index
#1842	2
#1831	2
C₄×#97	2
C₂×#335	2
C₂×#334	2

subgroup

index

C₄×#97

Subgroups — #8808

Conjugacy classes of subgroups

Complements

Maximal subgroups