Subgroups — #41⋊H(3,𝔽₃) #88692

Order 432.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃²×#41	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#41⋊C₃	C₃	1	3	yes	maximal, complemented
4	C₄×H(3,𝔽₃)	—	4	1	no	maximal
5	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
6	C₃²×D₄	—	3	1	no	—
7	C₂×C₃²×C₄	—	3	1	no	—
8	C₃×SL(2,𝔽₃)	C₂×C₃	1	3	yes	complemented
9	C₂×H(3,𝔽₃)	—	4	1	no	—
10	C₃×#41	—	3	1	no	—
11	C₃×#41	C₃²	1	1	yes	characteristic
12	#41⋊C₃	—	3	3	no	—
13	C₂²×C₃²	—	3	1	no	—
14	C₃²×C₄	—	3	1	no	—
15	C₃²×C₄	—	4	3	no	—
16	C₃²×C₄	A₄	1	1	yes	characteristic
17	H(3,𝔽₃)	—	4	1	no	Sylow
18	C₃×Q₈	—	3	1	no	—
19	C₃×Q₈	C₂×C₃²	1	1	yes	characteristic
20	C₃×D₄	—	3	1	no	—
21	C₃×D₄	—	3	3	no	—
22	C₂×C₃×C₄	—	3	3	no	—
23	C₂×C₃×C₄	—	3	1	no	—
24	SL(2,𝔽₃)	—	3	3	no	—
25	C₂×C₃²	—	4	3	no	—
26	C₂×C₃²	—	6	1	no	—
27	C₂×C₃²	C₂×A₄	1	1	yes	characteristic
28	#41	H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
29	C₂²×C₃	—	3	3	no	—
30	C₂²×C₃	—	3	1	no	—
31	C₃×C₄	—	3	3	no	—
32	C₃×C₄	—	3	1	no	—
33	C₃×C₄	—	3	1	no	—
34	C₃×C₄	—	12	3	no	—
35	C₃×C₄	C₃×A₄	1	1	yes	characteristic
36	C₃²	—	4	3	no	—
37	C₃²	#41⋊C₃	1	1	yes	characteristic, complemented
38	Q₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic
39	D₄	—	3	1	no	—
40	C₂×C₄	—	3	1	no	—
41	C₂×C₃	—	3	1	no	—
42	C₂×C₃	—	6	3	no	—
43	C₂×C₃	—	6	1	no	—
44	C₂×C₃	—	12	3	no	—
45	C₂×C₃	C₂×C₃×A₄	1	1	yes	characteristic
46	C₂²	—	3	1	no	—
47	C₄	—	3	1	no	—
48	C₄	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
49	C₃	—	3	1	no	—
50	C₃	—	12	3	no	—
51	C₃	C₃×#41⋊C₃	1	1	yes	characteristic
52	C₂	—	6	1	no	—
53	C₂	C₂×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₃×A₄
commutator, cosocle	C₃×Q₈	C₂×C₃²
Frattini, socle	C₂×C₃	C₂×C₃×A₄
Fitting	C₃²×#41	C₃
hypercenter	C₃²×C₄	A₄
hypocenter	Q₈	C₂×H(3,𝔽₃)
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₃×C₄ ◁ C₃²×C₄

Complements

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

N	H ≅ G/N	type
1. Q₈⋊H(3,𝔽₃)	C₂	semidirect
2. C₃²×#41	C₃	semidirect
3. C₃×#41⋊C₃	C₃	semidirect
8. C₃×SL(2,𝔽₃)	C₂×C₃	semidirect
28. #41	H(3,𝔽₃)	semidirect
37. C₃²	#41⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
Q₈⋊H(3,𝔽₃)	2
C₃²×#41	3
C₃×#41⋊C₃	3
C₄×H(3,𝔽₃)	4

Subgroups — #41⋊H(3,𝔽3) #88692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41⋊H(3,𝔽₃) #88692