Subgroups — C₂²×C₈×H(3,𝔽₃) #11721923

Order 864.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×H(3,𝔽₃)	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃²×C₈	C₃	1	4	yes	maximal, complemented
4	C₂³×H(3,𝔽₃)	C₄	1	1	yes	characteristic
5	C₂×C₄×H(3,𝔽₃)	C₄	1	3	yes	—
6	C₂×C₄×H(3,𝔽₃)	C₂²	1	3	yes	—
7	C₈×H(3,𝔽₃)	C₂²	1	4	yes	direct product factor
8	C₂²×C₃²×C₄	C₂×C₃	1	4	yes	—
9	C₂×C₃²×C₈	C₂×C₃	1	24	yes	complemented
10	C₂²×H(3,𝔽₃)	C₈	1	4	yes	direct product factor
11	C₂²×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
12	C₄×H(3,𝔽₃)	C₂×C₄	1	3	yes	—
13	C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
14	C₂²×C₃×C₈	—	3	4	no	—
15	C₂²×C₃×C₈	C₃²	1	1	yes	characteristic
16	C₂³×C₃²	C₃×C₄	1	4	yes	—
17	C₂×C₃²×C₄	C₃×C₄	1	12	yes	—
18	C₂×C₃²×C₄	C₂²×C₃	1	12	yes	—
19	C₃²×C₈	C₂²×C₃	1	16	yes	complemented
20	C₂×H(3,𝔽₃)	C₂×C₈	1	6	yes	direct product factor
21	C₂×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄	—	3	4	no	—
23	C₂²×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
24	C₂×C₃×C₈	—	3	24	no	—
25	C₂×C₃×C₈	C₂×C₃²	1	6	yes	—
26	C₂²×C₃²	C₃×C₈	1	16	yes	complemented
27	C₂²×C₃²	C₂×C₃×C₄	1	12	yes	—
28	C₃²×C₄	C₂×C₃×C₄	1	12	yes	—
29	C₃²×C₄	C₂³×C₃	1	4	yes	—
30	C₂²×C₈	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
31	H(3,𝔽₃)	C₂²×C₈	1	1	yes	characteristic, direct product factor, Sylow
32	C₂³×C₃	—	3	4	no	—
33	C₂³×C₃	C₃²×C₄	1	1	yes	characteristic
34	C₂×C₃×C₄	—	3	12	no	—
35	C₂×C₃×C₄	—	3	12	no	—
36	C₂×C₃×C₄	C₃²×C₄	1	3	yes	—
37	C₂×C₃×C₄	C₂²×C₃²	1	3	yes	—
38	C₃×C₈	—	3	16	no	—
39	C₃×C₈	C₂²×C₃²	1	4	yes	—
40	C₂×C₃²	C₂×C₃×C₈	1	24	yes	complemented
41	C₂×C₃²	C₂²×C₃×C₄	1	4	yes	—
42	C₂²×C₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic
43	C₂×C₈	C₂×H(3,𝔽₃)	1	6	yes	direct product factor
44	C₂²×C₃	—	3	16	no	—
45	C₂²×C₃	—	3	12	no	—
46	C₂²×C₃	C₃²×C₈	1	4	yes	—
47	C₂²×C₃	C₂×C₃²×C₄	1	3	yes	—
48	C₃×C₄	—	3	12	no	—
49	C₃×C₄	—	3	4	no	—
50	C₃×C₄	C₂×C₃²×C₄	1	3	yes	—
51	C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
52	C₃²	C₂²×C₃×C₈	1	4	yes	complemented
53	C₂³	C₄×H(3,𝔽₃)	1	1	yes	characteristic
54	C₂×C₄	C₄×H(3,𝔽₃)	1	3	yes	—
55	C₂×C₄	C₂²×H(3,𝔽₃)	1	3	yes	—
56	C₈	C₂²×H(3,𝔽₃)	1	4	yes	direct product factor
57	C₂×C₃	—	3	4	no	—
58	C₂×C₃	—	3	24	no	—
59	C₂×C₃	C₂×C₃²×C₈	1	6	yes	—
60	C₂×C₃	C₂²×C₃²×C₄	1	1	yes	characteristic
61	C₂²	C₈×H(3,𝔽₃)	1	4	yes	direct product factor
62	C₂²	C₂×C₄×H(3,𝔽₃)	1	3	yes	—
63	C₄	C₂×C₄×H(3,𝔽₃)	1	3	yes	—
64	C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃	—	3	4	no	—
66	C₃	C₂²×C₃²×C₈	1	1	yes	characteristic
67	C₂	C₂×C₈×H(3,𝔽₃)	1	6	yes	direct product factor
68	C₂	C₂²×C₄×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃×C₈	C₃²
commutator	C₃	C₂²×C₃²×C₈
Frattini, cosocle	C₃×C₄	C₂³×C₃²
socle	C₂³×C₃	C₃²×C₄
prime radical	C₂³×H(3,𝔽₃)	C₄
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×H(3,𝔽₃)	C₂	direct (N × H)
3. C₂²×C₃²×C₈	C₃	semidirect
7. C₈×H(3,𝔽₃)	C₂²	direct (N × H)
9. C₂×C₃²×C₈	C₂×C₃	semidirect
10. C₂²×H(3,𝔽₃)	C₈	direct (N × H)
19. C₃²×C₈	C₂²×C₃	semidirect
20. C₂×H(3,𝔽₃)	C₂×C₈	direct (N × H)
26. C₂²×C₃²	C₃×C₈	semidirect
30. C₂²×C₈	H(3,𝔽₃)	direct (N × H)
31. H(3,𝔽₃)	C₂²×C₈	direct (N × H)
40. C₂×C₃²	C₂×C₃×C₈	semidirect
43. C₂×C₈	C₂×H(3,𝔽₃)	direct (N × H)
52. C₃²	C₂²×C₃×C₈	semidirect
56. C₈	C₂²×H(3,𝔽₃)	direct (N × H)
61. C₂²	C₈×H(3,𝔽₃)	direct (N × H)
67. C₂	C₂×C₈×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
C₂²×C₄×H(3,𝔽₃)	2
C₂×C₈×H(3,𝔽₃)	2
C₂²×C₃²×C₈	3

Subgroups — C22×C8×H(3,𝔽3) #11721923

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×H(3,𝔽₃) #11721923