Group #9845 — subgroups

Subgroups — #9845

Order 256.

239 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 79 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	#3170	C₂	1	1	yes	maximal, characteristic
2	#1994	C₂	1	2	yes	maximal
3	C₄²×C₈	C₂	1	1	yes	maximal, characteristic
4	#1567	C₂	1	1	yes	maximal, characteristic
5	#1493	C₂	1	2	yes	maximal
6	#533	C₂²	1	2	yes	—
7	#500	C₄	1	1	yes	characteristic, complemented
8	#494	C₄	1	1	yes	characteristic, complemented
9	#487	C₄	1	2	yes	complemented
10	#467	—	2	2	no	—
11	C₄×Q₁₆	—	2	2	no	—
12	#423	C₂²	1	2	yes	—
13	C₂×C₄×C₈	—	2	2	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄³	C₂²	1	1	yes	characteristic
17	#339	—	2	4	no	—
18	#335	—	2	2	no	—
19	Q₈⋊C₈	—	2	4	no	—
20	C₂×Q₁₆	—	2	2	no	—
21	C₂×M₁₆	—	2	2	no	—
22	C₂²×C₈	—	2	1	no	—
23	C₂²×C₈	D₄	1	2	yes	—
24	C₄⋊Q₈	C₂×C₄	1	2	yes	—
25	C₄⋊Q₈	C₂×C₄	1	2	yes	—
26	#125	—	2	2	no	—
27	C₂²⋊Q₈	—	2	2	no	—
28	C₂²⋊Q₈	—	2	2	no	—
29	C₄×Q₈	—	2	2	no	—
30	C₄×Q₈	—	2	2	no	—
31	#117	—	2	2	no	—
32	C₂×C₄²	—	2	1	no	—
33	C₂×C₄²	—	2	1	no	—
34	C₂×C₄²	D₄	1	2	yes	—
35	C₂×C₄²	C₂³	1	1	yes	characteristic
36	#107	—	2	2	no	—
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	#103	—	2	4	no	—
40	C₄×C₈	—	2	8	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈	D₄	1	2	yes	—
45	C₄×C₈	D₄	1	2	yes	—
46	C₂×Q₈	—	2	2	no	—
47	C₂×Q₈	—	2	2	no	—
48	C₂²×C₄	—	2	2	no	—
49	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄	#41	1	1	yes	characteristic
52	Q₁₆	—	4	4	no	—
53	M₁₆	—	4	4	no	—
54	C₂×C₈	—	2	4	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	4	no	—
57	C₂×C₈	—	4	2	no	—
58	C₂×C₈	C₂²⋊C₄	1	2	yes	—
59	C₂×C₈	C₂²⋊C₄	1	2	yes	—
60	C₄⋊C₄	—	2	2	no	—
61	C₄⋊C₄	—	2	2	no	—
62	C₄⋊C₄	—	2	2	no	—
63	C₄⋊C₄	—	2	2	no	—
64	C₄⋊C₄	—	4	2	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₂²⋊C₄	—	4	2	no	—
67	C₄²	—	2	4	no	—
68	C₄²	—	2	4	no	—
69	C₄²	—	2	2	no	—
70	C₄²	—	4	2	no	—
71	C₄²	Q₁₆	1	4	yes	complemented
72	C₄²	C₂²×C₄	1	1	yes	characteristic
73	C₄²	C₂×D₄	1	1	yes	characteristic
74	C₄²	C₂×D₄	1	1	yes	characteristic
75	C₄²	#41	1	1	yes	characteristic
76	C₂³	#121	1	1	yes	characteristic
77	Q₈	—	4	4	no	—
78	Q₈	—	4	4	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	4	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄	C₄×D₄	1	1	yes	characteristic
88	C₂×C₄	C₄×D₄	1	1	yes	characteristic
89	C₂×C₄	#121	1	1	yes	characteristic
90	C₂×C₄	#124	1	1	yes	characteristic
91	C₂×C₄	#127	1	1	yes	characteristic
92	C₂×C₄	C₂×Q₁₆	1	2	yes	—
93	C₂×C₄	#135	1	2	yes	—
94	C₈	—	2	4	no	—
95	C₈	—	2	2	no	—
96	C₈	—	8	2	no	—
97	C₈	#104	1	4	yes	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#390	1	1	yes	characteristic
101	C₂²	#467	1	2	yes	—
102	C₄	—	2	8	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	—	8	2	no	—
106	C₄	—	8	2	no	—
107	C₄	C₂×#104	1	2	yes	—
108	C₄	C₄×Q₁₆	1	2	yes	—
109	C₄	#443	1	2	yes	—
110	C₄	#494	1	1	yes	characteristic
111	C₄	#495	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#1962	1	1	yes	characteristic
114	C₂	#1969	1	1	yes	characteristic
115	C₂	#1974	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3170

C₂

yes

maximal, characteristic

#1994

C₂

yes

maximal

C₄²×C₈

C₂

yes

maximal, characteristic

#1567

C₂

yes

maximal, characteristic

#1493

C₂

yes

maximal

#533

C₂²

yes

—

#500

C₄

yes

characteristic, complemented

#494

C₄

yes

characteristic, complemented

#487

C₄

yes

complemented

#467

—

C₄×Q₁₆

—

#423

C₂²

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄³

C₂²

yes

characteristic

#339

—

#335

—

Q₈⋊C₈

—

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

D₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Q₁₆

yes

complemented

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₂³

yes

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

characteristic

C₂×C₄

C₄×D₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×Q₁₆

yes

—

C₂×C₄

#135

yes

—

C₈

—

C₈

—

C₈

—

C₈

#104

yes

—

C₂²

—

C₂²

—

100

C₂²

#390

yes

characteristic

101

C₂²

#467

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

C₂×#104

yes

—

108

C₄

C₄×Q₁₆

yes

—

109

C₄

#443

yes

—

110

C₄

#494

yes

characteristic

111

C₄

#495

yes

characteristic

112

C₂

—

113

C₂

#1962

yes

characteristic

114

C₂

#1969

yes

characteristic

115

C₂

#1974

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#390
prime radical	C₂³	#121
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

C₂²

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
7. #500	C₄	semidirect
8. #494	C₄	semidirect
9. #487	C₄	semidirect
71. C₄²	Q₁₆	semidirect

H ≅ G/N

type

7. #500

C₄

semidirect

8. #494

C₄

semidirect

9. #487

C₄

semidirect

71. C₄²

Q₁₆

semidirect

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

subgroup	index
#3170	2
#1994	2
C₄²×C₈	2
#1567	2
#1493	2

subgroup

index

C₄²×C₈

Subgroups — #9845

Conjugacy classes of subgroups

Complements

Maximal subgroups