Subgroups — Q₈⋊H(3,𝔽₃) #6205

Order 216.

39 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×SL(2,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	C₂×H(3,𝔽₃)	—	4	1	no	maximal
4	C₃²×C₄	—	3	1	no	—
5	H(3,𝔽₃)	—	4	1	no	Sylow
6	C₃×Q₈	—	3	1	no	—
7	C₃×Q₈	C₃²	1	1	yes	characteristic
8	SL(2,𝔽₃)	—	3	3	no	—
9	C₂×C₃²	—	4	3	no	—
10	C₂×C₃²	A₄	1	1	yes	characteristic
11	C₃×C₄	—	3	3	no	—
12	C₃×C₄	—	3	1	no	—
13	C₃²	—	4	3	no	—
14	C₃²	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
15	Q₈	H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₃	—	3	1	no	—
17	C₂×C₃	—	12	3	no	—
18	C₂×C₃	C₃×A₄	1	1	yes	characteristic
19	C₄	—	3	1	no	—
20	C₃	—	3	1	no	—
21	C₃	—	12	3	no	—
22	C₃	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
23	C₂	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic

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.