Subgroups — #41×H(3,𝔽₃) #88763

Order 432.

220 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	D₄×H(3,𝔽₃)	C₂	1	3	yes	maximal, complemented
3	C₂×C₄×H(3,𝔽₃)	C₂	1	3	yes	maximal, complemented
4	C₃²×#41	C₃	1	4	yes	maximal, complemented
5	C₂²×H(3,𝔽₃)	C₂²	1	3	yes	—
6	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
8	C₃²×Q₈	C₂×C₃	1	4	yes	complemented
9	C₃²×D₄	C₂×C₃	1	12	yes	complemented
10	C₂×C₃²×C₄	C₂×C₃	1	12	yes	complemented
11	C₂×H(3,𝔽₃)	—	2	3	no	—
12	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
13	C₃×#41	—	3	4	no	—
14	C₃×#41	C₃²	1	1	yes	characteristic
15	C₂²×C₃²	C₂²×C₃	1	12	yes	—
16	C₃²×C₄	C₂²×C₃	1	12	yes	—
17	C₃²×C₄	C₂²×C₃	1	4	yes	—
18	H(3,𝔽₃)	#41	1	1	yes	characteristic, direct product factor, Sylow
19	C₃×Q₈	—	3	4	no	—
20	C₃×Q₈	C₂×C₃²	1	1	yes	characteristic
21	C₃×D₄	—	3	12	no	—
22	C₃×D₄	C₂×C₃²	1	3	yes	—
23	C₂×C₃×C₄	—	3	12	no	—
24	C₂×C₃×C₄	C₂×C₃²	1	3	yes	—
25	C₂×C₃²	—	2	12	no	—
26	C₂×C₃²	C₂³×C₃	1	4	yes	—
27	#41	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
28	C₂²×C₃	—	3	12	no	—
29	C₂²×C₃	C₂²×C₃²	1	3	yes	—
30	C₃×C₄	—	3	4	no	—
31	C₃×C₄	—	3	12	no	—
32	C₃×C₄	C₂²×C₃²	1	3	yes	—
33	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
34	C₃²	C₃×#41	1	4	yes	complemented
35	Q₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
36	D₄	C₂×H(3,𝔽₃)	1	3	yes	complemented
37	C₂×C₄	C₂×H(3,𝔽₃)	1	3	yes	complemented
38	C₂×C₃	—	2	3	no	—
39	C₂×C₃	—	3	4	no	—
40	C₂×C₃	—	6	12	no	—
41	C₂×C₃	C₂³×C₃²	1	1	yes	characteristic
42	C₂²	C₂²×H(3,𝔽₃)	1	3	yes	—
43	C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
44	C₄	C₂²×H(3,𝔽₃)	1	3	yes	—
45	C₃	—	3	4	no	—
46	C₃	C₃²×#41	1	1	yes	characteristic
47	C₂	—	2	3	no	—
48	C₂	C₂³×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₂²×C₃²
commutator, Frattini, socle, cosocle	C₂×C₃	C₂³×C₃²
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×H(3,𝔽₃)	C₂	semidirect
2. D₄×H(3,𝔽₃)	C₂	semidirect
3. C₂×C₄×H(3,𝔽₃)	C₂	semidirect
4. C₃²×#41	C₃	semidirect
8. C₃²×Q₈	C₂×C₃	semidirect
9. C₃²×D₄	C₂×C₃	semidirect
10. C₂×C₃²×C₄	C₂×C₃	semidirect
18. H(3,𝔽₃)	#41	direct (N × H)
27. #41	H(3,𝔽₃)	direct (N × H)
34. C₃²	C₃×#41	semidirect
35. Q₈	C₂×H(3,𝔽₃)	semidirect
36. D₄	C₂×H(3,𝔽₃)	semidirect
37. C₂×C₄	C₂×H(3,𝔽₃)	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
D₄×H(3,𝔽₃)	2
C₂×C₄×H(3,𝔽₃)	2
C₃²×#41	3

Subgroups — #41×H(3,𝔽3) #88763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #41×H(3,𝔽₃) #88763