Subgroups — C₂⁴×C₃⋊(C₃⋊D₃) #11724757

Order 864.

10472 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃³	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₃⋊(C₃⋊D₃)	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₃⋊D₃	—	3	13	no	maximal
4	C₂³×C₃³	C₂²	1	15	yes	complemented
5	C₂²×C₃⋊(C₃⋊D₃)	C₂²	1	140	yes	direct product factor
6	C₂⁴×C₃²	D₃	1	13	yes	complemented
7	C₂³×C₃⋊D₃	—	3	390	no	—
8	C₂²×C₃³	C₂³	1	35	yes	complemented
9	C₂×C₃⋊(C₃⋊D₃)	C₂³	1	120	yes	direct product factor
10	C₂⁴×D₃	—	9	13	no	—
11	C₂³×C₃²	C₂×D₃	1	195	yes	complemented
12	C₂²×C₃⋊D₃	—	3	1820	no	—
13	C₂×C₃³	C₂⁴	1	15	yes	complemented
14	C₃⋊(C₃⋊D₃)	C₂⁴	1	16	yes	direct product factor
15	C₂⁴×C₃	C₃⋊D₃	1	13	yes	complemented
16	C₂³×D₃	—	9	390	no	—
17	C₂²×C₃²	C₂²×D₃	1	455	yes	complemented
18	C₂×C₃⋊D₃	—	3	1560	no	—
19	C₂⁵	—	27	1	no	Sylow
20	C₃³	C₂⁵	1	1	yes	characteristic, complemented, Sylow
21	C₂³×C₃	C₂×C₃⋊D₃	1	195	yes	complemented
22	C₂²×D₃	—	9	1820	no	—
23	C₂×C₃²	C₂³×D₃	1	195	yes	complemented
24	C₃⋊D₃	—	3	208	no	—
25	C₂⁴	—	27	30	no	—
26	C₂⁴	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, direct product factor
27	C₂²×C₃	C₂²×C₃⋊D₃	1	455	yes	complemented
28	C₂×D₃	—	9	1560	no	—
29	C₃²	C₂⁴×D₃	1	13	yes	complemented
30	C₂³	—	27	140	no	—
31	C₂³	C₂×C₃⋊(C₃⋊D₃)	1	15	yes	direct product factor
32	C₂×C₃	C₂³×C₃⋊D₃	1	195	yes	complemented
33	D₃	—	9	208	no	—
34	C₂²	—	27	120	no	—
35	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	35	yes	direct product factor
36	C₃	C₂⁴×C₃⋊D₃	1	13	yes	complemented
37	C₂	—	27	16	no	—
38	C₂	C₂³×C₃⋊(C₃⋊D₃)	1	15	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂⁴	C₃⋊(C₃⋊D₃)
commutator, cosocle, hypocenter	C₃³	C₂⁵
Frattini	C₁	G
Fitting, socle	C₂⁴×C₃³	C₂
LCS	G ▷ C₃³
UCS	C₁ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃³	C₂	semidirect
2. C₂³×C₃⋊(C₃⋊D₃)	C₂	direct (N × H)
4. C₂³×C₃³	C₂²	semidirect
5. C₂²×C₃⋊(C₃⋊D₃)	C₂²	direct (N × H)
6. C₂⁴×C₃²	D₃	semidirect
8. C₂²×C₃³	C₂³	semidirect
9. C₂×C₃⋊(C₃⋊D₃)	C₂³	direct (N × H)
11. C₂³×C₃²	C₂×D₃	semidirect
13. C₂×C₃³	C₂⁴	semidirect
14. C₃⋊(C₃⋊D₃)	C₂⁴	direct (N × H)
15. C₂⁴×C₃	C₃⋊D₃	semidirect
17. C₂²×C₃²	C₂²×D₃	semidirect
20. C₃³	C₂⁵	semidirect
21. C₂³×C₃	C₂×C₃⋊D₃	semidirect
23. C₂×C₃²	C₂³×D₃	semidirect
26. C₂⁴	C₃⋊(C₃⋊D₃)	direct (N × H)
27. C₂²×C₃	C₂²×C₃⋊D₃	semidirect
29. C₃²	C₂⁴×D₃	semidirect
31. C₂³	C₂×C₃⋊(C₃⋊D₃)	direct (N × H)
32. C₂×C₃	C₂³×C₃⋊D₃	semidirect
35. C₂²	C₂²×C₃⋊(C₃⋊D₃)	direct (N × H)
36. C₃	C₂⁴×C₃⋊D₃	semidirect
38. C₂	C₂³×C₃⋊(C₃⋊D₃)	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×C₃³	2
C₂³×C₃⋊(C₃⋊D₃)	2
C₂⁴×C₃⋊D₃	3

Subgroups — C24×C3⋊(C3⋊D3) #11724757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃⋊(C₃⋊D₃) #11724757