Group #9836 — subgroups

Subgroups — #9836

Order 256.

255 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 255 conjugacy classes of subgroups — too many to draw. Only its 107 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	C₂×#500	C₂	1	1	yes	maximal, characteristic
2	#1990	C₂	1	2	yes	maximal
3	#1956	C₂	1	2	yes	maximal
4	#1844	C₂	1	1	yes	maximal, characteristic
5	C₂×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
7	#500	—	2	2	no	—
8	#500	C₄	1	4	yes	complemented
9	C₂×#107	—	2	2	no	—
10	C₂×#106	—	2	2	no	—
11	C₂×#103	—	2	4	no	—
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	#384	C₂²	1	2	yes	—
16	#340	—	2	4	no	—
17	#322	—	2	2	no	—
18	C₂²×Q₈	—	2	2	no	—
19	C₂²×C₈	D₄	1	2	yes	—
20	C₂²×C₈	D₄	1	2	yes	—
21	C₂²×C₈	D₄	1	2	yes	—
22	C₄⋊Q₈	—	2	2	no	—
23	C₄⋊Q₈	—	2	2	no	—
24	C₄⋊Q₈	—	2	2	no	—
25	C₄⋊Q₈	C₂×C₄	1	4	yes	—
26	C₂×C₄⋊C₄	—	2	2	no	—
27	C₂×C₄⋊C₄	—	2	2	no	—
28	C₂×C₄⋊C₄	—	2	2	no	—
29	C₂×C₄⋊C₄	—	2	2	no	—
30	C₂×C₄⋊C₄	—	2	2	no	—
31	C₂×C₄²	C₂³	1	1	yes	characteristic
32	#107	—	2	8	no	—
33	#106	—	4	4	no	—
34	#103	—	4	8	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	C₂×C₄	1	2	yes	—
40	C₄×C₈	D₄	1	2	yes	—
41	C₂×Q₈	—	2	4	no	—
42	C₂×Q₈	—	4	2	no	—
43	C₂×Q₈	—	4	4	no	—
44	C₂×Q₈	—	4	4	no	—
45	C₂²×C₄	—	4	2	no	—
46	C₂²×C₄	—	4	2	no	—
47	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂×C₈	—	2	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	2	4	no	—
53	C₂×C₈	—	2	4	no	—
54	C₂×C₈	C₂²⋊C₄	1	4	yes	—
55	C₂×C₈	SD₁₆	1	4	yes	—
56	C₂×C₈	Q₁₆	1	4	yes	—
57	C₄⋊C₄	—	2	4	no	—
58	C₄⋊C₄	—	2	4	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄⋊C₄	—	4	4	no	—
62	C₄⋊C₄	—	4	4	no	—
63	C₄⋊C₄	—	4	4	no	—
64	C₄²	C₂²×C₄	1	1	yes	characteristic
65	C₄²	C₂×D₄	1	1	yes	characteristic
66	C₄²	#41	1	1	yes	characteristic
67	C₄²	#41	1	1	yes	characteristic
68	C₂³	#127	1	1	yes	characteristic
69	Q₈	—	4	8	no	—
70	Q₈	—	8	4	no	—
71	C₂×C₄	—	4	4	no	—
72	C₂×C₄	—	8	2	no	—
73	C₂×C₄	—	8	2	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	—	8	2	no	—
76	C₂×C₄	—	8	2	no	—
77	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄	#121	1	1	yes	characteristic
81	C₂×C₄	#121	1	1	yes	characteristic
82	C₂×C₄	#124	1	1	yes	characteristic
83	C₂×C₄	C₂×SD₁₆	1	2	yes	—
84	C₂×C₄	C₂×Q₁₆	1	2	yes	—
85	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
86	C₂×C₄	#137	1	4	yes	—
87	C₈	—	4	2	no	—
88	C₈	—	4	2	no	—
89	C₈	#103	1	8	yes	complemented
90	C₂²	#390	1	1	yes	characteristic
91	C₂²	#492	1	1	yes	characteristic
92	C₂²	#494	1	1	yes	characteristic
93	C₂²	#496	1	2	yes	—
94	C₂²	#497	1	2	yes	—
95	C₄	—	8	4	no	—
96	C₄	—	16	2	no	—
97	C₄	C₂×#103	1	2	yes	—
98	C₄	#441	1	2	yes	—
99	C₄	#442	1	2	yes	—
100	C₄	#469	1	2	yes	—
101	C₄	#470	1	2	yes	—
102	C₄	#487	1	2	yes	—
103	C₂	#1714	1	2	yes	—
104	C₂	#1723	1	2	yes	—
105	C₂	#1956	1	1	yes	characteristic
106	C₂	#1971	1	1	yes	characteristic
107	C₂	#1972	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#500

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal

#1956

C₂

yes

maximal

#1844

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

#500

—

#500

C₄

yes

—

—

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

—

#340

—

#322

—

C₂²×Q₈

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

—

C₈

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#497

yes

—

C₄

—

C₄

—

C₄

C₂×#103

yes

—

C₄

#441

yes

—

C₄

#442

yes

—

100

C₄

#469

yes

—

101

C₄

#470

yes

—

102

C₄

#487

yes

—

103

C₂

#1714

yes

—

104

C₂

#1723

yes

—

105

C₂

#1956

yes

characteristic

106

C₂

#1971

yes

characteristic

107

C₂

#1972

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³

#127

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
8. #500	C₄	semidirect
89. C₈	#103	semidirect

H ≅ G/N

type

8. #500

C₄

semidirect

89. C₈

#103

semidirect

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

subgroup	index
C₂×#500	2
#1990	2
#1956	2
#1844	2
C₂×#322	2

subgroup

index

Subgroups — #9836

Conjugacy classes of subgroups

Complements

Maximal subgroups