Subgroups — C₂×#4207 #66455

Order 324.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#737	C₂	1	1	yes	maximal, characteristic, complemented
2	#4207	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃⋊D₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×AGL(1,ℤ/9)	—	3	2	no	maximal
5	C₂×(C₃⋊D₃)⋊C₃	—	3	1	no	maximal
6	C₂×C₃×D₉	—	3	1	no	maximal
7	#737	C₂²	1	1	yes	characteristic, complemented, Sylow
8	C₂×3₋¹⁺²	—	2	1	no	—
9	C₂×3₋¹⁺²	—	2	2	no	—
10	C₂×3₋¹⁺²	D₃	1	2	yes	complemented
11	C₂×H(3,𝔽₃)	D₃	1	1	yes	characteristic
12	C₂×C₃×C₉	—	2	1	no	—
13	C₂×C₃×C₉	D₃	1	1	yes	characteristic, complemented
14	C₂×C₃×C₉	C₂×C₃	1	1	yes	characteristic, complemented
15	C₃⋊D₉	C₂×C₃	1	2	yes	complemented
16	AGL(1,ℤ/9)	—	3	4	no	—
17	(C₃⋊D₃)⋊C₃	—	3	2	no	—
18	C₃×D₉	—	3	2	no	—
19	C₂×C₃⋊D₃	—	3	1	no	—
20	C₂×C₃×D₃	—	9	1	no	—
21	C₂×D₉	—	3	1	no	—
22	C₂×D₉	—	3	2	no	—
23	3₋¹⁺²	—	2	1	no	—
24	3₋¹⁺²	—	2	2	no	—
25	3₋¹⁺²	C₂×D₃	1	2	yes	complemented
26	H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic
27	C₃×C₉	—	2	1	no	—
28	C₃×C₉	C₂×D₃	1	1	yes	characteristic, complemented
29	C₃×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
30	C₂×C₃²	—	2	1	no	—
31	C₂×C₃²	C₃×D₃	1	1	yes	characteristic
32	C₂×C₃²	C₃⋊D₃	1	1	yes	characteristic
33	C₃⋊D₃	—	3	2	no	—
34	C₃×D₃	—	9	2	no	—
35	C₂×C₉	—	2	2	no	—
36	C₂×C₉	—	2	1	no	—
37	C₂×C₉	C₃×D₃	1	1	yes	characteristic
38	C₂×C₉	C₃×D₃	1	2	yes	—
39	D₉	—	3	2	no	—
40	D₉	—	3	4	no	—
41	C₂²×C₃	—	27	1	no	—
42	C₂×D₃	—	9	1	no	—
43	C₂×D₃	—	27	1	no	—
44	C₃²	—	2	1	no	—
45	C₃²	C₂×C₃×D₃	1	1	yes	characteristic
46	C₃²	C₂×C₃⋊D₃	1	1	yes	characteristic
47	C₉	—	2	2	no	—
48	C₉	—	2	1	no	—
49	C₉	C₂×C₃×D₃	1	1	yes	characteristic
50	C₉	C₂×C₃×D₃	1	2	yes	—
51	C₂×C₃	—	3	1	no	—
52	C₂×C₃	—	3	1	no	—
53	C₂×C₃	—	6	1	no	—
54	C₂×C₃	—	27	2	no	—
55	C₂×C₃	C₃×C₃⋊D₃	1	1	yes	characteristic
56	D₃	—	9	2	no	—
57	D₃	—	27	2	no	—
58	C₂²	—	27	1	no	Sylow
59	C₃	—	3	1	no	—
60	C₃	—	3	1	no	—
61	C₃	—	6	1	no	—
62	C₃	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
63	C₂	—	27	2	no	—
64	C₂	#4207	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	#4207
commutator, cosocle, hypocenter	C₃×C₉	C₂²×C₃
Frattini	C₃	C₂×C₃×C₃⋊D₃
Fitting	C₂×#737	C₂
socle	C₂×C₃	C₃×C₃⋊D₃
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×#737	C₂	semidirect
2. #4207	C₂	direct (N × H)
3. C₂×C₃⋊D₉	C₃	semidirect
7. #737	C₂²	semidirect
10. C₂×3₋¹⁺²	D₃	semidirect
13. C₂×C₃×C₉	D₃	semidirect
14. C₂×C₃×C₉	C₂×C₃	semidirect
15. C₃⋊D₉	C₂×C₃	semidirect
25. 3₋¹⁺²	C₂×D₃	semidirect
28. C₃×C₉	C₂×D₃	semidirect
29. C₃×C₉	C₂²×C₃	semidirect
64. C₂	#4207	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#737	2
#4207	2
C₂×C₃⋊D₉	3
C₂×AGL(1,ℤ/9)	3
C₂×(C₃⋊D₃)⋊C₃	3
C₂×C₃×D₉	3

Subgroups — C2×#4207 #66455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4207 #66455