Subgroups — Q₁₆×H(3,𝔽₃) #88575

Order 432.

99 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×H(3,𝔽₃)	C₂	1	2	yes	maximal
2	C₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
3	C₃²×Q₁₆	C₃	1	4	yes	maximal, complemented
4	C₄×H(3,𝔽₃)	—	2	2	no	—
5	C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₃²×Q₈	C₂×C₃	1	8	yes	—
7	C₃²×C₈	C₂×C₃	1	4	yes	—
8	C₂×H(3,𝔽₃)	D₄	1	1	yes	characteristic
9	C₃×Q₁₆	—	3	4	no	—
10	C₃×Q₁₆	C₃²	1	1	yes	characteristic
11	C₃²×C₄	—	2	8	no	—
12	C₃²×C₄	C₂²×C₃	1	4	yes	—
13	H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
14	C₃×Q₈	—	3	8	no	—
15	C₃×Q₈	C₂×C₃²	1	2	yes	—
16	C₃×C₈	—	3	4	no	—
17	C₃×C₈	C₂×C₃²	1	1	yes	characteristic
18	C₂×C₃²	C₃×D₄	1	4	yes	—
19	Q₁₆	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
20	C₃×C₄	—	2	2	no	—
21	C₃×C₄	—	3	4	no	—
22	C₃×C₄	—	6	8	no	—
23	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
24	C₃²	C₃×Q₁₆	1	4	yes	complemented
25	Q₈	C₂×H(3,𝔽₃)	1	2	yes	—
26	C₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic
27	C₂×C₃	—	3	4	no	—
28	C₂×C₃	C₃²×D₄	1	1	yes	characteristic
29	C₄	—	2	2	no	—
30	C₄	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
31	C₃	—	3	4	no	—
32	C₃	C₃²×Q₁₆	1	1	yes	characteristic
33	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
prime radical	C₂×H(3,𝔽₃)	D₄
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.