Subgroups — Q₈×3₋¹⁺² #6244

Order 216.

48 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×3₋¹⁺²	C₂	1	3	yes	maximal
2	C₃²×Q₈	C₃	1	1	yes	maximal, characteristic
3	Q₈×C₉	C₃	1	3	yes	maximal, complemented
4	C₂×3₋¹⁺²	C₂²	1	1	yes	characteristic
5	C₃²×C₄	C₂×C₃	1	3	yes	—
6	C₄×C₉	C₂×C₃	1	9	yes	—
7	3₋¹⁺²	Q₈	1	1	yes	characteristic, direct product factor, Sylow
8	C₃×Q₈	—	3	1	no	—
9	C₃×Q₈	C₃²	1	1	yes	characteristic
10	C₂×C₃²	C₂²×C₃	1	1	yes	characteristic
11	C₂×C₉	C₂²×C₃	1	3	yes	—
12	C₃×C₄	—	3	3	no	—
13	C₃×C₄	C₂×C₃²	1	3	yes	—
14	C₃²	C₃×Q₈	1	1	yes	characteristic
15	C₉	C₃×Q₈	1	3	yes	complemented
16	Q₈	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
17	C₂×C₃	—	3	1	no	—
18	C₂×C₃	C₂²×C₃²	1	1	yes	characteristic
19	C₄	C₂×3₋¹⁺²	1	3	yes	—
20	C₃	—	3	1	no	—
21	C₃	C₃²×Q₈	1	1	yes	characteristic
22	C₂	C₂²×3₋¹⁺²	1	1	yes	characteristic

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