Subgroups — H(3,𝔽₃)×C₂²⋊A₄ #13049338

Order 1296.

612 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂²⋊A₄	C₃	1	4	yes	maximal, complemented
2	C₂⁴×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₂²⋊A₄)	C₃	1	8	yes	maximal, complemented
4	A₄×H(3,𝔽₃)	—	4	5	no	maximal
5	C₂³×H(3,𝔽₃)	—	3	5	no	—
6	C₂⁴×C₃²	C₃²	1	4	yes	complemented
7	C₃×C₂²⋊A₄	—	3	12	no	—
8	C₃×C₂²⋊A₄	C₃²	1	1	yes	characteristic
9	C₃×C₂²⋊A₄	C₃²	1	8	yes	complemented
10	C₃²×A₄	—	4	20	no	—
11	C₂²×H(3,𝔽₃)	—	3	10	no	—
12	C₂²×H(3,𝔽₃)	A₄	1	5	yes	complemented
13	C₂²⋊H(3,𝔽₃)	—	4	40	no	—
14	C₃×H(3,𝔽₃)	—	16	1	no	Sylow
15	C₂³×C₃²	—	3	20	no	—
16	C₂×H(3,𝔽₃)	—	3	5	no	—
17	C₂⁴×C₃	—	3	4	no	—
18	C₂⁴×C₃	C₃³	1	1	yes	characteristic
19	C₂²⋊A₄	—	3	8	no	—
20	C₂²⋊A₄	H(3,𝔽₃)	1	3	yes	direct product factor
21	C₂²×C₃²	—	3	40	no	—
22	C₂²×C₃²	C₃×A₄	1	20	yes	complemented
23	C₃×A₄	—	4	5	no	—
24	C₃×A₄	—	4	40	no	—
25	C₃×A₄	—	12	60	no	—
26	C₃³	—	16	4	no	—
27	H(3,𝔽₃)	—	16	8	no	—
28	H(3,𝔽₃)	C₂²⋊A₄	1	1	yes	characteristic, direct product factor
29	C₂³×C₃	—	3	5	no	—
30	C₂³×C₃	—	9	20	no	—
31	C₂×C₃²	—	3	20	no	—
32	C₂⁴	C₃×H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
33	C₂²×C₃	—	3	10	no	—
34	C₂²×C₃	—	3	20	no	—
35	C₂²×C₃	—	9	40	no	—
36	C₂²×C₃	C₃²×A₄	1	5	yes	—
37	A₄	—	4	15	no	—
38	A₄	—	12	40	no	—
39	C₃²	—	16	1	no	—
40	C₃²	—	16	8	no	—
41	C₃²	—	48	12	no	—
42	C₃²	C₃×C₂²⋊A₄	1	4	yes	complemented
43	C₂³	—	3	5	no	—
44	C₂×C₃	—	3	5	no	—
45	C₂×C₃	—	9	20	no	—
46	C₂²	—	3	10	no	—
47	C₂²	A₄×H(3,𝔽₃)	1	5	yes	complemented
48	C₃	—	3	4	no	—
49	C₃	—	16	3	no	—
50	C₃	—	48	8	no	—
51	C₃	C₃²×C₂²⋊A₄	1	1	yes	characteristic
52	C₂	—	3	5	no	—

	subgroup	G/H
center, Frattini	C₃	C₃²×C₂²⋊A₄
commutator, socle, cosocle	C₂⁴×C₃	C₃³
Fitting	C₂⁴×H(3,𝔽₃)	C₃
hypercenter	H(3,𝔽₃)	C₂²⋊A₄
hypocenter	C₂⁴	C₃×H(3,𝔽₃)
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃ ◁ H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×C₂²⋊A₄	C₃	semidirect
2. C₂⁴×H(3,𝔽₃)	C₃	semidirect
3. C₃²⋊(C₂²⋊A₄)	C₃	semidirect
6. C₂⁴×C₃²	C₃²	semidirect
9. C₃×C₂²⋊A₄	C₃²	semidirect
12. C₂²×H(3,𝔽₃)	A₄	semidirect
20. C₂²⋊A₄	H(3,𝔽₃)	direct (N × H)
22. C₂²×C₃²	C₃×A₄	semidirect
28. H(3,𝔽₃)	C₂²⋊A₄	direct (N × H)
32. C₂⁴	C₃×H(3,𝔽₃)	semidirect
42. C₃²	C₃×C₂²⋊A₄	semidirect
47. C₂²	A₄×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×C₂²⋊A₄	3
C₂⁴×H(3,𝔽₃)	3
C₃²⋊(C₂²⋊A₄)	3
A₄×H(3,𝔽₃)	4

Subgroups — H(3,𝔽3)×C22⋊A4 #13049338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×C₂²⋊A₄ #13049338