Subgroups — #10619613

Order 648.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#731	C₂	1	1	yes	maximal, characteristic
2	#66323	C₂	1	2	yes	maximal
3	#6229	C₃	1	1	yes	maximal, characteristic
4	C₉×Dic₆	—	3	1	no	maximal
5	C₂×#731	C₂²	1	1	yes	characteristic
6	C₄×3₋¹⁺²	—	2	1	no	—
7	C₄×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
8	C₃×C₄×C₉	D₃	1	1	yes	characteristic
9	#1084	C₂×C₃	1	2	yes	—
10	C₉×Dic₃	—	3	2	no	—
11	#731	Q₈	1	1	yes	characteristic, complemented, Sylow
12	C₃×Dic₆	—	3	1	no	—
13	C₃×Dic₆	—	9	1	no	—
14	Q₈×C₉	—	9	1	no	—
15	C₂×3₋¹⁺²	—	2	1	no	—
16	C₂×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
17	C₂×C₃×C₉	C₂×D₃	1	1	yes	characteristic
18	C₃²×C₄	—	3	1	no	—
19	C₃²×C₄	C₃×D₃	1	1	yes	characteristic
20	C₃×Dic₃	—	3	2	no	—
21	C₃×Dic₃	—	9	2	no	—
22	C₄×C₉	—	6	1	no	—
23	C₄×C₉	—	6	1	no	—
24	C₄×C₉	—	9	2	no	—
25	3₋¹⁺²	—	2	1	no	—
26	H(3,𝔽₃)	C₃×Q₈	1	1	yes	characteristic
27	C₃×C₉	Dic₆	1	1	yes	characteristic, complemented
28	C₃×Q₈	—	9	1	no	—
29	Dic₆	—	3	1	no	—
30	Dic₆	—	9	1	no	—
31	C₂×C₃²	—	3	1	no	—
32	C₂×C₃²	C₂×C₃×D₃	1	1	yes	characteristic
33	C₂×C₉	—	6	1	no	—
34	C₂×C₉	—	6	1	no	—
35	C₃×C₄	—	6	1	no	—
36	C₃×C₄	—	9	2	no	—
37	C₃×C₄	—	18	1	no	—
38	C₃×C₄	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
39	Dic₃	—	3	2	no	—
40	Dic₃	—	9	2	no	—
41	C₃²	—	3	1	no	—
42	C₃²	C₃×Dic₆	1	1	yes	characteristic
43	C₉	—	6	1	no	—
44	C₉	—	6	1	no	—
45	Q₈	—	9	1	no	Sylow
46	C₂×C₃	—	6	1	no	—
47	C₂×C₃	—	18	1	no	—
48	C₂×C₃	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
49	C₄	—	9	2	no	—
50	C₄	#4176	1	1	yes	characteristic
51	C₃	—	6	1	no	—
52	C₃	—	18	1	no	—
53	C₃	#634⋊C₃	1	1	yes	characteristic
54	C₂	C₂×#4176	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂×(C₃⋊D₃)⋊C₃
commutator, prime radical, cosocle	C₂×H(3,𝔽₃)	C₂²×C₃
Frattini	C₂×C₃²	C₂×C₃×D₃
Fitting	C₄×#731	C₂
hypercenter	C₃×C₄	(C₃⋊D₃)⋊C₃
hypocenter	H(3,𝔽₃)	C₃×Q₈
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

Complements

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

N	H ≅ G/N	type
11. #731	Q₈	semidirect
27. 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
C₄×#731	2
#66323	2
#6229	3
C₉×Dic₆	3