Subgroups — #13091498

Order 1458.

671 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(3,𝔽₃)²	C₂	1	1	yes	maximal, characteristic, complemented, Sylow
2	#92300	C₃	1	4	yes	maximal, complemented
3	(C₃⋊D₃)⋊H(3,𝔽₃)	—	3	4	no	maximal
4	C₃²×H(3,𝔽₃)	—	2	4	no	—
5	C₃²⋊H(3,𝔽₃)	—	2	12	no	—
6	C₃²⋊H(3,𝔽₃)	D₃	1	4	yes	complemented
7	C₃²⋊H(3,𝔽₃)	C₂×C₃	1	4	yes	complemented
8	C₃²×C₃⋊D₃	—	3	4	no	—
9	C₃⋊#272	C₃²	1	1	yes	characteristic, complemented
10	D₃×H(3,𝔽₃)	—	9	1	no	—
11	C₃×(C₃⋊D₃)⋊C₃	—	3	12	no	—
12	C₃⁴	—	2	6	no	—
13	C₃⁴	C₃×D₃	1	4	yes	complemented
14	C₃×H(3,𝔽₃)	—	2	1	no	—
15	C₃×H(3,𝔽₃)	—	2	8	no	—
16	C₃×H(3,𝔽₃)	—	2	12	no	—
17	C₃×H(3,𝔽₃)	—	2	12	no	—
18	C₃×H(3,𝔽₃)	—	2	3	no	—
19	C₃×H(3,𝔽₃)	—	2	6	no	—
20	C₃×H(3,𝔽₃)	—	2	4	no	—
21	C₃×H(3,𝔽₃)	—	2	4	no	—
22	C₃×H(3,𝔽₃)	—	6	36	no	—
23	C₃×H(3,𝔽₃)	C₃×D₃	1	12	yes	complemented
24	C₃×H(3,𝔽₃)	C₃⋊D₃	1	1	yes	characteristic
25	C₃×H(3,𝔽₃)	C₂×C₃²	1	1	yes	characteristic
26	C₃×C₃⋊D₃	—	3	4	no	—
27	C₃×C₃⋊D₃	—	9	4	no	—
28	C₃²×D₃	—	9	4	no	—
29	C₃²×D₃	—	27	4	no	—
30	C₂×H(3,𝔽₃)	—	27	1	no	—
31	#272	—	27	1	no	—
32	(C₃⋊D₃)⋊C₃	—	9	12	no	—
33	C₃³	—	2	4	no	—
34	C₃³	—	2	12	no	—
35	C₃³	—	6	12	no	—
36	C₃³	—	6	36	no	—
37	C₃³	—	9	4	no	—
38	C₃³	—	9	4	no	—
39	C₃³	—	18	6	no	—
40	C₃³	—	18	6	no	—
41	C₃³	C₃²×D₃	1	4	yes	—
42	C₃³	C₃×C₃⋊D₃	1	4	yes	—
43	H(3,𝔽₃)	—	2	9	no	—
44	H(3,𝔽₃)	—	6	36	no	—
45	H(3,𝔽₃)	—	6	36	no	—
46	H(3,𝔽₃)	—	6	24	no	—
47	H(3,𝔽₃)	—	9	1	no	—
48	H(3,𝔽₃)	—	9	1	no	—
49	H(3,𝔽₃)	—	9	12	no	—
50	H(3,𝔽₃)	—	18	4	no	—
51	H(3,𝔽₃)	—	18	4	no	—
52	H(3,𝔽₃)	—	18	3	no	—
53	H(3,𝔽₃)	—	18	6	no	—
54	C₂×C₃²	—	27	4	no	—
55	C₃⋊D₃	—	3	4	no	—
56	C₃×D₃	—	9	1	no	—
57	C₃×D₃	—	27	4	no	—
58	C₃×D₃	—	27	4	no	—
59	C₃×D₃	—	27	12	no	—
60	C₃²	—	2	12	no	—
61	C₃²	—	3	4	no	—
62	C₃²	—	3	4	no	—
63	C₃²	—	3	4	no	—
64	C₃²	—	3	4	no	—
65	C₃²	—	6	4	no	—
66	C₃²	—	6	4	no	—
67	C₃²	—	6	4	no	—
68	C₃²	—	6	24	no	—
69	C₃²	—	6	4	no	—
70	C₃²	—	6	4	no	—
71	C₃²	—	6	12	no	—
72	C₃²	—	6	12	no	—
73	C₃²	—	9	12	no	—
74	C₃²	—	18	12	no	—
75	C₃²	—	18	54	no	—
76	C₃²	C₃²×C₃⋊D₃	1	1	yes	characteristic
77	C₂×C₃	—	27	1	no	—
78	C₂×C₃	—	81	4	no	—
79	D₃	—	9	1	no	—
80	D₃	—	27	4	no	—
81	C₃	—	2	1	no	—
82	C₃	—	6	12	no	—
83	C₃	—	9	4	no	—
84	C₃	—	9	4	no	—
85	C₃	—	18	12	no	—
86	C₃	#92391	1	1	yes	characteristic
87	C₃	#92392	1	1	yes	characteristic
88	C₂	—	27	1	no	Sylow

	subgroup	G/H
center, hypercenter	C₃	#92391
commutator, cosocle, hypocenter	C₃×H(3,𝔽₃)	C₂×C₃²
Frattini, socle	C₃²	C₃²×C₃⋊D₃
Fitting	H(3,𝔽₃)²	C₂
LCS	G ▷ C₃×H(3,𝔽₃)
UCS	C₁ ◁ C₃

Complements

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

N	H ≅ G/N	type
1. H(3,𝔽₃)²	C₂	semidirect
2. #92300	C₃	semidirect
6. C₃²⋊H(3,𝔽₃)	D₃	semidirect
7. C₃²⋊H(3,𝔽₃)	C₂×C₃	semidirect
9. C₃⋊#272	C₃²	semidirect
13. C₃⁴	C₃×D₃	semidirect
23. C₃×H(3,𝔽₃)	C₃×D₃	semidirect

Maximal subgroups

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

subgroup	index
H(3,𝔽₃)²	2
#92300	3
(C₃⋊D₃)⋊H(3,𝔽₃)	3