Subgroups — C₂⁴×#732 #13048210

Order 1296.

1072 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#732	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	C₂⁴×C₃×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂²×#732	C₂²	1	35	yes	direct product factor
5	C₂³×H(3,𝔽₃)	C₂×C₃	1	45	yes	complemented
6	C₂³×C₃×C₉	C₂×C₃	1	15	yes	complemented
7	C₂×#732	C₂³	1	15	yes	direct product factor
8	C₂⁴×C₃²	—	3	3	no	—
9	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
10	C₂⁴×C₉	—	3	1	no	—
11	C₂²×H(3,𝔽₃)	C₂²×C₃	1	105	yes	complemented
12	C₂²×C₃×C₉	C₂²×C₃	1	35	yes	complemented
13	#732	C₂⁴	1	1	yes	characteristic, direct product factor, Sylow
14	C₂³×C₃²	—	3	45	no	—
15	C₂³×C₃²	C₂×C₃²	1	15	yes	—
16	C₂³×C₉	—	3	15	no	—
17	C₂×H(3,𝔽₃)	C₂³×C₃	1	45	yes	complemented
18	C₂×C₃×C₉	C₂³×C₃	1	15	yes	complemented
19	C₂⁴×C₃	—	3	1	no	—
20	C₂⁴×C₃	—	9	3	no	—
21	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₂²×C₃²	—	3	105	no	—
23	C₂²×C₃²	C₂²×C₃²	1	35	yes	—
24	C₂²×C₉	—	3	35	no	—
25	H(3,𝔽₃)	C₂⁴×C₃	1	3	yes	complemented
26	C₃×C₉	C₂⁴×C₃	1	1	yes	characteristic, complemented
27	C₂³×C₃	—	3	15	no	—
28	C₂³×C₃	—	9	45	no	—
29	C₂³×C₃	C₂×H(3,𝔽₃)	1	15	yes	—
30	C₂×C₃²	—	3	45	no	—
31	C₂×C₃²	C₂³×C₃²	1	15	yes	—
32	C₂×C₉	—	3	15	no	—
33	C₂⁴	#732	1	1	yes	characteristic, direct product factor, Sylow
34	C₂²×C₃	—	3	35	no	—
35	C₂²×C₃	—	9	105	no	—
36	C₂²×C₃	C₂²×H(3,𝔽₃)	1	35	yes	—
37	C₃²	—	3	3	no	—
38	C₃²	C₂⁴×C₃²	1	1	yes	characteristic
39	C₉	—	3	1	no	—
40	C₂³	C₂×#732	1	15	yes	direct product factor
41	C₂×C₃	—	3	15	no	—
42	C₂×C₃	—	9	45	no	—
43	C₂×C₃	C₂³×H(3,𝔽₃)	1	15	yes	—
44	C₂²	C₂²×#732	1	35	yes	direct product factor
45	C₃	—	3	1	no	—
46	C₃	—	9	3	no	—
47	C₃	C₂⁴×H(3,𝔽₃)	1	1	yes	characteristic
48	C₂	C₂³×#732	1	15	yes	direct product factor

	subgroup	G/H
center, socle	C₂⁴×C₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂⁴×C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ C₂⁴×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#732	C₂	direct (N × H)
2. C₂⁴×H(3,𝔽₃)	C₃	semidirect
3. C₂⁴×C₃×C₉	C₃	semidirect
4. C₂²×#732	C₂²	direct (N × H)
5. C₂³×H(3,𝔽₃)	C₂×C₃	semidirect
6. C₂³×C₃×C₉	C₂×C₃	semidirect
7. C₂×#732	C₂³	direct (N × H)
11. C₂²×H(3,𝔽₃)	C₂²×C₃	semidirect
12. C₂²×C₃×C₉	C₂²×C₃	semidirect
13. #732	C₂⁴	direct (N × H)
17. C₂×H(3,𝔽₃)	C₂³×C₃	semidirect
18. C₂×C₃×C₉	C₂³×C₃	semidirect
25. H(3,𝔽₃)	C₂⁴×C₃	semidirect
26. C₃×C₉	C₂⁴×C₃	semidirect
33. C₂⁴	#732	direct (N × H)
40. C₂³	C₂×#732	direct (N × H)
44. C₂²	C₂²×#732	direct (N × H)
48. C₂	C₂³×#732	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×#732	2
C₂⁴×H(3,𝔽₃)	3
C₂⁴×C₃×C₉	3

Subgroups — C24×#732 #13048210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#732 #13048210