Subgroups — #10683089

Order 768.

66 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	#67591	C₂	1	2	yes	maximal
3	#7546	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
5	#4504	—	2	2	no	—
6	C₃×C₆₄	C₄	1	2	yes	complemented
7	C₂×C₆₄	D₃	1	1	yes	characteristic
8	#1423	—	3	2	no	—
9	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
10	#818	—	4	2	no	—
11	C₃×C₃₂	C₂×C₄	1	1	yes	characteristic
12	C₃×C₃₂	Q₈	1	1	yes	characteristic
13	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
14	#366	—	6	2	no	—
15	C₆₄	Dic₃	1	2	yes	complemented
16	C₂×C₃×C₈	D₈	1	1	yes	characteristic
17	C₄⋊Dic₃	—	8	2	no	—
18	C₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
19	C₃×C₁₆	Q₁₆	1	1	yes	characteristic
20	C₂×C₁₆	D₁₂	1	1	yes	characteristic
21	#107	—	12	2	no	—
22	C₃₂	Dic₆	1	1	yes	characteristic
23	C₃₂	C₂×Dic₃	1	1	yes	characteristic
24	C₂×C₃×C₄	D₁₆	1	1	yes	characteristic
25	C₂×Dic₃	—	16	2	no	—
26	C₃×C₈	#107	1	1	yes	characteristic
27	C₃×C₈	Q₃₂	1	1	yes	characteristic
28	C₂×C₈	D₂₄	1	1	yes	characteristic
29	C₄⋊C₄	—	24	2	no	—
30	C₁₆	Dic₁₂	1	1	yes	characteristic
31	C₁₆	C₄⋊Dic₃	1	1	yes	characteristic
32	C₂²×C₃	D₃₂	1	1	yes	characteristic
33	C₃×C₄	#366	1	1	yes	characteristic
34	C₃×C₄	Q₆₄	1	1	yes	characteristic
35	Dic₃	—	32	2	no	—
36	C₂×C₄	—	48	2	no	—
37	C₂×C₄	D₄₈	1	1	yes	characteristic
38	C₈	Dic₂₄	1	1	yes	characteristic
39	C₈	#818	1	1	yes	characteristic
40	C₂×C₃	#1423	1	1	yes	characteristic
41	C₂×C₃	SD₁₂₈	1	2	yes	—
42	C₂²	D₉₆	1	1	yes	characteristic
43	C₄	—	96	2	no	—
44	C₄	Dic₄₈	1	1	yes	characteristic
45	C₄	#4504	1	1	yes	characteristic
46	C₃	#7546	1	1	yes	characteristic, complemented, Sylow
47	C₂	#67434	1	2	yes	—
48	C₂	#67591	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₉₆
commutator	C₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₃
Fitting	C₂×C₃×C₆₄	C₂
socle, prime radical	C₂²×C₃	D₃₂
cosocle	C₂×C₃×C₃₂	C₂²
hypercenter	C₂×C₆₄	D₃
hypocenter	C₃	#7546
LCS	G ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
6. C₃×C₆₄	C₄	semidirect
15. C₆₄	Dic₃	semidirect
46. C₃	#7546	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
#67591	2
#7546	3