Subgroups — C₂²×C₃³×C₄ #89083

Order 432.

756 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃³×C₄	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃²×C₄	C₃	1	13	yes	maximal, direct product factor
4	C₂²×C₃³	C₄	1	4	yes	direct product factor
5	C₂²×C₃³	C₂²	1	3	yes	—
6	C₃³×C₄	C₂²	1	4	yes	direct product factor
7	C₂³×C₃²	C₂×C₃	1	13	yes	—
8	C₂×C₃²×C₄	C₂×C₃	1	78	yes	direct product factor
9	C₂×C₃³	C₂×C₄	1	6	yes	direct product factor
10	C₂×C₃³	C₂³	1	1	yes	characteristic
11	C₂²×C₃×C₄	C₃²	1	13	yes	direct product factor
12	C₂²×C₃²	C₃×C₄	1	52	yes	direct product factor
13	C₂²×C₃²	C₂²×C₃	1	39	yes	—
14	C₃²×C₄	C₂²×C₃	1	52	yes	direct product factor
15	C₃³	C₂²×C₄	1	1	yes	characteristic, direct product factor, Sylow
16	C₂³×C₃	C₂×C₃²	1	13	yes	—
17	C₂×C₃×C₄	C₂×C₃²	1	78	yes	direct product factor
18	C₂×C₃²	C₂×C₃×C₄	1	78	yes	direct product factor
19	C₂×C₃²	C₂³×C₃	1	13	yes	—
20	C₂²×C₄	C₃³	1	1	yes	characteristic, direct product factor, Sylow
21	C₂²×C₃	C₃²×C₄	1	52	yes	direct product factor
22	C₂²×C₃	C₂²×C₃²	1	39	yes	—
23	C₃×C₄	C₂²×C₃²	1	52	yes	direct product factor
24	C₃²	C₂²×C₃×C₄	1	13	yes	direct product factor
25	C₂³	C₂×C₃³	1	1	yes	characteristic
26	C₂×C₄	C₂×C₃³	1	6	yes	direct product factor
27	C₂×C₃	C₂×C₃²×C₄	1	78	yes	direct product factor
28	C₂×C₃	C₂³×C₃²	1	13	yes	—
29	C₂²	C₃³×C₄	1	4	yes	direct product factor
30	C₂²	C₂²×C₃³	1	3	yes	—
31	C₄	C₂²×C₃³	1	4	yes	direct product factor
32	C₃	C₂²×C₃²×C₄	1	13	yes	direct product factor
33	C₂	C₂×C₃³×C₄	1	6	yes	direct product factor
34	C₂	C₂³×C₃³	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂³×C₃³
socle, prime radical	C₂³×C₃³	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃³×C₄	C₂	direct (N × H)
3. C₂²×C₃²×C₄	C₃	direct (N × H)
4. C₂²×C₃³	C₄	direct (N × H)
6. C₃³×C₄	C₂²	direct (N × H)
8. C₂×C₃²×C₄	C₂×C₃	direct (N × H)
9. C₂×C₃³	C₂×C₄	direct (N × H)
11. C₂²×C₃×C₄	C₃²	direct (N × H)
12. C₂²×C₃²	C₃×C₄	direct (N × H)
14. C₃²×C₄	C₂²×C₃	direct (N × H)
15. C₃³	C₂²×C₄	direct (N × H)
17. C₂×C₃×C₄	C₂×C₃²	direct (N × H)
18. C₂×C₃²	C₂×C₃×C₄	direct (N × H)
20. C₂²×C₄	C₃³	direct (N × H)
21. C₂²×C₃	C₃²×C₄	direct (N × H)
23. C₃×C₄	C₂²×C₃²	direct (N × H)
24. C₃²	C₂²×C₃×C₄	direct (N × H)
26. C₂×C₄	C₂×C₃³	direct (N × H)
27. C₂×C₃	C₂×C₃²×C₄	direct (N × H)
29. C₂²	C₃³×C₄	direct (N × H)
31. C₄	C₂²×C₃³	direct (N × H)
32. C₃	C₂²×C₃²×C₄	direct (N × H)
33. C₂	C₂×C₃³×C₄	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×C₃³	2
C₂×C₃³×C₄	2
C₂²×C₃²×C₄	3

Subgroups — C22×C33×C4 #89083

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃³×C₄ #89083