Subgroups — C₂²×C₂²⋊H(3,𝔽₃) #88903

Order 432.

224 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊H(3,𝔽₃)	C₂	direct (N × H)
2. C₂⁴×C₃²	C₃	semidirect
3. C₂²×C₃×A₄	C₃	semidirect
5. C₂²⋊H(3,𝔽₃)	C₂²	direct (N × H)
8. C₂³×C₃²	C₂×C₃	semidirect
9. C₂×C₃×A₄	C₂×C₃	semidirect
19. C₂²×C₃²	A₄	semidirect
20. C₂²×C₃²	C₂²×C₃	semidirect
21. C₃×A₄	C₂²×C₃	semidirect
33. C₂×C₃²	C₂×A₄	semidirect
34. C₂⁴	H(3,𝔽₃)	semidirect
50. C₃²	C₂²×A₄	semidirect
53. C₂³	C₂×H(3,𝔽₃)	semidirect
65. C₂²	C₂²⋊H(3,𝔽₃)	direct (N × H)
66. C₂²	C₂²×H(3,𝔽₃)	semidirect
72. C₂	C₂×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
C₂⁴×C₃²	3
C₂²×C₃×A₄	3
C₂²×H(3,𝔽₃)	4

Subgroups — C22×C22⋊H(3,𝔽3) #88903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂²⋊H(3,𝔽₃) #88903