Subgroups — (C₃×SL(2,𝔽₃))⋊C₉ #10619577

Order 648.

79 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×Q₈×C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈⋊C₉	C₃	1	2	yes	maximal, complemented
4	C₂×C₃²⋊C₉	—	4	1	no	maximal
5	C₃×C₄×C₉	—	3	1	no	—
6	C₃²⋊C₉	—	4	1	no	Sylow
7	C₃²×Q₈	C₃²	1	1	yes	characteristic
8	C₃×SL(2,𝔽₃)	—	3	1	no	—
9	C₃×SL(2,𝔽₃)	—	3	2	no	—
10	C₃×SL(2,𝔽₃)	C₉	1	3	yes	complemented
11	Q₈×C₉	—	3	1	no	—
12	Q₈⋊C₉	—	3	2	no	—
13	C₂×C₃³	—	4	1	no	—
14	C₂×C₃×C₉	—	4	2	no	—
15	C₂×C₃×C₉	A₄	1	1	yes	characteristic
16	C₃²×C₄	—	3	1	no	—
17	C₄×C₉	—	3	3	no	—
18	C₃³	—	4	1	no	—
19	C₃×C₉	—	4	2	no	—
20	C₃×C₉	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
21	C₃×Q₈	C₃×C₉	1	1	yes	characteristic
22	C₃×Q₈	H(3,𝔽₃)	1	1	yes	characteristic
23	C₃×Q₈	3₋¹⁺²	1	2	yes	—
24	SL(2,𝔽₃)	—	3	3	no	—
25	C₂×C₃²	—	4	3	no	—
26	C₂×C₃²	—	12	2	no	—
27	C₂×C₃²	—	12	1	no	—
28	C₂×C₃²	C₃×A₄	1	1	yes	characteristic
29	C₂×C₉	—	3	1	no	—
30	C₂×C₉	—	12	2	no	—
31	C₃×C₄	—	3	1	no	—
32	C₃×C₄	—	3	2	no	—
33	C₃×C₄	—	3	1	no	—
34	C₃²	—	4	3	no	—
35	C₃²	—	12	2	no	—
36	C₃²	—	12	1	no	—
37	C₃²	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
38	C₉	—	3	1	no	—
39	C₉	—	12	2	no	—
40	Q₈	C₃²⋊C₉	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃	—	12	3	no	—
42	C₂×C₃	C₉×A₄	1	1	yes	characteristic
43	C₂×C₃	C₉⋊A₄	1	2	yes	—
44	C₂×C₃	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
45	C₄	—	3	1	no	—
46	C₃	—	12	3	no	—
47	C₃	C₉×SL(2,𝔽₃)	1	1	yes	characteristic
48	C₃	C₉⋊SL(2,𝔽₃)	1	2	yes	—
49	C₃	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
50	C₂	(C₃×A₄)⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle	C₂×C₃²	C₃×A₄
commutator	C₃×Q₈	C₃×C₉
Fitting	C₃×Q₈×C₉	C₃
prime radical	C₃²×SL(2,𝔽₃)	C₃
cosocle	C₃²×Q₈	C₃²
hypercenter	C₂×C₃×C₉	A₄
hypocenter	Q₈	C₃²⋊C₉
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃×C₉

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×C₉	C₃	semidirect
3. C₃×Q₈⋊C₉	C₃	semidirect
10. C₃×SL(2,𝔽₃)	C₉	semidirect
20. C₃×C₉	SL(2,𝔽₃)	semidirect
40. Q₈	C₃²⋊C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃²×SL(2,𝔽₃)	3
C₃×Q₈×C₉	3
C₃×Q₈⋊C₉	3
C₂×C₃²⋊C₉	4

Subgroups — (C3×SL(2,𝔽3))⋊C9 #10619577

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₃×SL(2,𝔽₃))⋊C₉ #10619577