Subgroups — D₈×3₋¹⁺² #88570

Order 432.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
2	C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₈	C₃	1	1	yes	maximal, characteristic
4	C₉×D₈	C₃	1	3	yes	maximal, complemented
5	C₂²×3₋¹⁺²	—	2	2	no	—
6	C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₃²×D₄	C₂×C₃	1	2	yes	—
8	C₃²×C₈	C₂×C₃	1	1	yes	characteristic
9	D₄×C₉	C₂×C₃	1	6	yes	complemented
10	C₈×C₉	C₂×C₃	1	3	yes	complemented
11	C₂×3₋¹⁺²	—	4	2	no	—
12	C₂×3₋¹⁺²	D₄	1	1	yes	characteristic
13	C₃×D₈	—	3	1	no	—
14	C₃×D₈	C₃²	1	1	yes	characteristic
15	C₂²×C₃²	—	2	2	no	—
16	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
17	C₂²×C₉	—	2	6	no	—
18	C₄×C₉	C₂²×C₃	1	3	yes	—
19	3₋¹⁺²	D₈	1	1	yes	characteristic, direct product factor, Sylow
20	C₃×D₄	—	3	2	no	—
21	C₃×D₄	C₂×C₃²	1	2	yes	—
22	C₃×C₈	—	3	1	no	—
23	C₃×C₈	C₂×C₃²	1	1	yes	characteristic
24	C₂×C₃²	—	4	2	no	—
25	C₂×C₃²	C₃×D₄	1	1	yes	characteristic
26	C₂×C₉	—	4	6	no	—
27	C₂×C₉	C₃×D₄	1	3	yes	—
28	D₈	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
29	C₂²×C₃	—	2	2	no	—
30	C₂²×C₃	—	6	2	no	—
31	C₃×C₄	—	3	1	no	—
32	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
33	C₃²	C₃×D₈	1	1	yes	characteristic
34	C₉	C₃×D₈	1	3	yes	complemented
35	D₄	C₂×3₋¹⁺²	1	2	yes	complemented
36	C₈	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
37	C₂×C₃	—	3	1	no	—
38	C₂×C₃	—	4	2	no	—
39	C₂×C₃	—	12	2	no	—
40	C₂×C₃	C₃²×D₄	1	1	yes	characteristic
41	C₂²	—	2	2	no	—
42	C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
43	C₃	—	3	1	no	—
44	C₃	C₃²×D₈	1	1	yes	characteristic
45	C₂	—	4	2	no	—
46	C₂	D₄×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×D₄
commutator, Frattini, cosocle	C₃×C₄	C₂²×C₃²
prime radical	C₃²×D₈	C₃
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺² ◁ G

Complements

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

N	H ≅ G/N	type
1. D₄×3₋¹⁺²	C₂	semidirect
2. C₈×3₋¹⁺²	C₂	semidirect
4. C₉×D₈	C₃	semidirect
9. D₄×C₉	C₂×C₃	semidirect
10. C₈×C₉	C₂×C₃	semidirect
19. 3₋¹⁺²	D₈	direct (N × H)
28. D₈	3₋¹⁺²	direct (N × H)
34. C₉	C₃×D₈	semidirect
35. D₄	C₂×3₋¹⁺²	semidirect
36. C₈	C₂×3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
D₄×3₋¹⁺²	2
C₈×3₋¹⁺²	2
C₃²×D₈	3
C₉×D₈	3

Subgroups — D8×3−1+2 #88570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×3₋¹⁺² #88570