Group #13048365 — subgroups

Subgroups — 3₋¹⁺²×GL(2,𝔽₃) #13048365

Order 1296.

161 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 161 conjugacy classes of subgroups — too many to draw. Only its 35 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	SL(2,𝔽₃)×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×GL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₉×GL(2,𝔽₃)	C₃	1	3	yes	maximal, complemented
4	SD₁₆×3₋¹⁺²	—	3	1	no	maximal
5	C₂×D₃×3₋¹⁺²	—	4	1	no	maximal
6	C₃²×SL(2,𝔽₃)	C₂×C₃	1	1	yes	characteristic
7	Q₈×3₋¹⁺²	D₃	1	1	yes	characteristic, complemented
8	D₄×3₋¹⁺²	—	3	1	no	—
9	#6204	—	2	1	no	—
10	C₉⋊SL(2,𝔽₃)	—	2	3	no	—
11	C₉×SL(2,𝔽₃)	C₂×C₃	1	3	yes	complemented
12	C₈×3₋¹⁺²	—	3	1	no	—
13	C₂×C₃×3₋¹⁺²	—	4	1	no	—
14	D₃×3₋¹⁺²	—	4	2	no	—
15	C₃×GL(2,𝔽₃)	—	3	1	no	—
16	C₃×GL(2,𝔽₃)	C₃²	1	1	yes	characteristic
17	C₃²×SD₁₆	—	3	1	no	—
18	C₉×SD₁₆	—	3	3	no	—
19	C₂×C₃²×D₃	—	4	1	no	—
20	C₂²×3₋¹⁺²	—	6	1	no	—
21	C₂×D₃×C₉	—	4	3	no	—
22	C₄×3₋¹⁺²	—	3	1	no	—
23	C₃×3₋¹⁺²	—	4	1	no	Sylow
24	C₃²×Q₈	C₃×D₃	1	1	yes	characteristic
25	C₃²×D₄	—	3	1	no	—
26	C₃×SL(2,𝔽₃)	—	2	1	no	—
27	C₃×SL(2,𝔽₃)	—	3	1	no	—
28	C₃×SL(2,𝔽₃)	—	6	1	no	—
29	C₃×SL(2,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
30	C₃²×C₈	—	3	1	no	—
31	Q₈×C₉	C₃×D₃	1	3	yes	complemented
32	D₄×C₉	—	3	3	no	—
33	Q₈⋊C₉	—	2	3	no	—
34	C₈×C₉	—	3	3	no	—
35	C₂×C₃³	—	4	1	no	—
36	C₃²×D₃	—	4	2	no	—
37	C₂×3₋¹⁺²	—	8	1	no	—
38	C₂×3₋¹⁺²	—	8	3	no	—
39	C₂×3₋¹⁺²	—	12	1	no	—
40	C₂×3₋¹⁺²	S₄	1	1	yes	characteristic
41	C₂×C₃×C₉	—	4	3	no	—
42	D₃×C₉	—	4	6	no	—
43	GL(2,𝔽₃)	3₋¹⁺²	1	1	yes	characteristic, direct product factor
44	C₃×SD₁₆	—	3	1	no	—
45	C₃×SD₁₆	—	9	1	no	—
46	C₂²×C₃²	—	6	1	no	—
47	C₂×C₃×D₃	—	4	1	no	—
48	C₂×C₃×D₃	—	12	1	no	—
49	C₃²×C₄	—	3	1	no	—
50	C₂²×C₉	—	6	3	no	—
51	C₄×C₉	—	3	3	no	—
52	C₃³	—	4	1	no	—
53	3₋¹⁺²	—	8	1	no	—
54	3₋¹⁺²	—	8	3	no	—
55	3₋¹⁺²	GL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
56	C₃×C₉	—	4	3	no	—
57	C₃×Q₈	—	3	1	no	—
58	C₃×Q₈	C₃²×D₃	1	1	yes	characteristic
59	C₃×D₄	—	3	1	no	—
60	C₃×D₄	—	9	1	no	—
61	SL(2,𝔽₃)	—	2	1	no	—
62	SL(2,𝔽₃)	—	6	1	no	—
63	SL(2,𝔽₃)	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
64	C₃×C₈	—	3	1	no	—
65	C₃×C₈	—	9	1	no	—
66	C₂×C₃²	—	4	1	no	—
67	C₂×C₃²	—	8	1	no	—
68	C₂×C₃²	—	12	1	no	—
69	C₂×C₃²	—	12	1	no	—
70	C₂×C₃²	—	24	1	no	—
71	C₂×C₃²	C₃×S₄	1	1	yes	characteristic
72	C₃×D₃	—	4	2	no	—
73	C₃×D₃	—	12	2	no	—
74	C₂×C₉	—	8	3	no	—
75	C₂×C₉	—	12	3	no	—
76	C₂×C₉	C₃×S₄	1	3	yes	—
77	SD₁₆	—	3	1	no	Sylow
78	C₂²×C₃	—	6	1	no	—
79	C₂²×C₃	—	18	1	no	—
80	C₂×D₃	—	4	1	no	—
81	C₃×C₄	—	3	1	no	—
82	C₃×C₄	—	9	1	no	—
83	C₃²	—	4	1	no	—
84	C₃²	—	8	1	no	—
85	C₃²	—	12	1	no	—
86	C₃²	—	24	1	no	—
87	C₃²	C₃×GL(2,𝔽₃)	1	1	yes	characteristic
88	C₉	—	8	3	no	—
89	C₉	C₃×GL(2,𝔽₃)	1	3	yes	complemented
90	Q₈	D₃×3₋¹⁺²	1	1	yes	characteristic, complemented
91	D₄	—	3	1	no	—
92	C₈	—	3	1	no	—
93	C₂×C₃	—	3	1	no	—
94	C₂×C₃	—	4	1	no	—
95	C₂×C₃	—	8	1	no	—
96	C₂×C₃	—	12	1	no	—
97	C₂×C₃	—	24	1	no	—
98	C₂×C₃	—	36	1	no	—
99	C₂×C₃	C₃²×S₄	1	1	yes	characteristic
100	D₃	—	4	2	no	—
101	C₂²	—	6	1	no	—
102	C₄	—	3	1	no	—
103	C₃	—	3	1	no	—
104	C₃	—	4	1	no	—
105	C₃	—	8	1	no	—
106	C₃	—	24	1	no	—
107	C₃	C₃²×GL(2,𝔽₃)	1	1	yes	characteristic
108	C₂	—	12	1	no	—
109	C₂	S₄×3₋¹⁺²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

SL(2,𝔽₃)×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

C₃²×GL(2,𝔽₃)

C₃

yes

maximal, characteristic

C₉×GL(2,𝔽₃)

C₃

yes

maximal, complemented

SD₁₆×3₋¹⁺²

—

maximal

C₂×D₃×3₋¹⁺²

—

maximal

C₃²×SL(2,𝔽₃)

C₂×C₃

yes

characteristic

Q₈×3₋¹⁺²

D₃

yes

characteristic, complemented

D₄×3₋¹⁺²

—

#6204

—

C₉⋊SL(2,𝔽₃)

—

C₉×SL(2,𝔽₃)

C₂×C₃

yes

complemented

C₈×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

D₃×3₋¹⁺²

—

C₃×GL(2,𝔽₃)

—

C₃×GL(2,𝔽₃)

C₃²

yes

characteristic

C₃²×SD₁₆

—

C₉×SD₁₆

—

C₂×C₃²×D₃

—

C₂²×3₋¹⁺²

—

C₂×D₃×C₉

—

C₄×3₋¹⁺²

—

C₃×3₋¹⁺²

—

Sylow

C₃²×Q₈

C₃×D₃

yes

characteristic

C₃²×D₄

—

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

—

C₃×SL(2,𝔽₃)

C₂×C₃²

yes

characteristic

C₃²×C₈

—

Q₈×C₉

C₃×D₃

yes

complemented

D₄×C₉

—

Q₈⋊C₉

—

C₈×C₉

—

C₂×C₃³

—

C₃²×D₃

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

S₄

yes

characteristic

C₂×C₃×C₉

—

D₃×C₉

—

GL(2,𝔽₃)

3₋¹⁺²

yes

characteristic, direct product factor

C₃×SD₁₆

—

C₃×SD₁₆

—

C₂²×C₃²

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₂²×C₉

—

C₄×C₉

—

C₃³

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

GL(2,𝔽₃)

yes

characteristic, direct product factor

C₃×C₉

—

C₃×Q₈

—

C₃×Q₈

C₃²×D₃

yes

characteristic

C₃×D₄

—

C₃×D₄

—

SL(2,𝔽₃)

—

SL(2,𝔽₃)

—

SL(2,𝔽₃)

C₂×3₋¹⁺²

yes

characteristic, complemented

C₃×C₈

—

C₃×C₈

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₃×S₄

yes

characteristic

C₃×D₃

—

C₃×D₃

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₃×S₄

yes

—

SD₁₆

—

Sylow

C₂²×C₃

—

C₂²×C₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

C₃×GL(2,𝔽₃)

yes

characteristic

C₉

—

C₉

C₃×GL(2,𝔽₃)

yes

complemented

Q₈

D₃×3₋¹⁺²

yes

characteristic, complemented

D₄

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₃²×S₄

yes

characteristic

100

D₃

—

101

C₂²

—

102

C₄

—

103

C₃

—

104

C₃

—

105

C₃

—

106

C₃

—

107

C₃

C₃²×GL(2,𝔽₃)

yes

characteristic

108

C₂

—

109

C₂

S₄×3₋¹⁺²

yes

characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₃²×S₄
commutator, cosocle	C₃×SL(2,𝔽₃)	C₂×C₃²
Fitting	Q₈×3₋¹⁺²	D₃
prime radical	C₃²×GL(2,𝔽₃)	C₃
hypercenter	C₂×3₋¹⁺²	S₄
hypocenter	SL(2,𝔽₃)	C₂×3₋¹⁺²
LCS	G ▷ C₃×SL(2,𝔽₃) ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₂×3₋¹⁺²

subgroup

G/H

center, Frattini, socle

C₂×C₃

C₃²×S₄

commutator, cosocle

C₃×SL(2,𝔽₃)

C₂×C₃²

Fitting

Q₈×3₋¹⁺²

D₃

prime radical

C₃²×GL(2,𝔽₃)

C₃

hypercenter

C₂×3₋¹⁺²

S₄

hypocenter

SL(2,𝔽₃)

C₂×3₋¹⁺²

LCS

G ▷ C₃×SL(2,𝔽₃) ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂×C₃ ◁ C₂×3₋¹⁺²

N	H ≅ G/N	type
1. SL(2,𝔽₃)×3₋¹⁺²	C₂	semidirect
3. C₉×GL(2,𝔽₃)	C₃	semidirect
7. Q₈×3₋¹⁺²	D₃	semidirect
11. C₉×SL(2,𝔽₃)	C₂×C₃	semidirect
31. Q₈×C₉	C₃×D₃	semidirect
43. GL(2,𝔽₃)	3₋¹⁺²	direct (N × H)
55. 3₋¹⁺²	GL(2,𝔽₃)	direct (N × H)
63. SL(2,𝔽₃)	C₂×3₋¹⁺²	semidirect
89. C₉	C₃×GL(2,𝔽₃)	semidirect
90. Q₈	D₃×3₋¹⁺²	semidirect

H ≅ G/N

type

1. SL(2,𝔽₃)×3₋¹⁺²

C₂

semidirect

3. C₉×GL(2,𝔽₃)

C₃

semidirect

7. Q₈×3₋¹⁺²

D₃

semidirect

11. C₉×SL(2,𝔽₃)

C₂×C₃

semidirect

31. Q₈×C₉

C₃×D₃

semidirect

43. GL(2,𝔽₃)

3₋¹⁺²

direct (N × H)

55. 3₋¹⁺²

GL(2,𝔽₃)

direct (N × H)

63. SL(2,𝔽₃)

C₂×3₋¹⁺²

semidirect

89. C₉

C₃×GL(2,𝔽₃)

semidirect

90. Q₈

D₃×3₋¹⁺²

semidirect

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

subgroup	index
SL(2,𝔽₃)×3₋¹⁺²	2
C₃²×GL(2,𝔽₃)	3
C₉×GL(2,𝔽₃)	3
SD₁₆×3₋¹⁺²	3
C₂×D₃×3₋¹⁺²	4

subgroup

index

SL(2,𝔽₃)×3₋¹⁺²

C₃²×GL(2,𝔽₃)

C₉×GL(2,𝔽₃)

SD₁₆×3₋¹⁺²

C₂×D₃×3₋¹⁺²

Subgroups — 3₋¹⁺²×GL(2,𝔽₃) #13048365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×GL(2,𝔽3) #13048365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×GL(2,𝔽₃) #13048365