Group #8807 — subgroups

Subgroups — #8807

Order 256.

211 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 211 conjugacy classes of subgroups — too many to draw. Only its 127 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	#1831	C₂	1	2	yes	maximal
2	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#335	C₂	1	2	yes	maximal
4	C₂×#334	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₄³	C₂²	1	1	yes	characteristic
13	#336	—	2	4	no	—
14	#335	—	2	4	no	—
15	#334	—	2	4	no	—
16	C₂²×C₈	—	2	4	no	—
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
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	4	no	—
27	C₄⋊C₈	—	2	4	no	—
28	C₄×C₈	—	2	2	no	—
29	C₄×C₈	—	2	2	no	—
30	C₄×C₈	—	2	2	no	—
31	C₄×C₈	C₈	1	4	yes	complemented
32	C₄×C₈	D₄	1	2	yes	—
33	C₄×C₈	D₄	1	2	yes	—
34	C₄×C₈	Q₈	1	2	yes	—
35	C₄×C₈	Q₈	1	2	yes	—
36	C₂²×C₄	C₄⋊C₄	1	2	yes	—
37	C₂²×C₄	C₄⋊C₄	1	2	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	1	no	—
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	—	2	1	no	—
45	C₂×C₈	—	4	4	no	—
46	C₂×C₈	—	4	8	no	—
47	C₂×C₈	C₄⋊C₄	1	2	yes	—
48	C₂×C₈	C₄⋊C₄	1	2	yes	—
49	C₂×C₈	C₄⋊C₄	1	2	yes	—
50	C₂×C₈	C₄⋊C₄	1	2	yes	—
51	C₂×C₈	C₂×C₈	1	2	yes	—
52	C₂×C₈	C₂×C₈	1	2	yes	—
53	C₂×C₈	M₁₆	1	2	yes	—
54	C₂×C₈	M₁₆	1	2	yes	—
55	C₄²	—	2	4	no	—
56	C₄²	—	2	1	no	—
57	C₄²	—	2	1	no	—
58	C₄²	—	2	4	no	—
59	C₄²	C₂×C₈	1	2	yes	—
60	C₄²	M₁₆	1	2	yes	—
61	C₄²	C₂×D₄	1	1	yes	characteristic
62	C₄²	C₂×D₄	1	1	yes	characteristic
63	C₄²	C₂×Q₈	1	1	yes	characteristic
64	C₄²	C₂×Q₈	1	1	yes	characteristic
65	C₂³	C₂×C₄⋊C₄	1	1	yes	characteristic
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	—	2	2	no	—
71	C₂×C₄	C₄⋊C₈	1	4	yes	—
72	C₂×C₄	#108	1	4	yes	—
73	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄	#127	1	1	yes	characteristic
76	C₂×C₄	C₄⋊Q₈	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	C₂²×C₈	1	1	yes	characteristic
80	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄	#135	1	4	yes	—
84	C₈	—	2	2	no	—
85	C₈	—	2	2	no	—
86	C₈	—	8	4	no	—
87	C₈	C₄⋊C₈	1	4	yes	—
88	C₈	C₄⋊C₈	1	4	yes	—
89	C₂²	#384	1	1	yes	characteristic
90	C₂²	C₂×C₄⋊C₈	1	1	yes	characteristic
91	C₂²	#423	1	1	yes	characteristic
92	C₂²	#427	1	2	yes	—
93	C₂²	C₂×#108	1	2	yes	—
94	C₄	—	2	4	no	—
95	C₄	—	2	4	no	—
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
99	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
100	C₄	#423	1	1	yes	characteristic
101	C₄	#423	1	1	yes	characteristic
102	C₄	#495	1	2	yes	—
103	C₄	#499	1	2	yes	—
104	C₂	#1564	1	2	yes	—
105	C₂	#1568	1	2	yes	—
106	C₂	#1842	1	1	yes	characteristic
107	C₂	#1846	1	1	yes	characteristic
108	C₂	#1848	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1831

C₂

yes

maximal

C₄²×C₈

C₂

yes

maximal, characteristic

C₂×#335

C₂

yes

maximal

C₂×#334

C₂

yes

maximal

C₂×C₄⋊C₈

C₂²

yes

—

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

—

—

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

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

—

C₄×C₈

C₈

yes

complemented

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

Q₈

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

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

yes

—

C₂×C₈

yes

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₈

yes

—

C₄²

M₁₆

yes

—

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₂³

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊C₈

yes

—

C₂×C₄

#108

yes

—

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

C₄⋊Q₈

yes

characteristic

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

characteristic

C₂×C₄

#135

yes

—

C₈

—

C₈

—

C₈

—

C₈

C₄⋊C₈

yes

—

C₈

C₄⋊C₈

yes

—

C₂²

#384

yes

characteristic

C₂²

C₂×C₄⋊C₈

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×#108

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×C₄⋊C₈

yes

characteristic

C₄

C₂×C₄⋊C₈

yes

characteristic

100

C₄

#423

yes

characteristic

101

C₄

#423

yes

characteristic

102

C₄

#495

yes

—

103

C₄

#499

yes

—

104

C₂

#1564

yes

—

105

C₂

#1568

yes

—

106

C₂

#1842

yes

characteristic

107

C₂

#1846

yes

characteristic

108

C₂

#1848

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
31. C₄×C₈	C₈	semidirect

H ≅ G/N

type

31. C₄×C₈

C₈

semidirect

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

subgroup	index
#1831	2
C₄²×C₈	2
C₂×#335	2
C₂×#334	2

subgroup

index

#1831

C₄²×C₈

C₂×#335

C₂×#334

Subgroups — #8807

Conjugacy classes of subgroups

Complements

Maximal subgroups