Subgroups — C₂²×C₄×H(3,𝔽₃) #88754

Order 432.

297 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×H(3,𝔽₃)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃²×C₄	C₃	1	4	yes	maximal, complemented
4	C₂²×H(3,𝔽₃)	C₄	1	4	yes	direct product factor
5	C₂²×H(3,𝔽₃)	C₂²	1	3	yes	—
6	C₄×H(3,𝔽₃)	C₂²	1	4	yes	direct product factor
7	C₂³×C₃²	C₂×C₃	1	4	yes	—
8	C₂×C₃²×C₄	C₂×C₃	1	24	yes	complemented
9	C₂×H(3,𝔽₃)	C₂×C₄	1	6	yes	direct product factor
10	C₂×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
11	C₂²×C₃×C₄	—	3	4	no	—
12	C₂²×C₃×C₄	C₃²	1	1	yes	characteristic
13	C₂²×C₃²	C₃×C₄	1	16	yes	complemented
14	C₂²×C₃²	C₂²×C₃	1	12	yes	—
15	C₃²×C₄	C₂²×C₃	1	16	yes	complemented
16	H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic, direct product factor, Sylow
17	C₂³×C₃	—	3	4	no	—
18	C₂³×C₃	C₂×C₃²	1	1	yes	characteristic
19	C₂×C₃×C₄	—	3	24	no	—
20	C₂×C₃×C₄	C₂×C₃²	1	6	yes	—
21	C₂×C₃²	C₂×C₃×C₄	1	24	yes	complemented
22	C₂×C₃²	C₂³×C₃	1	4	yes	—
23	C₂²×C₄	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
24	C₂²×C₃	—	3	16	no	—
25	C₂²×C₃	—	3	12	no	—
26	C₂²×C₃	C₃²×C₄	1	4	yes	—
27	C₂²×C₃	C₂²×C₃²	1	3	yes	—
28	C₃×C₄	—	3	16	no	—
29	C₃×C₄	C₂²×C₃²	1	4	yes	—
30	C₃²	C₂²×C₃×C₄	1	4	yes	complemented
31	C₂³	C₂×H(3,𝔽₃)	1	1	yes	characteristic
32	C₂×C₄	C₂×H(3,𝔽₃)	1	6	yes	direct product factor
33	C₂×C₃	—	3	24	no	—
34	C₂×C₃	—	3	4	no	—
35	C₂×C₃	C₂×C₃²×C₄	1	6	yes	—
36	C₂×C₃	C₂³×C₃²	1	1	yes	characteristic
37	C₂²	C₄×H(3,𝔽₃)	1	4	yes	direct product factor
38	C₂²	C₂²×H(3,𝔽₃)	1	3	yes	—
39	C₄	C₂²×H(3,𝔽₃)	1	4	yes	direct product factor
40	C₃	—	3	4	no	—
41	C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
42	C₂	C₂×C₄×H(3,𝔽₃)	1	6	yes	direct product factor
43	C₂	C₂³×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₄	C₃²
commutator	C₃	C₂²×C₃²×C₄
Frattini, cosocle	C₂×C₃	C₂³×C₃²
socle	C₂³×C₃	C₂×C₃²
prime radical	C₂³×H(3,𝔽₃)	C₂
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×H(3,𝔽₃)	C₂	direct (N × H)
3. C₂²×C₃²×C₄	C₃	semidirect
4. C₂²×H(3,𝔽₃)	C₄	direct (N × H)
6. C₄×H(3,𝔽₃)	C₂²	direct (N × H)
8. C₂×C₃²×C₄	C₂×C₃	semidirect
9. C₂×H(3,𝔽₃)	C₂×C₄	direct (N × H)
13. C₂²×C₃²	C₃×C₄	semidirect
15. C₃²×C₄	C₂²×C₃	semidirect
16. H(3,𝔽₃)	C₂²×C₄	direct (N × H)
21. C₂×C₃²	C₂×C₃×C₄	semidirect
23. C₂²×C₄	H(3,𝔽₃)	direct (N × H)
30. C₃²	C₂²×C₃×C₄	semidirect
32. C₂×C₄	C₂×H(3,𝔽₃)	direct (N × H)
37. C₂²	C₄×H(3,𝔽₃)	direct (N × H)
39. C₄	C₂²×H(3,𝔽₃)	direct (N × H)
42. C₂	C₂×C₄×H(3,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
C₂³×H(3,𝔽₃)	2
C₂×C₄×H(3,𝔽₃)	2
C₂²×C₃²×C₄	3

Subgroups — C22×C4×H(3,𝔽3) #88754

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄×H(3,𝔽₃) #88754