Subgroups — C₂²⋊C₄×H(3,𝔽₃) #88557

Order 432.

187 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×H(3,𝔽₃)	C₂	1	2	yes	maximal, complemented
3	C₃²×C₂²⋊C₄	C₃	1	4	yes	maximal, complemented
4	C₂²×H(3,𝔽₃)	—	2	2	no	—
5	C₂²×H(3,𝔽₃)	C₄	1	2	yes	complemented
6	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
7	C₄×H(3,𝔽₃)	—	2	2	no	—
8	C₂³×C₃²	C₂×C₃	1	4	yes	—
9	C₂×C₃²×C₄	C₂×C₃	1	8	yes	complemented
10	C₂×H(3,𝔽₃)	—	2	2	no	—
11	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
12	C₂×H(3,𝔽₃)	D₄	1	2	yes	—
13	C₃×C₂²⋊C₄	—	3	4	no	—
14	C₃×C₂²⋊C₄	C₃²	1	1	yes	characteristic
15	C₂²×C₃²	—	2	8	no	—
16	C₂²×C₃²	C₃×C₄	1	8	yes	complemented
17	C₂²×C₃²	C₂²×C₃	1	4	yes	—
18	C₃²×C₄	—	2	8	no	—
19	H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic, direct product factor, Sylow
20	C₂³×C₃	—	3	4	no	—
21	C₂³×C₃	C₂×C₃²	1	1	yes	characteristic
22	C₂×C₃×C₄	—	3	8	no	—
23	C₂×C₃×C₄	C₂×C₃²	1	2	yes	—
24	C₂×C₃²	—	2	8	no	—
25	C₂×C₃²	C₂×C₃×C₄	1	4	yes	—
26	C₂×C₃²	C₃×D₄	1	8	yes	—
27	C₂²⋊C₄	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
28	C₂²×C₃	—	2	2	no	—
29	C₂²×C₃	—	3	8	no	—
30	C₂²×C₃	—	3	4	no	—
31	C₂²×C₃	—	6	8	no	—
32	C₂²×C₃	C₃²×C₄	1	2	yes	—
33	C₂²×C₃	C₂²×C₃²	1	1	yes	characteristic
34	C₃×C₄	—	2	2	no	—
35	C₃×C₄	—	6	8	no	—
36	C₃²	C₃×C₂²⋊C₄	1	4	yes	complemented
37	C₂³	C₂×H(3,𝔽₃)	1	1	yes	characteristic
38	C₂×C₄	C₂×H(3,𝔽₃)	1	2	yes	complemented
39	C₂×C₃	—	2	2	no	—
40	C₂×C₃	—	3	4	no	—
41	C₂×C₃	—	3	8	no	—
42	C₂×C₃	—	6	8	no	—
43	C₂×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
44	C₂×C₃	C₃²×D₄	1	2	yes	—
45	C₂²	—	2	2	no	—
46	C₂²	C₄×H(3,𝔽₃)	1	2	yes	complemented
47	C₂²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
48	C₄	—	2	2	no	—
49	C₃	—	3	4	no	—
50	C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
51	C₂	—	2	2	no	—
52	C₂	C₂×C₄×H(3,𝔽₃)	1	1	yes	characteristic
53	C₂	D₄×H(3,𝔽₃)	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×H(3,𝔽₃)	C₂	semidirect
3. C₃²×C₂²⋊C₄	C₃	semidirect
5. C₂²×H(3,𝔽₃)	C₄	semidirect
9. C₂×C₃²×C₄	C₂×C₃	semidirect
16. C₂²×C₃²	C₃×C₄	semidirect
19. H(3,𝔽₃)	C₂²⋊C₄	direct (N × H)
27. C₂²⋊C₄	H(3,𝔽₃)	direct (N × H)
36. C₃²	C₃×C₂²⋊C₄	semidirect
38. C₂×C₄	C₂×H(3,𝔽₃)	semidirect
46. C₂²	C₄×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
C₂³×H(3,𝔽₃)	2
C₂×C₄×H(3,𝔽₃)	2
C₃²×C₂²⋊C₄	3

Subgroups — C22⋊C4×H(3,𝔽3) #88557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₄×H(3,𝔽₃) #88557