Subgroups — C₈×#731 #10619519

Order 648.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#731	C₂	1	1	yes	maximal, characteristic
2	C₈×3₋¹⁺²	C₃	1	2	yes	maximal, complemented
3	C₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃×C₈×C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×#731	C₄	1	1	yes	characteristic
6	C₄×3₋¹⁺²	C₂×C₃	1	2	yes	—
7	C₄×H(3,𝔽₃)	C₂×C₃	1	1	yes	characteristic
8	C₃×C₄×C₉	C₂×C₃	1	1	yes	characteristic
9	#731	C₈	1	1	yes	characteristic, direct product factor, Sylow
10	C₃²×C₈	—	3	1	no	—
11	C₃²×C₈	C₃²	1	1	yes	characteristic
12	C₈×C₉	—	3	1	no	—
13	C₈×C₉	—	3	2	no	—
14	C₂×3₋¹⁺²	C₃×C₄	1	2	yes	—
15	C₂×H(3,𝔽₃)	C₃×C₄	1	1	yes	characteristic
16	C₂×C₃×C₉	C₃×C₄	1	1	yes	characteristic
17	C₃²×C₄	—	3	1	no	—
18	C₃²×C₄	C₂×C₃²	1	1	yes	characteristic
19	C₄×C₉	—	3	1	no	—
20	C₄×C₉	—	3	2	no	—
21	3₋¹⁺²	C₃×C₈	1	2	yes	complemented
22	H(3,𝔽₃)	C₃×C₈	1	1	yes	characteristic
23	C₃×C₉	C₃×C₈	1	1	yes	characteristic, complemented
24	C₃×C₈	—	3	1	no	—
25	C₃×C₈	—	9	1	no	—
26	C₃×C₈	H(3,𝔽₃)	1	1	yes	characteristic
27	C₂×C₃²	—	3	1	no	—
28	C₂×C₃²	C₃²×C₄	1	1	yes	characteristic
29	C₂×C₉	—	3	1	no	—
30	C₂×C₉	—	3	2	no	—
31	C₃×C₄	—	3	1	no	—
32	C₃×C₄	—	9	1	no	—
33	C₃×C₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic
34	C₃²	—	3	1	no	—
35	C₃²	C₃²×C₈	1	1	yes	characteristic
36	C₉	—	3	1	no	—
37	C₉	—	3	2	no	—
38	C₈	#731	1	1	yes	characteristic, direct product factor, Sylow
39	C₂×C₃	—	3	1	no	—
40	C₂×C₃	—	9	1	no	—
41	C₂×C₃	C₄×H(3,𝔽₃)	1	1	yes	characteristic
42	C₄	C₂×#731	1	1	yes	characteristic
43	C₃	—	3	1	no	—
44	C₃	—	9	1	no	—
45	C₃	C₈×H(3,𝔽₃)	1	1	yes	characteristic
46	C₂	C₄×#731	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	H(3,𝔽₃)
commutator	C₃²	C₃²×C₈
Frattini, cosocle	C₃²×C₄	C₂×C₃²
socle	C₂×C₃	C₄×H(3,𝔽₃)
prime radical	C₂×H(3,𝔽₃)	C₃×C₄
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₃²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×3₋¹⁺²	C₃	semidirect
4. C₃×C₈×C₉	C₃	semidirect
9. #731	C₈	direct (N × H)
21. 3₋¹⁺²	C₃×C₈	semidirect
23. C₃×C₉	C₃×C₈	semidirect
38. C₈	#731	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×#731	2
C₈×3₋¹⁺²	3
C₈×H(3,𝔽₃)	3
C₃×C₈×C₉	3

Subgroups — C8×#731 #10619519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#731 #10619519