Subgroups — Dic₃⋊C₆₄ #10683082

Order 768.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₆₄	C₂	1	1	yes	maximal, characteristic
2	Dic₃×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊C₆₄	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₆₄	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
6	Dic₃×C₁₆	C₄	1	1	yes	characteristic
7	C₂×C₃⋊C₃₂	C₄	1	1	yes	characteristic
8	C₃×C₆₄	—	2	1	no	—
9	C₃⋊C₆₄	—	2	1	no	—
10	C₂×C₆₄	—	3	1	no	—
11	C₂×C₆₄	D₃	1	1	yes	characteristic
12	C₄×C₃₂	—	3	1	no	—
13	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
14	C₈×Dic₃	C₈	1	1	yes	characteristic
15	C₂×C₃⋊C₁₆	C₈	1	1	yes	characteristic
16	C₃×C₃₂	D₄	1	1	yes	characteristic
17	C₃×C₃₂	Q₈	1	1	yes	characteristic
18	C₃⋊C₃₂	—	2	1	no	—
19	C₂×C₃₂	—	3	1	no	—
20	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
21	C₄×C₁₆	—	3	1	no	—
22	C₆₄	—	2	1	no	—
23	C₆₄	—	6	1	no	—
24	C₂×C₃×C₈	C₂×C₈	1	1	yes	characteristic
25	C₄×Dic₃	C₁₆	1	1	yes	characteristic
26	C₂×C₃⋊C₈	C₁₆	1	1	yes	characteristic
27	C₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
28	C₃×C₁₆	M₁₆	1	1	yes	characteristic
29	C₃⋊C₁₆	—	2	1	no	—
30	C₂×C₁₆	—	3	1	no	—
31	C₂×C₁₆	C₄×D₃	1	1	yes	characteristic
32	C₄×C₈	—	3	1	no	—
33	C₃₂	—	6	1	no	—
34	C₃₂	Dic₆	1	1	yes	characteristic
35	C₃₂	#67	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
37	C₂×Dic₃	C₃₂	1	1	yes	characteristic
38	C₂×Dic₃	C₃₂	1	1	yes	characteristic
39	C₃×C₈	C₄⋊C₈	1	1	yes	characteristic
40	C₃×C₈	M₃₂	1	1	yes	characteristic
41	C₃⋊C₈	—	2	1	no	—
42	C₂×C₈	—	3	1	no	—
43	C₂×C₈	D₃×C₈	1	1	yes	characteristic
44	C₄²	—	3	1	no	—
45	C₁₆	—	6	1	no	—
46	C₁₆	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
47	C₁₆	Dic₃⋊C₄	1	1	yes	characteristic
48	C₂²×C₃	C₂×C₃₂	1	1	yes	characteristic
49	C₃×C₄	C₄⋊C₁₆	1	1	yes	characteristic
50	C₃×C₄	M₆₄	1	1	yes	characteristic
51	Dic₃	—	2	1	no	—
52	Dic₃	C₆₄	1	2	yes	complemented
53	C₂×C₄	—	3	1	no	—
54	C₂×C₄	—	3	1	no	—
55	C₂×C₄	D₃×C₁₆	1	1	yes	characteristic
56	C₈	—	6	1	no	—
57	C₈	#798	1	1	yes	characteristic
58	C₈	Dic₃⋊C₈	1	1	yes	characteristic
59	C₂×C₃	C₄⋊C₃₂	1	1	yes	characteristic
60	C₂×C₃	C₂×C₆₄	1	1	yes	characteristic
61	C₂×C₃	M₁₂₈	1	1	yes	characteristic
62	C₂²	D₃×C₃₂	1	1	yes	characteristic
63	C₄	—	3	2	no	—
64	C₄	—	6	1	no	—
65	C₄	#4447	1	1	yes	characteristic
66	C₄	Dic₃⋊C₁₆	1	1	yes	characteristic
67	C₃	C₄⋊C₆₄	1	1	yes	characteristic, complemented, Sylow
68	C₂	D₃×C₆₄	1	1	yes	characteristic
69	C₂	#67432	1	1	yes	characteristic
70	C₂	Dic₃⋊C₃₂	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₃₂	C₂×D₃
commutator	C₂×C₃	C₂×C₆₄
Fitting	C₂×C₃×C₆₄	C₂
socle, prime radical	C₂²×C₃	C₂×C₃₂
cosocle	C₂×C₃×C₃₂	C₂²
hypercenter	C₂×C₆₄	D₃
hypocenter	C₃	C₄⋊C₆₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
52. Dic₃	C₆₄	semidirect
67. C₃	C₄⋊C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₆₄	2
Dic₃×C₃₂	2
C₂×C₃⋊C₆₄	2
C₄⋊C₆₄	3

Subgroups — Dic3⋊C64 #10683082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₃⋊C₆₄ #10683082