Subgroups — C₂×C₉×SL(2,𝔽₃) #88680

Order 432.

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

	subgroup	G/H
center, hypercenter	C₂²×C₉	A₄
commutator, hypocenter	Q₈	C₂×C₃×C₉
Frattini	C₂×C₃	C₂×C₃×A₄
Fitting	C₂×Q₈×C₉	C₃
socle	C₂²×C₃	C₃×A₄
prime radical	C₂×C₃×SL(2,𝔽₃)	C₃
cosocle	C₃×Q₈	C₂×C₃²
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂²×C₉

Complements

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

N	H ≅ G/N	type
1. C₉×SL(2,𝔽₃)	C₂	direct (N × H)
3. C₂×Q₈×C₉	C₃	semidirect
4. C₂×Q₈⋊C₉	C₃	semidirect
8. Q₈×C₉	C₂×C₃	semidirect
10. Q₈⋊C₉	C₂×C₃	semidirect
14. C₂×SL(2,𝔽₃)	C₉	direct (N × H)
24. SL(2,𝔽₃)	C₂×C₉	direct (N × H)
29. C₂×C₉	SL(2,𝔽₃)	direct (N × H)
31. C₂×Q₈	C₃×C₉	semidirect
38. C₉	C₂×SL(2,𝔽₃)	direct (N × H)
40. Q₈	C₂×C₃×C₉	semidirect
51. C₂	C₉×SL(2,𝔽₃)	direct (N × H)

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
C₂×C₃×SL(2,𝔽₃)	3
C₂×Q₈×C₉	3
C₂×Q₈⋊C₉	3
C₂²×C₃×C₉	4

Subgroups — C2×C9×SL(2,𝔽3) #88680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×SL(2,𝔽₃) #88680