Subgroups — (C₃⋊D₃)⋊H(3,𝔽₃) #92257

Order 486.

153 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	C₃²×C₃⋊D₃	C₃	1	1	yes	maximal, characteristic, complemented
3	D₃×H(3,𝔽₃)	—	3	1	no	maximal
4	C₃×(C₃⋊D₃)⋊C₃	C₃	1	3	yes	maximal, complemented
5	C₃⁴	C₂×C₃	1	1	yes	characteristic, complemented
6	C₃×H(3,𝔽₃)	—	2	1	no	—
7	C₃×H(3,𝔽₃)	—	2	3	no	—
8	C₃×H(3,𝔽₃)	D₃	1	1	yes	characteristic, complemented
9	C₃×H(3,𝔽₃)	C₂×C₃	1	3	yes	complemented
10	C₃×C₃⋊D₃	—	3	1	no	—
11	C₃×C₃⋊D₃	C₃²	1	1	yes	characteristic
12	C₃²×D₃	—	3	1	no	—
13	C₃²×D₃	—	3	3	no	—
14	C₃²×D₃	—	9	1	no	—
15	C₂×H(3,𝔽₃)	—	9	1	no	—
16	(C₃⋊D₃)⋊C₃	—	3	3	no	—
17	C₃³	—	2	1	no	—
18	C₃³	—	2	3	no	—
19	C₃³	—	3	1	no	—
20	C₃³	—	3	1	no	—
21	C₃³	—	6	3	no	—
22	C₃³	—	6	1	no	—
23	C₃³	—	6	1	no	—
24	C₃³	C₃×D₃	1	1	yes	characteristic
25	C₃³	C₃×D₃	1	3	yes	complemented
26	C₃³	C₂×C₃²	1	1	yes	characteristic
27	H(3,𝔽₃)	—	3	1	no	—
28	H(3,𝔽₃)	—	3	3	no	—
29	H(3,𝔽₃)	—	6	1	no	—
30	H(3,𝔽₃)	—	6	3	no	—
31	H(3,𝔽₃)	—	6	9	no	—
32	H(3,𝔽₃)	—	6	2	no	—
33	H(3,𝔽₃)	—	6	1	no	—
34	C₂×C₃²	—	9	3	no	—
35	C₂×C₃²	—	9	1	no	—
36	C₃⋊D₃	H(3,𝔽₃)	1	1	yes	characteristic, complemented
37	C₃×D₃	—	3	1	no	—
38	C₃×D₃	—	9	3	no	—
39	C₃×D₃	—	9	1	no	—
40	C₃×D₃	—	9	1	no	—
41	C₃×D₃	—	9	3	no	—
42	C₃²	—	2	3	no	—
43	C₃²	—	2	1	no	—
44	C₃²	—	2	1	no	—
45	C₃²	—	3	3	no	—
46	C₃²	—	3	1	no	—
47	C₃²	—	3	1	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	3	3	no	—
50	C₃²	—	6	1	no	—
51	C₃²	—	6	1	no	—
52	C₃²	—	6	1	no	—
53	C₃²	—	6	3	no	—
54	C₃²	—	6	6	no	—
55	C₃²	—	6	3	no	—
56	C₃²	—	6	3	no	—
57	C₃²	—	6	3	no	—
58	C₃²	—	6	3	no	—
59	C₃²	—	6	1	no	—
60	C₃²	—	6	1	no	—
61	C₃²	—	6	3	no	—
62	C₃²	—	6	3	no	—
63	C₃²	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic, complemented
64	C₃²	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
65	C₃²	C₃²×D₃	1	1	yes	characteristic
66	C₂×C₃	—	9	1	no	—
67	C₂×C₃	—	27	1	no	—
68	C₂×C₃	—	27	3	no	—
69	D₃	—	3	1	no	—
70	D₃	—	9	1	no	—
71	C₃	—	2	1	no	—
72	C₃	—	3	1	no	—
73	C₃	—	3	1	no	—
74	C₃	—	6	3	no	—
75	C₃	—	6	1	no	—
76	C₃	—	6	1	no	—
77	C₃	—	9	3	no	—
78	C₃	—	18	3	no	—
79	C₃	C₃×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
80	C₃	D₃×H(3,𝔽₃)	1	1	yes	characteristic
81	C₂	—	9	1	no	Sylow

	subgroup	G/H
center	C₃	C₃×(C₃⋊D₃)⋊C₃
commutator, cosocle	C₃³	C₂×C₃²
Frattini, socle	C₃²	C₃²×D₃
Fitting	C₃²⋊H(3,𝔽₃)	C₂
hypercenter	C₃²	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₂×H(3,𝔽₃)
LCS	G ▷ C₃³ ▷ C₃²
UCS	C₁ ◁ C₃ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. C₃²⋊H(3,𝔽₃)	C₂	semidirect
2. C₃²×C₃⋊D₃	C₃	semidirect
4. C₃×(C₃⋊D₃)⋊C₃	C₃	semidirect
5. C₃⁴	C₂×C₃	semidirect
8. C₃×H(3,𝔽₃)	D₃	semidirect
9. C₃×H(3,𝔽₃)	C₂×C₃	semidirect
25. C₃³	C₃×D₃	semidirect
36. C₃⋊D₃	H(3,𝔽₃)	semidirect
63. C₃²	(C₃⋊D₃)⋊C₃	semidirect
64. C₃²	C₂×H(3,𝔽₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃²⋊H(3,𝔽₃)	2
C₃²×C₃⋊D₃	3
D₃×H(3,𝔽₃)	3
C₃×(C₃⋊D₃)⋊C₃	3

Subgroups — (C3⋊D3)⋊H(3,𝔽3) #92257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — (C₃⋊D₃)⋊H(3,𝔽₃) #92257