Subgroups — C₂³×C₃²×C₄ #64370

Order 288.

708 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃²	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×C₄	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²	C₄	1	8	yes	direct product factor
5	C₂³×C₃²	C₂²	1	7	yes	—
6	C₂×C₃²×C₄	C₂²	1	28	yes	direct product factor
7	C₂⁴×C₃	C₂×C₃	1	4	yes	—
8	C₂²×C₃×C₄	C₂×C₃	1	56	yes	direct product factor
9	C₂²×C₃²	C₂×C₄	1	28	yes	direct product factor
10	C₂²×C₃²	C₂³	1	7	yes	—
11	C₃²×C₄	C₂³	1	8	yes	direct product factor
12	C₂³×C₄	C₃²	1	1	yes	characteristic, direct product factor, Sylow
13	C₂³×C₃	C₃×C₄	1	32	yes	direct product factor
14	C₂³×C₃	C₂²×C₃	1	28	yes	—
15	C₂×C₃×C₄	C₂²×C₃	1	112	yes	direct product factor
16	C₂×C₃²	C₂²×C₄	1	14	yes	direct product factor
17	C₂×C₃²	C₂⁴	1	1	yes	characteristic
18	C₂⁴	C₂×C₃²	1	1	yes	characteristic
19	C₂²×C₄	C₂×C₃²	1	14	yes	direct product factor
20	C₂²×C₃	C₂×C₃×C₄	1	112	yes	direct product factor
21	C₂²×C₃	C₂³×C₃	1	28	yes	—
22	C₃×C₄	C₂³×C₃	1	32	yes	direct product factor
23	C₃²	C₂³×C₄	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³	C₃²×C₄	1	8	yes	direct product factor
25	C₂³	C₂²×C₃²	1	7	yes	—
26	C₂×C₄	C₂²×C₃²	1	28	yes	direct product factor
27	C₂×C₃	C₂²×C₃×C₄	1	56	yes	direct product factor
28	C₂×C₃	C₂⁴×C₃	1	4	yes	—
29	C₂²	C₂×C₃²×C₄	1	28	yes	direct product factor
30	C₂²	C₂³×C₃²	1	7	yes	—
31	C₄	C₂³×C₃²	1	8	yes	direct product factor
32	C₃	C₂³×C₃×C₄	1	4	yes	direct product factor
33	C₂	C₂²×C₃²×C₄	1	14	yes	direct product factor
34	C₂	C₂⁴×C₃²	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂⁴×C₃²
socle, prime radical	C₂⁴×C₃²	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₄	C₂	direct (N × H)
3. C₂³×C₃×C₄	C₃	direct (N × H)
4. C₂³×C₃²	C₄	direct (N × H)
6. C₂×C₃²×C₄	C₂²	direct (N × H)
8. C₂²×C₃×C₄	C₂×C₃	direct (N × H)
9. C₂²×C₃²	C₂×C₄	direct (N × H)
11. C₃²×C₄	C₂³	direct (N × H)
12. C₂³×C₄	C₃²	direct (N × H)
13. C₂³×C₃	C₃×C₄	direct (N × H)
15. C₂×C₃×C₄	C₂²×C₃	direct (N × H)
16. C₂×C₃²	C₂²×C₄	direct (N × H)
19. C₂²×C₄	C₂×C₃²	direct (N × H)
20. C₂²×C₃	C₂×C₃×C₄	direct (N × H)
22. C₃×C₄	C₂³×C₃	direct (N × H)
23. C₃²	C₂³×C₄	direct (N × H)
24. C₂³	C₃²×C₄	direct (N × H)
26. C₂×C₄	C₂²×C₃²	direct (N × H)
27. C₂×C₃	C₂²×C₃×C₄	direct (N × H)
29. C₂²	C₂×C₃²×C₄	direct (N × H)
31. C₄	C₂³×C₃²	direct (N × H)
32. C₃	C₂³×C₃×C₄	direct (N × H)
33. C₂	C₂²×C₃²×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
C₂⁴×C₃²	2
C₂²×C₃²×C₄	2
C₂³×C₃×C₄	3

Subgroups — C23×C32×C4 #64370

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃²×C₄ #64370