Subgroups — C₃²×Ω⁺(4,𝔽₃) #72252948

Order 2592.

1320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#997	C₃	1	2	yes	maximal, complemented
2	C₃²×#994	C₃	1	2	yes	maximal, complemented
3	C₃×Ω⁺(4,𝔽₃)	C₃	1	36	yes	maximal, direct product factor
4	C₃³×SL(2,𝔽₃)	—	4	2	no	maximal
5	C₃²×#41⋊C₃	—	3	2	no	—
6	C₃²×H(5,𝔽₂)	C₃²	1	1	yes	characteristic, complemented
7	C₃×#997	C₃²	1	24	yes	complemented
8	C₃×#994	C₃²	1	24	yes	complemented
9	Ω⁺(4,𝔽₃)	C₃²	1	81	yes	direct product factor
10	C₂×C₃²×A₄	—	12	2	no	—
11	C₃³×Q₈	—	4	2	no	—
12	C₃²×SL(2,𝔽₃)	—	4	4	no	—
13	C₃²×SL(2,𝔽₃)	—	4	72	no	—
14	C₃²×SL(2,𝔽₃)	A₄	1	2	yes	complemented
15	C₂×C₃⁴	—	16	1	no	—
16	C₃²×#41	—	3	2	no	—
17	C₂×C₃²×D₄	—	9	1	no	—
18	C₃×#41⋊C₃	—	3	24	no	—
19	C₃²×A₄	—	12	2	no	—
20	C₃³×C₄	—	12	2	no	—
21	C₃×H(5,𝔽₂)	C₃³	1	4	yes	complemented
22	#997	C₃³	1	18	yes	complemented
23	#994	C₃³	1	18	yes	complemented
24	C₃⁴	—	16	1	no	Sylow
25	C₂³×C₃²	—	3	2	no	—
26	C₂×C₃×A₄	—	12	24	no	—
27	C₃²×Q₈	—	4	24	no	—
28	C₃²×Q₈	C₃×A₄	1	2	yes	—
29	C₃²×D₄	—	9	2	no	—
30	C₂×C₃²×C₄	—	9	1	no	—
31	C₃×SL(2,𝔽₃)	—	4	48	no	—
32	C₃×SL(2,𝔽₃)	—	4	162	no	—
33	C₃×SL(2,𝔽₃)	C₃×A₄	1	24	yes	complemented
34	C₂×C₃³	—	4	2	no	—
35	C₂×C₃³	—	16	36	no	—
36	C₂×C₃³	—	16	2	no	—
37	C₃×#41	—	3	8	no	—
38	C₂×C₃×D₄	—	9	4	no	—
39	#41⋊C₃	—	3	18	no	—
40	C₂²×C₃²	—	9	1	no	—
41	C₂²×C₃²	—	36	2	no	—
42	C₃×A₄	—	12	24	no	—
43	C₃²×C₄	—	3	2	no	—
44	C₃²×C₄	—	12	24	no	—
45	H(5,𝔽₂)	C₃⁴	1	1	yes	characteristic, complemented, Sylow
46	C₃³	—	4	2	no	—
47	C₃³	—	16	36	no	—
48	C₃³	—	16	2	no	—
49	C₂³×C₃	—	3	8	no	—
50	C₂×A₄	—	12	18	no	—
51	C₃×Q₈	—	4	18	no	—
52	C₃×Q₈	C₃²×A₄	1	8	yes	—
53	C₃×D₄	—	9	8	no	—
54	C₂×C₃×C₄	—	9	4	no	—
55	SL(2,𝔽₃)	—	4	36	no	—
56	SL(2,𝔽₃)	C₃²×A₄	1	18	yes	complemented
57	C₂×C₃²	—	4	24	no	—
58	C₂×C₃²	—	16	81	no	—
59	C₂×C₃²	—	16	24	no	—
60	C₂×C₃²	—	18	1	no	—
61	C₂×C₃²	A₄²	1	1	yes	characteristic
62	#41	—	3	2	no	—
63	C₂×D₄	—	9	1	no	—
64	C₂²×C₃	—	9	4	no	—
65	C₂²×C₃	—	36	8	no	—
66	A₄	—	12	18	no	—
67	C₃×C₄	—	3	8	no	—
68	C₃×C₄	—	12	18	no	—
69	C₃²	—	4	24	no	—
70	C₃²	—	16	81	no	—
71	C₃²	—	16	24	no	—
72	C₃²	Ω⁺(4,𝔽₃)	1	1	yes	characteristic, direct product factor
73	C₂³	—	3	2	no	—
74	Q₈	C₃³×A₄	1	2	yes	—
75	D₄	—	9	2	no	—
76	C₂×C₄	—	9	1	no	—
77	C₂×C₃	—	4	18	no	—
78	C₂×C₃	—	16	18	no	—
79	C₂×C₃	—	18	4	no	—
80	C₂×C₃	C₃×A₄²	1	4	yes	—
81	C₂²	—	9	1	no	—
82	C₂²	—	36	2	no	—
83	C₄	—	3	2	no	—
84	C₃	—	4	18	no	—
85	C₃	—	16	18	no	—
86	C₃	C₃×Ω⁺(4,𝔽₃)	1	4	yes	direct product factor
87	C₂	—	18	1	no	—
88	C₂	C₃²×A₄²	1	1	yes	characteristic

	subgroup	G/H
center, socle, hypercenter	C₂×C₃²	A₄²
commutator, cosocle, hypocenter	H(5,𝔽₂)	C₃⁴
Frattini	C₂	C₃²×A₄²
Fitting	C₃²×H(5,𝔽₂)	C₃²
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂×C₃²

Complements

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

N	H ≅ G/N	type
1. C₃²×#997	C₃	semidirect
2. C₃²×#994	C₃	semidirect
3. C₃×Ω⁺(4,𝔽₃)	C₃	direct (N × H)
6. C₃²×H(5,𝔽₂)	C₃²	semidirect
7. C₃×#997	C₃²	semidirect
8. C₃×#994	C₃²	semidirect
9. Ω⁺(4,𝔽₃)	C₃²	direct (N × H)
14. C₃²×SL(2,𝔽₃)	A₄	semidirect
21. C₃×H(5,𝔽₂)	C₃³	semidirect
22. #997	C₃³	semidirect
23. #994	C₃³	semidirect
33. C₃×SL(2,𝔽₃)	C₃×A₄	semidirect
45. H(5,𝔽₂)	C₃⁴	semidirect
56. SL(2,𝔽₃)	C₃²×A₄	semidirect
72. C₃²	Ω⁺(4,𝔽₃)	direct (N × H)
86. C₃	C₃×Ω⁺(4,𝔽₃)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#997	3
C₃²×#994	3
C₃×Ω⁺(4,𝔽₃)	3
C₃³×SL(2,𝔽₃)	4

Subgroups — C32×Ω+(4,𝔽3) #72252948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×Ω⁺(4,𝔽₃) #72252948