Subgroups — C₂×C₈×3₋¹⁺² #88564

Order 432.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₈×3₋¹⁺²	C₂	1	2	yes	maximal, direct product factor
3	C₂×C₃²×C₈	C₃	1	1	yes	maximal, characteristic
4	C₂×C₈×C₉	C₃	1	3	yes	maximal, complemented
5	C₂²×3₋¹⁺²	C₄	1	1	yes	characteristic
6	C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
7	C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄	C₂×C₃	1	1	yes	characteristic
9	C₃²×C₈	C₂×C₃	1	2	yes	—
10	C₂×C₄×C₉	C₂×C₃	1	3	yes	—
11	C₈×C₉	C₂×C₃	1	6	yes	complemented
12	C₂×3₋¹⁺²	C₈	1	2	yes	direct product factor
13	C₂×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₈	—	3	1	no	—
15	C₂×C₃×C₈	C₃²	1	1	yes	characteristic
16	C₂²×C₃²	C₃×C₄	1	1	yes	characteristic
17	C₃²×C₄	C₃×C₄	1	1	yes	characteristic
18	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
19	C₂²×C₉	C₃×C₄	1	3	yes	—
20	C₄×C₉	C₃×C₄	1	3	yes	—
21	C₄×C₉	C₂²×C₃	1	3	yes	—
22	3₋¹⁺²	C₂×C₈	1	1	yes	characteristic, direct product factor, Sylow
23	C₂×C₃×C₄	—	3	1	no	—
24	C₂×C₃×C₄	C₂×C₃²	1	1	yes	characteristic
25	C₃×C₈	—	3	2	no	—
26	C₃×C₈	C₂×C₃²	1	2	yes	—
27	C₂×C₃²	C₃×C₈	1	2	yes	—
28	C₂×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
29	C₂×C₉	C₃×C₈	1	6	yes	complemented
30	C₂×C₉	C₂×C₃×C₄	1	3	yes	—
31	C₂×C₈	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
32	C₂²×C₃	—	3	1	no	—
33	C₂²×C₃	C₃²×C₄	1	1	yes	characteristic
34	C₃×C₄	—	3	1	no	—
35	C₃×C₄	—	3	1	no	—
36	C₃×C₄	C₃²×C₄	1	1	yes	characteristic
37	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
38	C₃²	C₂×C₃×C₈	1	1	yes	characteristic
39	C₉	C₂×C₃×C₈	1	3	yes	complemented
40	C₂×C₄	C₂×3₋¹⁺²	1	1	yes	characteristic
41	C₈	C₂×3₋¹⁺²	1	2	yes	direct product factor
42	C₂×C₃	—	3	2	no	—
43	C₂×C₃	—	3	1	no	—
44	C₂×C₃	C₃²×C₈	1	2	yes	—
45	C₂×C₃	C₂×C₃²×C₄	1	1	yes	characteristic
46	C₂²	C₄×3₋¹⁺²	1	1	yes	characteristic
47	C₄	C₄×3₋¹⁺²	1	1	yes	characteristic
48	C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
49	C₃	—	3	1	no	—
50	C₃	C₂×C₃²×C₈	1	1	yes	characteristic
51	C₂	C₈×3₋¹⁺²	1	2	yes	direct product factor
52	C₂	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₈	C₃²
commutator	C₃	C₂×C₃²×C₈
Frattini, cosocle	C₃×C₄	C₂²×C₃²
socle	C₂²×C₃	C₃²×C₄
prime radical	C₂²×C₃²	C₃×C₄
LCS	G ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈×3₋¹⁺²	C₂	direct (N × H)
4. C₂×C₈×C₉	C₃	semidirect
11. C₈×C₉	C₂×C₃	semidirect
12. C₂×3₋¹⁺²	C₈	direct (N × H)
22. 3₋¹⁺²	C₂×C₈	direct (N × H)
29. C₂×C₉	C₃×C₈	semidirect
31. C₂×C₈	3₋¹⁺²	direct (N × H)
39. C₉	C₂×C₃×C₈	semidirect
41. C₈	C₂×3₋¹⁺²	direct (N × H)
51. C₂	C₈×3₋¹⁺²	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×C₄×3₋¹⁺²	2
C₈×3₋¹⁺²	2
C₂×C₃²×C₈	3
C₂×C₈×C₉	3

Subgroups — C2×C8×3−1+2 #88564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈×3₋¹⁺² #88564