Subgroups — #10619566

Order 648.

69 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66332	C₂	1	1	yes	maximal, characteristic
2	#66326	C₂	1	1	yes	maximal, characteristic
3	#66325	C₂	1	1	yes	maximal, characteristic
4	#6196	—	3	1	no	maximal
5	#6189	—	3	1	no	maximal
6	C₂×#732	C₂²	1	1	yes	characteristic
7	#1084	D₃	1	1	yes	characteristic
8	C₃⋊Dic₉	D₃	1	1	yes	characteristic
9	(C₃⋊Dic₃)⋊C₃	—	3	1	no	—
10	(C₃⋊Dic₃)⋊C₃	—	3	1	no	—
11	(C₃⋊Dic₃)⋊C₃	—	6	1	no	—
12	C₉×Dic₃	—	3	1	no	—
13	C₃×Dic₉	—	3	1	no	—
14	#732	Q₈	1	1	yes	characteristic, complemented, Sylow
15	#627	—	9	1	no	—
16	#627	—	9	1	no	—
17	Dic₁₈	—	9	1	no	—
18	C₂×H(3,𝔽₃)	—	2	1	no	—
19	C₂×H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃×C₉	C₂×D₃	1	1	yes	characteristic
21	C₃⋊Dic₃	—	3	1	no	—
22	C₃⋊Dic₃	—	3	1	no	—
23	C₃⋊Dic₃	—	6	1	no	—
24	C₃×Dic₃	—	3	1	no	—
25	C₃×Dic₃	—	9	1	no	—
26	C₃×Dic₃	—	9	1	no	—
27	C₃×Dic₃	—	18	1	no	—
28	C₄×C₉	—	9	1	no	—
29	Dic₉	—	3	1	no	—
30	Dic₉	—	18	1	no	—
31	H(3,𝔽₃)	—	2	1	no	—
32	H(3,𝔽₃)	Dic₆	1	1	yes	characteristic
33	C₃×C₉	Dic₆	1	1	yes	characteristic, complemented
34	Dic₆	—	27	1	no	—
35	Dic₆	—	27	1	no	—
36	Dic₆	—	27	1	no	—
37	C₂×C₃²	—	6	1	no	—
38	C₂×C₃²	—	12	1	no	—
39	C₂×C₃²	D₃²	1	1	yes	characteristic
40	C₂×C₉	—	6	1	no	—
41	C₃×C₄	—	9	1	no	—
42	C₃×C₄	—	27	1	no	—
43	C₃×C₄	—	27	1	no	—
44	Dic₃	—	3	1	no	—
45	Dic₃	—	9	1	no	—
46	Dic₃	—	9	1	no	—
47	Dic₃	—	18	1	no	—
48	Dic₃	—	27	1	no	—
49	Dic₃	—	54	1	no	—
50	Dic₃	—	54	1	no	—
51	C₃²	—	6	1	no	—
52	C₃²	—	12	1	no	—
53	C₃²	#627	1	1	yes	characteristic
54	C₉	—	6	1	no	—
55	Q₈	—	81	1	no	Sylow
56	C₂×C₃	—	3	1	no	—
57	C₂×C₃	—	36	1	no	—
58	C₂×C₃	—	36	1	no	—
59	C₂×C₃	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
60	C₄	—	9	1	no	—
61	C₄	—	27	1	no	—
62	C₄	—	27	1	no	—
63	C₃	—	3	1	no	—
64	C₃	—	36	1	no	—
65	C₃	—	36	1	no	—
66	C₃	#6196	1	1	yes	characteristic
67	C₂	#4178⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#4178⋊C₂
commutator, Fitting, prime radical, cosocle	C₂×#732	C₂²
Frattini	C₂×C₃²	D₃²
socle	C₂×C₃	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	#732	Q₈
LCS	G ▷ C₂×#732 ▷ #732
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
14. #732	Q₈	semidirect
33. C₃×C₉	Dic₆	semidirect

Maximal subgroups

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

subgroup	index
#66332	2
#66326	2
#66325	2
#6196	3
#6189	3