Subgroups — D₉×SL(2,𝔽₃) #88617

Order 432.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×SL(2,𝔽₃)	—	3	1	no	maximal
3	Q₈×D₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×D₉	—	4	1	no	maximal
5	C₃×SL(2,𝔽₃)	D₃	1	1	yes	characteristic
6	Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
7	C₄×D₉	—	3	1	no	—
8	Dic₁₈	—	3	1	no	—
9	Q₈⋊C₉	—	2	1	no	—
10	C₂×C₃×C₉	—	4	1	no	—
11	C₃×D₉	—	4	2	no	—
12	D₃×Q₈	—	3	1	no	—
13	C₂×SL(2,𝔽₃)	—	9	1	no	—
14	C₂×C₃×D₃	—	12	1	no	—
15	C₂×D₉	A₄	1	1	yes	characteristic
16	C₄×C₉	—	3	1	no	—
17	Dic₉	—	3	1	no	—
18	C₃×C₉	—	4	1	no	Sylow
19	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
20	C₄×D₃	—	9	1	no	—
21	Dic₆	—	9	1	no	—
22	SL(2,𝔽₃)	—	2	1	no	—
23	SL(2,𝔽₃)	D₉	1	1	yes	characteristic, direct product factor
24	C₂×C₃²	—	4	1	no	—
25	C₃×D₃	—	12	2	no	—
26	C₂×C₉	—	8	1	no	—
27	C₂×C₉	C₂×A₄	1	1	yes	characteristic
28	D₉	SL(2,𝔽₃)	1	2	yes	direct product factor
29	C₂×Q₈	—	9	1	no	Sylow
30	C₂²×C₃	—	36	1	no	—
31	C₂×D₃	—	3	1	no	—
32	C₃×C₄	—	3	1	no	—
33	Dic₃	—	9	1	no	—
34	C₃²	—	4	1	no	—
35	C₉	—	8	1	no	—
36	C₉	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
37	Q₈	—	27	1	no	—
38	Q₈	C₃×D₉	1	1	yes	characteristic, complemented
39	C₂×C₄	—	27	1	no	—
40	C₂×C₃	—	4	1	no	—
41	C₂×C₃	—	8	1	no	—
42	C₂×C₃	—	36	2	no	—
43	C₂×C₃	D₃×A₄	1	1	yes	characteristic
44	D₃	—	3	2	no	—
45	C₂²	—	9	1	no	—
46	C₄	—	3	1	no	—
47	C₄	—	27	1	no	—
48	C₃	—	4	1	no	—
49	C₃	—	8	1	no	—
50	C₃	D₃×SL(2,𝔽₃)	1	1	yes	characteristic
51	C₂	—	9	2	no	—
52	C₂	A₄×D₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	A₄×D₉
commutator, Fitting, cosocle, hypocenter	Q₈×C₉	C₂×C₃
Frattini, socle	C₂×C₃	D₃×A₄
LCS	G ▷ Q₈×C₉
UCS	C₁ ◁ C₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₉×SL(2,𝔽₃)	C₂	semidirect
3. Q₈×D₉	C₃	semidirect
6. Q₈×C₉	C₂×C₃	semidirect
23. SL(2,𝔽₃)	D₉	direct (N × H)
28. D₉	SL(2,𝔽₃)	direct (N × H)
36. C₉	C₂×SL(2,𝔽₃)	semidirect
38. Q₈	C₃×D₉	semidirect

Maximal subgroups

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

subgroup	index
C₉×SL(2,𝔽₃)	2
D₃×SL(2,𝔽₃)	3
Q₈×D₉	3
C₂×C₃×D₉	4

Subgroups — D9×SL(2,𝔽3) #88617

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₉×SL(2,𝔽₃) #88617