Subgroups — Q₁₆×3₋¹⁺² #88576

Order 432.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×3₋¹⁺²	C₂	1	2	yes	maximal
2	C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	C₃²×Q₁₆	C₃	1	1	yes	maximal, characteristic
4	C₉×Q₁₆	C₃	1	3	yes	maximal, complemented
5	C₄×3₋¹⁺²	—	2	2	no	—
6	C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₃²×Q₈	C₂×C₃	1	2	yes	—
8	C₃²×C₈	C₂×C₃	1	1	yes	characteristic
9	Q₈×C₉	C₂×C₃	1	6	yes	—
10	C₈×C₉	C₂×C₃	1	3	yes	—
11	C₂×3₋¹⁺²	D₄	1	1	yes	characteristic
12	C₃×Q₁₆	—	3	1	no	—
13	C₃×Q₁₆	C₃²	1	1	yes	characteristic
14	C₃²×C₄	—	2	2	no	—
15	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
16	C₄×C₉	—	2	6	no	—
17	C₄×C₉	C₂²×C₃	1	3	yes	—
18	3₋¹⁺²	Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
19	C₃×Q₈	—	3	2	no	—
20	C₃×Q₈	C₂×C₃²	1	2	yes	—
21	C₃×C₈	—	3	1	no	—
22	C₃×C₈	C₂×C₃²	1	1	yes	characteristic
23	C₂×C₃²	C₃×D₄	1	1	yes	characteristic
24	C₂×C₉	C₃×D₄	1	3	yes	—
25	Q₁₆	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
26	C₃×C₄	—	2	2	no	—
27	C₃×C₄	—	3	1	no	—
28	C₃×C₄	—	6	2	no	—
29	C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
30	C₃²	C₃×Q₁₆	1	1	yes	characteristic
31	C₉	C₃×Q₁₆	1	3	yes	complemented
32	Q₈	C₂×3₋¹⁺²	1	2	yes	—
33	C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
34	C₂×C₃	—	3	1	no	—
35	C₂×C₃	C₃²×D₄	1	1	yes	characteristic
36	C₄	—	2	2	no	—
37	C₄	C₂²×3₋¹⁺²	1	1	yes	characteristic
38	C₃	—	3	1	no	—
39	C₃	C₃²×Q₁₆	1	1	yes	characteristic
40	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	C₂×C₃²	C₃×D₄
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺² ◁ 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.