Subgroups — C₂⁴×Dic₆ #87552

Order 384.

6264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₄	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₃	C₂	1	2	yes	maximal
3	C₂³×Dic₆	C₂	1	60	yes	maximal, direct product factor
4	C₂⁴×Q₈	—	3	1	no	maximal, Sylow
5	C₂⁵×C₃	C₂²	1	1	yes	characteristic
6	C₂³×C₃×C₄	C₂²	1	30	yes	—
7	C₂³×Dic₃	C₂²	1	60	yes	—
8	C₂²×Dic₆	C₂²	1	560	yes	direct product factor
9	C₂³×Q₈	—	3	60	no	—
10	C₂⁴×C₄	—	3	2	no	—
11	C₂⁴×C₄	D₃	1	1	yes	characteristic
12	C₂⁴×C₃	Q₈	1	16	yes	complemented
13	C₂⁴×C₃	C₂³	1	15	yes	—
14	C₂²×C₃×C₄	C₂³	1	140	yes	—
15	C₂²×Dic₃	C₂³	1	280	yes	—
16	C₂×Dic₆	C₂³	1	960	yes	direct product factor
17	C₂⁵	C₂×D₃	1	1	yes	characteristic
18	C₂²×Q₈	—	3	560	no	—
19	C₂³×C₄	—	3	60	no	—
20	C₂³×C₄	C₂×D₃	1	30	yes	—
21	C₂³×C₃	C₂×Q₈	1	120	yes	complemented
22	C₂³×C₃	C₂⁴	1	35	yes	—
23	C₂×C₃×C₄	C₂⁴	1	120	yes	—
24	C₂×Dic₃	C₂⁴	1	240	yes	—
25	Dic₆	C₂⁴	1	256	yes	direct product factor
26	C₂⁴	Dic₆	1	16	yes	direct product factor
27	C₂⁴	C₂²×D₃	1	15	yes	—
28	C₂×Q₈	—	3	960	no	—
29	C₂²×C₄	—	3	280	no	—
30	C₂²×C₄	C₂²×D₃	1	140	yes	—
31	C₂²×C₃	C₂²×Q₈	1	140	yes	complemented
32	C₂²×C₃	C₂⁵	1	15	yes	—
33	C₃×C₄	C₂⁵	1	16	yes	—
34	Dic₃	C₂⁵	1	32	yes	—
35	C₂³	C₂×Dic₆	1	120	yes	direct product factor
36	C₂³	C₂³×D₃	1	35	yes	—
37	Q₈	—	3	256	no	—
38	C₂×C₄	—	3	240	no	—
39	C₂×C₄	C₂³×D₃	1	120	yes	—
40	C₂×C₃	C₂³×Q₈	1	30	yes	complemented
41	C₂×C₃	C₂⁶	1	1	yes	characteristic
42	C₂²	C₂²×Dic₆	1	140	yes	direct product factor
43	C₂²	C₂⁴×D₃	1	15	yes	—
44	C₄	—	3	32	no	—
45	C₄	C₂⁴×D₃	1	16	yes	—
46	C₃	C₂⁴×Q₈	1	1	yes	characteristic, complemented, Sylow
47	C₂	C₂³×Dic₆	1	30	yes	direct product factor
48	C₂	C₂⁵×D₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂×D₃
commutator, cosocle	C₂×C₃	C₂⁶
Frattini	C₂	C₂⁵×D₃
Fitting	C₂⁴×C₃×C₄	C₂
socle, prime radical	C₂⁵×C₃	C₂²
hypercenter	C₂⁴×C₄	D₃
hypocenter	C₃	C₂⁴×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
3. C₂³×Dic₆	C₂	direct (N × H)
8. C₂²×Dic₆	C₂²	direct (N × H)
12. C₂⁴×C₃	Q₈	semidirect
16. C₂×Dic₆	C₂³	direct (N × H)
21. C₂³×C₃	C₂×Q₈	semidirect
25. Dic₆	C₂⁴	direct (N × H)
26. C₂⁴	Dic₆	direct (N × H)
31. C₂²×C₃	C₂²×Q₈	semidirect
35. C₂³	C₂×Dic₆	direct (N × H)
40. C₂×C₃	C₂³×Q₈	semidirect
42. C₂²	C₂²×Dic₆	direct (N × H)
46. C₃	C₂⁴×Q₈	semidirect
47. C₂	C₂³×Dic₆	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×C₄	2
C₂⁴×Dic₃	2
C₂³×Dic₆	2
C₂⁴×Q₈	3

Subgroups — C24×Dic6 #87552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×Dic₆ #87552