Subgroups — C₂⁴×#731 #13048209

Order 1296.

938 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#731	C₂	1	15	yes	maximal, direct product factor
2	C₂⁴×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
3	C₂⁴×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₂⁴×C₃×C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×#731	C₂²	1	35	yes	direct product factor
6	C₂³×3₋¹⁺²	C₂×C₃	1	30	yes	complemented
7	C₂³×H(3,𝔽₃)	C₂×C₃	1	15	yes	—
8	C₂³×C₃×C₉	C₂×C₃	1	15	yes	complemented
9	C₂×#731	C₂³	1	15	yes	direct product factor
10	C₂⁴×C₃²	—	3	1	no	—
11	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
12	C₂⁴×C₉	—	3	1	no	—
13	C₂⁴×C₉	—	3	2	no	—
14	C₂²×3₋¹⁺²	C₂²×C₃	1	70	yes	complemented
15	C₂²×H(3,𝔽₃)	C₂²×C₃	1	35	yes	—
16	C₂²×C₃×C₉	C₂²×C₃	1	35	yes	complemented
17	#731	C₂⁴	1	1	yes	characteristic, direct product factor, Sylow
18	C₂³×C₃²	—	3	15	no	—
19	C₂³×C₃²	C₂×C₃²	1	15	yes	—
20	C₂³×C₉	—	3	15	no	—
21	C₂³×C₉	—	3	30	no	—
22	C₂×3₋¹⁺²	C₂³×C₃	1	30	yes	complemented
23	C₂×H(3,𝔽₃)	C₂³×C₃	1	15	yes	—
24	C₂×C₃×C₉	C₂³×C₃	1	15	yes	complemented
25	C₂⁴×C₃	—	3	1	no	—
26	C₂⁴×C₃	—	9	1	no	—
27	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
28	C₂²×C₃²	—	3	35	no	—
29	C₂²×C₃²	C₂²×C₃²	1	35	yes	—
30	C₂²×C₉	—	3	35	no	—
31	C₂²×C₉	—	3	70	no	—
32	3₋¹⁺²	C₂⁴×C₃	1	2	yes	complemented
33	H(3,𝔽₃)	C₂⁴×C₃	1	1	yes	characteristic
34	C₃×C₉	C₂⁴×C₃	1	1	yes	characteristic, complemented
35	C₂³×C₃	—	3	15	no	—
36	C₂³×C₃	—	9	15	no	—
37	C₂³×C₃	C₂×H(3,𝔽₃)	1	15	yes	—
38	C₂×C₃²	—	3	15	no	—
39	C₂×C₃²	C₂³×C₃²	1	15	yes	—
40	C₂×C₉	—	3	15	no	—
41	C₂×C₉	—	3	30	no	—
42	C₂⁴	#731	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²×C₃	—	3	35	no	—
44	C₂²×C₃	—	9	35	no	—
45	C₂²×C₃	C₂²×H(3,𝔽₃)	1	35	yes	—
46	C₃²	—	3	1	no	—
47	C₃²	C₂⁴×C₃²	1	1	yes	characteristic
48	C₉	—	3	1	no	—
49	C₉	—	3	2	no	—
50	C₂³	C₂×#731	1	15	yes	direct product factor
51	C₂×C₃	—	3	15	no	—
52	C₂×C₃	—	9	15	no	—
53	C₂×C₃	C₂³×H(3,𝔽₃)	1	15	yes	—
54	C₂²	C₂²×#731	1	35	yes	direct product factor
55	C₃	—	3	1	no	—
56	C₃	—	9	1	no	—
57	C₃	C₂⁴×H(3,𝔽₃)	1	1	yes	characteristic
58	C₂	C₂³×#731	1	15	yes	direct product factor

	subgroup	G/H
center, socle	C₂⁴×C₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂⁴×C₃²
prime radical	C₂⁴×H(3,𝔽₃)	C₃
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×#731	C₂	direct (N × H)
2. C₂⁴×3₋¹⁺²	C₃	semidirect
4. C₂⁴×C₃×C₉	C₃	semidirect
5. C₂²×#731	C₂²	direct (N × H)
6. C₂³×3₋¹⁺²	C₂×C₃	semidirect
8. C₂³×C₃×C₉	C₂×C₃	semidirect
9. C₂×#731	C₂³	direct (N × H)
14. C₂²×3₋¹⁺²	C₂²×C₃	semidirect
16. C₂²×C₃×C₉	C₂²×C₃	semidirect
17. #731	C₂⁴	direct (N × H)
22. C₂×3₋¹⁺²	C₂³×C₃	semidirect
24. C₂×C₃×C₉	C₂³×C₃	semidirect
32. 3₋¹⁺²	C₂⁴×C₃	semidirect
34. C₃×C₉	C₂⁴×C₃	semidirect
42. C₂⁴	#731	direct (N × H)
50. C₂³	C₂×#731	direct (N × H)
54. C₂²	C₂²×#731	direct (N × H)
58. C₂	C₂³×#731	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×#731	2
C₂⁴×3₋¹⁺²	3
C₂⁴×H(3,𝔽₃)	3
C₂⁴×C₃×C₉	3

Subgroups — C24×#731 #13048209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×#731 #13048209