Subgroups — C₄×C₂²⋊H(3,𝔽₃) #88686

Order 432.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃²×C₄	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×A₄	C₃	1	3	yes	maximal, complemented
4	C₂²⋊H(3,𝔽₃)	C₄	1	1	yes	characteristic, direct product factor
5	C₄×H(3,𝔽₃)	—	4	1	no	maximal
6	C₂³×C₃²	C₂×C₃	1	1	yes	characteristic
7	C₂×C₃×A₄	C₂×C₃	1	3	yes	—
8	C₂×C₃²×C₄	—	3	1	no	—
9	C₂×C₃²×C₄	—	3	1	no	—
10	C₂×H(3,𝔽₃)	—	4	1	no	—
11	C₂²×C₃×C₄	—	3	1	no	—
12	C₂²×C₃×C₄	C₃²	1	1	yes	characteristic
13	C₄×A₄	—	3	3	no	—
14	C₂²×C₃²	—	3	1	no	—
15	C₂²×C₃²	—	3	1	no	—
16	C₂²×C₃²	C₃×C₄	1	1	yes	characteristic, complemented
17	C₃×A₄	C₃×C₄	1	3	yes	complemented
18	C₃²×C₄	—	3	1	no	—
19	C₃²×C₄	—	4	3	no	—
20	C₃²×C₄	A₄	1	1	yes	characteristic, complemented
21	H(3,𝔽₃)	—	4	1	no	Sylow
22	C₂³×C₃	—	3	1	no	—
23	C₂³×C₃	C₂×C₃²	1	1	yes	characteristic
24	C₂×A₄	—	3	3	no	—
25	C₂×C₃×C₄	—	3	3	no	—
26	C₂×C₃×C₄	—	3	3	no	—
27	C₂×C₃×C₄	—	3	1	no	—
28	C₂×C₃×C₄	—	3	1	no	—
29	C₂×C₃²	—	3	1	no	—
30	C₂×C₃²	—	3	1	no	—
31	C₂×C₃²	—	4	3	no	—
32	C₂×C₃²	C₂×A₄	1	1	yes	characteristic
33	C₂²×C₄	H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
34	C₂²×C₃	—	3	1	no	—
35	C₂²×C₃	—	3	3	no	—
36	C₂²×C₃	—	3	3	no	—
37	C₂²×C₃	—	3	1	no	—
38	C₂²×C₃	—	3	1	no	—
39	C₂²×C₃	C₃²×C₄	1	1	yes	characteristic
40	A₄	—	3	3	no	—
41	C₃×C₄	—	3	3	no	—
42	C₃×C₄	—	3	1	no	—
43	C₃×C₄	—	3	1	no	—
44	C₃×C₄	—	12	3	no	—
45	C₃×C₄	C₃×A₄	1	1	yes	characteristic
46	C₃²	—	4	3	no	—
47	C₃²	C₄×A₄	1	1	yes	characteristic, complemented
48	C₂³	C₂×H(3,𝔽₃)	1	1	yes	characteristic
49	C₂×C₄	—	3	1	no	—
50	C₂×C₄	—	3	1	no	—
51	C₂×C₃	—	3	3	no	—
52	C₂×C₃	—	3	3	no	—
53	C₂×C₃	—	3	1	no	—
54	C₂×C₃	—	3	1	no	—
55	C₂×C₃	—	3	1	no	—
56	C₂×C₃	—	12	3	no	—
57	C₂×C₃	C₂×C₃×A₄	1	1	yes	characteristic
58	C₂²	—	3	1	no	—
59	C₂²	—	3	1	no	—
60	C₂²	C₄×H(3,𝔽₃)	1	1	yes	characteristic, complemented
61	C₄	—	3	1	no	—
62	C₄	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic, direct product factor
63	C₃	—	3	1	no	—
64	C₃	—	12	3	no	—
65	C₃	C₃×C₄×A₄	1	1	yes	characteristic
66	C₂	—	3	1	no	—
67	C₂	—	3	1	no	—
68	C₂	C₂×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	C₃×A₄
commutator	C₂²×C₃	C₃²×C₄
Frattini	C₂×C₃	C₂×C₃×A₄
Fitting	C₂²×C₃²×C₄	C₃
socle, cosocle	C₂³×C₃	C₂×C₃²
prime radical	C₂×C₂²⋊H(3,𝔽₃)	C₂
hypercenter	C₃²×C₄	A₄
hypocenter	C₂²	C₄×H(3,𝔽₃)
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃×C₄ ◁ C₃²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₄	C₃	semidirect
3. C₃×C₄×A₄	C₃	semidirect
4. C₂²⋊H(3,𝔽₃)	C₄	direct (N × H)
16. C₂²×C₃²	C₃×C₄	semidirect
17. C₃×A₄	C₃×C₄	semidirect
20. C₃²×C₄	A₄	semidirect
33. C₂²×C₄	H(3,𝔽₃)	semidirect
47. C₃²	C₄×A₄	semidirect
60. C₂²	C₄×H(3,𝔽₃)	semidirect
62. C₄	C₂²⋊H(3,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
C₂×C₂²⋊H(3,𝔽₃)	2
C₂²×C₃²×C₄	3
C₃×C₄×A₄	3
C₄×H(3,𝔽₃)	4

Subgroups — C4×C22⋊H(3,𝔽3) #88686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₂²⋊H(3,𝔽₃) #88686