Group #9841 — subgroups

Subgroups — #9841

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×#498	C₂	1	1	yes	maximal, characteristic
2	#1990	C₂	1	2	yes	maximal
3	#1956	C₂	1	2	yes	maximal
4	#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#322	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
7	#498	—	2	2	no	—
8	#498	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	2	yes	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	#384	C₂²	1	1	yes	characteristic
15	#384	C₂²	1	1	yes	characteristic
16	#340	—	2	4	no	—
17	#322	—	2	2	no	—
18	C₂²×Q₈	—	2	2	no	—
19	C₂²×C₈	D₄	1	4	yes	—
20	C₂²×C₈	D₄	1	2	yes	—
21	C₄⋊Q₈	—	2	2	no	—
22	C₄⋊Q₈	—	2	2	no	—
23	C₄⋊Q₈	—	2	2	no	—
24	C₄⋊Q₈	C₂×C₄	1	4	yes	—
25	C₂×C₄⋊C₄	—	2	2	no	—
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	1	no	—
30	C₂×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄²	C₂³	1	1	yes	characteristic
32	#107	—	4	4	no	—
33	#106	—	2	8	no	—
34	#103	—	4	8	no	—
35	C₄×C₈	—	2	2	no	—
36	C₄×C₈	—	2	2	no	—
37	C₄×C₈	C₂×C₄	1	2	yes	—
38	C₄×C₈	D₄	1	2	yes	—
39	C₂×Q₈	—	2	4	no	—
40	C₂×Q₈	—	4	2	no	—
41	C₂×Q₈	—	4	4	no	—
42	C₂×Q₈	—	4	4	no	—
43	C₂²×C₄	—	4	2	no	—
44	C₂²×C₄	—	4	1	no	—
45	C₂²×C₄	—	4	1	no	—
46	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄	C₂×D₄	1	2	yes	—
48	C₂×C₈	—	2	4	no	—
49	C₂×C₈	—	2	4	no	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	C₂²⋊C₄	1	4	yes	—
52	C₂×C₈	SD₁₆	1	4	yes	—
53	C₂×C₈	Q₁₆	1	4	yes	—
54	C₄⋊C₄	—	2	4	no	—
55	C₄⋊C₄	—	2	4	no	—
56	C₄⋊C₄	—	4	2	no	—
57	C₄⋊C₄	—	4	2	no	—
58	C₄⋊C₄	—	4	2	no	—
59	C₄⋊C₄	—	4	2	no	—
60	C₄⋊C₄	—	4	4	no	—
61	C₄⋊C₄	—	4	4	no	—
62	C₄²	C₂²×C₄	1	1	yes	characteristic
63	C₄²	C₂×D₄	1	1	yes	characteristic
64	C₄²	#41	1	2	yes	—
65	C₂³	#127	1	1	yes	characteristic
66	Q₈	—	4	8	no	—
67	Q₈	—	8	4	no	—
68	C₂×C₄	—	4	4	no	—
69	C₂×C₄	—	8	2	no	—
70	C₂×C₄	—	8	2	no	—
71	C₂×C₄	—	8	1	no	—
72	C₂×C₄	—	8	1	no	—
73	C₂×C₄	—	8	2	no	—
74	C₂×C₄	—	8	2	no	—
75	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₄	C₄×D₄	1	2	yes	—
77	C₂×C₄	#121	1	2	yes	—
78	C₂×C₄	#124	1	1	yes	characteristic
79	C₂×C₄	C₂×SD₁₆	1	2	yes	—
80	C₂×C₄	C₂×Q₁₆	1	2	yes	—
81	C₂×C₄	AGL(1,ℤ/8)	1	4	yes	—
82	C₂×C₄	#137	1	4	yes	—
83	C₈	—	4	4	no	—
84	C₈	#103	1	8	yes	complemented
85	C₂²	#390	1	1	yes	characteristic
86	C₂²	#492	1	1	yes	characteristic
87	C₂²	#494	1	1	yes	characteristic
88	C₂²	#496	1	2	yes	—
89	C₂²	#497	1	2	yes	—
90	C₄	—	8	4	no	—
91	C₄	—	16	1	no	—
92	C₄	—	16	1	no	—
93	C₄	C₂×#103	1	2	yes	—
94	C₄	#440	1	4	yes	—
95	C₄	#468	1	4	yes	—
96	C₄	#487	1	2	yes	—
97	C₂	#1715	1	1	yes	characteristic
98	C₂	#1716	1	1	yes	characteristic
99	C₂	#1723	1	2	yes	—
100	C₂	#1956	1	1	yes	characteristic
101	C₂	#1973	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#498

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal

#1956

C₂

yes

maximal

#1845

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

#498

—

#498

C₄

yes

—

—

—

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#340

—

#322

—

C₂²×Q₈

—

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×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

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

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

—

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

—

C₄⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

#41

yes

—

C₂³

#127

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₄×D₄

yes

—

C₂×C₄

#121

yes

—

C₂×C₄

#124

yes

characteristic

C₂×C₄

C₂×SD₁₆

yes

—

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

AGL(1,ℤ/8)

yes

—

C₂×C₄

#137

yes

—

C₈

—

C₈

yes

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#497

yes

—

C₄

—

C₄

—

C₄

—

C₄

C₂×#103

yes

—

C₄

#440

yes

—

C₄

#468

yes

—

C₄

#487

yes

—

C₂

yes

C₂

yes

C₂

yes

—

100

C₂

#1956

yes

characteristic

101

C₂

#1973

yes

—

	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. #498	C₄	semidirect
84. C₈	#103	semidirect

H ≅ G/N

type

8. #498

C₄

semidirect

84. C₈

#103

semidirect

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

subgroup	index
C₂×#498	2
#1990	2
#1956	2
#1845	2
C₂×#322	2

subgroup

index

Subgroups — #9841

Conjugacy classes of subgroups

Complements

Maximal subgroups