Subgroups — C₂⁴×C₃⋊C₁₆ #11701645

Order 768.

2590 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₃⋊C₁₆	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₁₆	—	3	1	no	maximal, Sylow
4	C₂⁴×C₃×C₄	C₄	1	1	yes	characteristic
5	C₂³×C₃×C₈	C₄	1	15	yes	—
6	C₂³×C₃×C₈	C₂²	1	15	yes	—
7	C₂²×C₃⋊C₁₆	C₂²	1	140	yes	direct product factor
8	C₂⁴×C₈	D₃	1	1	yes	characteristic
9	C₂³×C₁₆	—	3	30	no	—
10	C₂⁵×C₃	C₈	1	1	yes	characteristic
11	C₂³×C₃×C₄	C₈	1	15	yes	—
12	C₂³×C₃×C₄	C₂×C₄	1	15	yes	—
13	C₂²×C₃×C₈	C₂×C₄	1	105	yes	—
14	C₂²×C₃×C₈	C₂³	1	35	yes	—
15	C₂×C₃⋊C₁₆	C₂³	1	120	yes	direct product factor
16	C₂⁴×C₄	Dic₃	1	1	yes	characteristic
17	C₂³×C₈	Dic₃	1	15	yes	—
18	C₂³×C₈	C₂×D₃	1	15	yes	—
19	C₂²×C₁₆	—	3	140	no	—
20	C₂⁴×C₃	C₁₆	1	16	yes	complemented
21	C₂⁴×C₃	C₂×C₈	1	15	yes	—
22	C₂²×C₃×C₄	C₂×C₈	1	105	yes	—
23	C₂²×C₃×C₄	C₂²×C₄	1	35	yes	—
24	C₂×C₃×C₈	C₂²×C₄	1	105	yes	—
25	C₂×C₃×C₈	C₂⁴	1	15	yes	—
26	C₃⋊C₁₆	C₂⁴	1	16	yes	direct product factor
27	C₂⁵	C₃⋊C₈	1	1	yes	characteristic
28	C₂³×C₄	C₃⋊C₈	1	15	yes	—
29	C₂³×C₄	C₂×Dic₃	1	15	yes	—
30	C₂²×C₈	C₂×Dic₃	1	105	yes	—
31	C₂²×C₈	C₂²×D₃	1	35	yes	—
32	C₂×C₁₆	—	3	120	no	—
33	C₂³×C₃	C₂×C₁₆	1	120	yes	complemented
34	C₂³×C₃	C₂²×C₈	1	35	yes	—
35	C₂×C₃×C₄	C₂²×C₈	1	105	yes	—
36	C₂×C₃×C₄	C₂³×C₄	1	15	yes	—
37	C₃×C₈	C₂³×C₄	1	15	yes	—
38	C₃×C₈	C₂⁵	1	1	yes	characteristic
39	C₂⁴	C₃⋊C₁₆	1	16	yes	direct product factor
40	C₂⁴	C₂×C₃⋊C₈	1	15	yes	—
41	C₂²×C₄	C₂×C₃⋊C₈	1	105	yes	—
42	C₂²×C₄	C₂²×Dic₃	1	35	yes	—
43	C₂×C₈	C₂²×Dic₃	1	105	yes	—
44	C₂×C₈	C₂³×D₃	1	15	yes	—
45	C₁₆	—	3	16	no	—
46	C₂²×C₃	C₂²×C₁₆	1	140	yes	complemented
47	C₂²×C₃	C₂³×C₈	1	15	yes	—
48	C₃×C₄	C₂³×C₈	1	15	yes	—
49	C₃×C₄	C₂⁴×C₄	1	1	yes	characteristic
50	C₂³	C₂×C₃⋊C₁₆	1	120	yes	direct product factor
51	C₂³	C₂²×C₃⋊C₈	1	35	yes	—
52	C₂×C₄	C₂²×C₃⋊C₈	1	105	yes	—
53	C₂×C₄	C₂³×Dic₃	1	15	yes	—
54	C₈	C₂³×Dic₃	1	15	yes	—
55	C₈	C₂⁴×D₃	1	1	yes	characteristic
56	C₂×C₃	C₂³×C₁₆	1	30	yes	complemented
57	C₂×C₃	C₂⁴×C₈	1	1	yes	characteristic
58	C₂²	C₂²×C₃⋊C₁₆	1	140	yes	direct product factor
59	C₂²	C₂³×C₃⋊C₈	1	15	yes	—
60	C₄	C₂³×C₃⋊C₈	1	15	yes	—
61	C₄	C₂⁴×Dic₃	1	1	yes	characteristic
62	C₃	C₂⁴×C₁₆	1	1	yes	characteristic, complemented, Sylow
63	C₂	C₂³×C₃⋊C₁₆	1	30	yes	direct product factor
64	C₂	C₂⁴×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂⁴×C₈	D₃
commutator, hypocenter	C₃	C₂⁴×C₁₆
Frattini	C₈	C₂⁴×D₃
Fitting	C₂⁴×C₃×C₈	C₂
socle, prime radical	C₂⁵×C₃	C₈
cosocle	C₃×C₈	C₂⁵
LCS	G ▷ C₃
UCS	C₁ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂³×C₃⋊C₁₆	C₂	direct (N × H)
7. C₂²×C₃⋊C₁₆	C₂²	direct (N × H)
15. C₂×C₃⋊C₁₆	C₂³	direct (N × H)
20. C₂⁴×C₃	C₁₆	semidirect
26. C₃⋊C₁₆	C₂⁴	direct (N × H)
33. C₂³×C₃	C₂×C₁₆	semidirect
39. C₂⁴	C₃⋊C₁₆	direct (N × H)
46. C₂²×C₃	C₂²×C₁₆	semidirect
50. C₂³	C₂×C₃⋊C₁₆	direct (N × H)
56. C₂×C₃	C₂³×C₁₆	semidirect
58. C₂²	C₂²×C₃⋊C₁₆	direct (N × H)
62. C₃	C₂⁴×C₁₆	semidirect
63. C₂	C₂³×C₃⋊C₁₆	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×C₃×C₈	2
C₂³×C₃⋊C₁₆	2
C₂⁴×C₁₆	3

Subgroups — C24×C3⋊C16 #11701645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃⋊C₁₆ #11701645