Subgroups — C₂⁴×C₃×D₄ #87578

Order 384.

10552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₃	C₂	1	2	yes	maximal, complemented
2	C₂³×C₃×D₄	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
5	C₂⁵×C₃	—	2	32	no	—
6	C₂⁵×C₃	C₂²	1	1	yes	characteristic
7	C₂⁵×C₃	C₂²	1	60	yes	complemented
8	C₂²×C₃×D₄	C₂²	1	560	yes	direct product factor
9	C₂³×C₃×C₄	C₂²	1	30	yes	complemented
10	C₂⁶	C₂×C₃	1	2	yes	complemented
11	C₂³×D₄	C₂×C₃	1	60	yes	direct product factor
12	C₂⁴×C₄	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂⁴×C₃	—	2	480	no	—
14	C₂⁴×C₃	D₄	1	16	yes	direct product factor
15	C₂⁴×C₃	C₂³	1	15	yes	—
16	C₂⁴×C₃	C₂³	1	280	yes	complemented
17	C₂×C₃×D₄	C₂³	1	960	yes	direct product factor
18	C₂²×C₃×C₄	C₂³	1	140	yes	complemented
19	C₂⁵	—	2	32	no	—
20	C₂⁵	C₂²×C₃	1	1	yes	characteristic
21	C₂⁵	C₂²×C₃	1	60	yes	complemented
22	C₂²×D₄	C₂²×C₃	1	560	yes	direct product factor
23	C₂³×C₄	C₂²×C₃	1	30	yes	complemented
24	C₂³×C₃	—	2	1120	no	—
25	C₂³×C₃	C₂×D₄	1	120	yes	direct product factor
26	C₂³×C₃	C₂⁴	1	35	yes	—
27	C₂³×C₃	C₂⁴	1	240	yes	complemented
28	C₃×D₄	C₂⁴	1	256	yes	direct product factor
29	C₂×C₃×C₄	C₂⁴	1	120	yes	complemented
30	C₂⁴	—	2	480	no	—
31	C₂⁴	C₃×D₄	1	16	yes	direct product factor
32	C₂⁴	C₂³×C₃	1	15	yes	—
33	C₂⁴	C₂³×C₃	1	280	yes	complemented
34	C₂×D₄	C₂³×C₃	1	960	yes	direct product factor
35	C₂²×C₄	C₂³×C₃	1	140	yes	complemented
36	C₂²×C₃	—	2	480	no	—
37	C₂²×C₃	C₂²×D₄	1	140	yes	direct product factor
38	C₂²×C₃	C₂⁵	1	15	yes	—
39	C₂²×C₃	C₂⁵	1	32	yes	complemented
40	C₃×C₄	C₂⁵	1	16	yes	complemented
41	C₂³	—	2	1120	no	—
42	C₂³	C₂×C₃×D₄	1	120	yes	direct product factor
43	C₂³	C₂⁴×C₃	1	35	yes	—
44	C₂³	C₂⁴×C₃	1	240	yes	complemented
45	D₄	C₂⁴×C₃	1	256	yes	direct product factor
46	C₂×C₄	C₂⁴×C₃	1	120	yes	complemented
47	C₂×C₃	—	2	32	no	—
48	C₂×C₃	C₂³×D₄	1	30	yes	direct product factor
49	C₂×C₃	C₂⁶	1	1	yes	characteristic
50	C₂²	—	2	480	no	—
51	C₂²	C₂²×C₃×D₄	1	140	yes	direct product factor
52	C₂²	C₂⁵×C₃	1	15	yes	—
53	C₂²	C₂⁵×C₃	1	32	yes	complemented
54	C₄	C₂⁵×C₃	1	16	yes	complemented
55	C₃	C₂⁴×D₄	1	1	yes	characteristic, direct product factor, Sylow
56	C₂	—	2	32	no	—
57	C₂	C₂³×C₃×D₄	1	30	yes	direct product factor
58	C₂	C₂⁶×C₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵×C₃	C₂²
commutator, Frattini, cosocle	C₂	C₂⁶×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁶×C₃	C₂	semidirect
2. C₂³×C₃×D₄	C₂	direct (N × H)
3. C₂⁴×C₃×C₄	C₂	semidirect
4. C₂⁴×D₄	C₃	direct (N × H)
7. C₂⁵×C₃	C₂²	semidirect
8. C₂²×C₃×D₄	C₂²	direct (N × H)
9. C₂³×C₃×C₄	C₂²	semidirect
10. C₂⁶	C₂×C₃	semidirect
11. C₂³×D₄	C₂×C₃	direct (N × H)
12. C₂⁴×C₄	C₂×C₃	semidirect
14. C₂⁴×C₃	D₄	direct (N × H)
16. C₂⁴×C₃	C₂³	semidirect
17. C₂×C₃×D₄	C₂³	direct (N × H)
18. C₂²×C₃×C₄	C₂³	semidirect
21. C₂⁵	C₂²×C₃	semidirect
22. C₂²×D₄	C₂²×C₃	direct (N × H)
23. C₂³×C₄	C₂²×C₃	semidirect
25. C₂³×C₃	C₂×D₄	direct (N × H)
27. C₂³×C₃	C₂⁴	semidirect
28. C₃×D₄	C₂⁴	direct (N × H)
29. C₂×C₃×C₄	C₂⁴	semidirect
31. C₂⁴	C₃×D₄	direct (N × H)
33. C₂⁴	C₂³×C₃	semidirect
34. C₂×D₄	C₂³×C₃	direct (N × H)
35. C₂²×C₄	C₂³×C₃	semidirect
37. C₂²×C₃	C₂²×D₄	direct (N × H)
39. C₂²×C₃	C₂⁵	semidirect
40. C₃×C₄	C₂⁵	semidirect
42. C₂³	C₂×C₃×D₄	direct (N × H)
44. C₂³	C₂⁴×C₃	semidirect
45. D₄	C₂⁴×C₃	direct (N × H)
46. C₂×C₄	C₂⁴×C₃	semidirect
48. C₂×C₃	C₂³×D₄	direct (N × H)
51. C₂²	C₂²×C₃×D₄	direct (N × H)
53. C₂²	C₂⁵×C₃	semidirect
54. C₄	C₂⁵×C₃	semidirect
55. C₃	C₂⁴×D₄	direct (N × H)
57. C₂	C₂³×C₃×D₄	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁶×C₃	2
C₂³×C₃×D₄	2
C₂⁴×C₃×C₄	2
C₂⁴×D₄	3

Subgroups — C24×C3×D4 #87578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃×D₄ #87578