Subgroups — C₂×D₄×H(3,𝔽₃) #88757

Order 432.

297 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂²×C₃	C₂²×C₃²
commutator, Frattini, cosocle	C₂×C₃	C₂³×C₃²
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
1. C₂³×H(3,𝔽₃)	C₂	semidirect
2. D₄×H(3,𝔽₃)	C₂	direct (N × H)
3. C₂×C₄×H(3,𝔽₃)	C₂	semidirect
4. C₂×C₃²×D₄	C₃	semidirect
7. C₂²×H(3,𝔽₃)	C₂²	semidirect
8. C₄×H(3,𝔽₃)	C₂²	semidirect
9. C₂³×C₃²	C₂×C₃	semidirect
10. C₃²×D₄	C₂×C₃	semidirect
11. C₂×C₃²×C₄	C₂×C₃	semidirect
13. C₂×H(3,𝔽₃)	D₄	direct (N × H)
19. C₂²×C₃²	C₂²×C₃	semidirect
20. C₃²×C₄	C₂²×C₃	semidirect
21. H(3,𝔽₃)	C₂×D₄	direct (N × H)
29. C₂×C₃²	C₃×D₄	semidirect
31. C₂×D₄	H(3,𝔽₃)	direct (N × H)
40. C₃²	C₂×C₃×D₄	semidirect
41. C₂³	C₂×H(3,𝔽₃)	semidirect
42. D₄	C₂×H(3,𝔽₃)	direct (N × H)
43. C₂×C₄	C₂×H(3,𝔽₃)	semidirect
52. C₂²	C₂²×H(3,𝔽₃)	semidirect
53. C₄	C₂²×H(3,𝔽₃)	semidirect
57. C₂	D₄×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
C₂³×H(3,𝔽₃)	2
D₄×H(3,𝔽₃)	2
C₂×C₄×H(3,𝔽₃)	2
C₂×C₃²×D₄	3

Subgroups — C2×D4×H(3,𝔽3) #88757

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×H(3,𝔽₃) #88757