Subgroups — D₈×H(3,𝔽₃) #88569

Order 432.

121 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₄×H(3,𝔽₃)	C₂	1	2	yes	maximal, complemented
2	C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×D₈	C₃	1	4	yes	maximal, complemented
4	C₂²×H(3,𝔽₃)	—	2	2	no	—
5	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₃²×D₄	C₂×C₃	1	8	yes	complemented
7	C₃²×C₈	C₂×C₃	1	4	yes	complemented
8	C₂×H(3,𝔽₃)	—	4	2	no	—
9	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
10	C₃×D₈	—	3	4	no	—
11	C₃×D₈	C₃²	1	1	yes	characteristic
12	C₂²×C₃²	—	2	8	no	—
13	C₃²×C₄	C₂²×C₃	1	4	yes	—
14	H(3,𝔽₃)	D₈	1	1	yes	characteristic, direct product factor, Sylow
15	C₃×D₄	—	3	8	no	—
16	C₃×D₄	C₂×C₃²	1	2	yes	—
17	C₃×C₈	—	3	4	no	—
18	C₃×C₈	C₂×C₃²	1	1	yes	characteristic
19	C₂×C₃²	—	4	8	no	—
20	C₂×C₃²	C₃×D₄	1	4	yes	—
21	D₈	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
22	C₂²×C₃	—	2	2	no	—
23	C₂²×C₃	—	6	8	no	—
24	C₃×C₄	—	3	4	no	—
25	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
26	C₃²	C₃×D₈	1	4	yes	complemented
27	D₄	C₂×H(3,𝔽₃)	1	2	yes	complemented
28	C₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic, complemented
29	C₂×C₃	—	3	4	no	—
30	C₂×C₃	—	4	2	no	—
31	C₂×C₃	—	12	8	no	—
32	C₂×C₃	C₃²×D₄	1	1	yes	characteristic
33	C₂²	—	2	2	no	—
34	C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
35	C₃	—	3	4	no	—
36	C₃	C₃²×D₈	1	1	yes	characteristic
37	C₂	—	4	2	no	—
38	C₂	D₄×H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₃²×D₄
commutator, Frattini, cosocle	C₃×C₄	C₂²×C₃²
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₄×H(3,𝔽₃) ◁ G

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

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