Subgroups — C₃×C₂²⋊SL(2,𝔽₃) #64338

Order 288.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
2	C₂²⋊SL(2,𝔽₃)	C₃	1	3	yes	maximal, direct product factor
3	C₂×C₃×A₄	—	4	1	no	maximal
4	C₃×SL(2,𝔽₃)	—	4	4	no	maximal
5	C₂×C₃×Q₈	—	3	4	no	—
6	C₂²×C₃×C₄	—	3	1	no	—
7	C₃×A₄	—	4	1	no	—
8	C₂²×Q₈	C₃²	1	1	yes	characteristic, complemented, Sylow
9	C₂³×C₃	A₄	1	1	yes	characteristic
10	C₂×A₄	—	4	3	no	—
11	C₃×Q₈	—	3	4	no	—
12	C₃×Q₈	A₄	1	4	yes	complemented
13	C₂×C₃×C₄	—	3	6	no	—
14	SL(2,𝔽₃)	—	4	12	no	—
15	C₂×C₃²	—	16	1	no	—
16	C₂×Q₈	—	3	4	no	—
17	C₂²×C₄	—	3	1	no	—
18	C₂²×C₃	—	3	1	no	—
19	C₂²×C₃	—	3	1	no	—
20	C₂²×C₃	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
21	A₄	—	4	3	no	—
22	C₃×C₄	—	3	4	no	—
23	C₃²	—	16	1	no	Sylow
24	C₂³	C₃×A₄	1	1	yes	characteristic
25	Q₈	—	3	4	no	—
26	Q₈	C₃×A₄	1	4	yes	complemented
27	C₂×C₄	—	3	6	no	—
28	C₂×C₃	—	3	1	no	—
29	C₂×C₃	—	3	1	no	—
30	C₂×C₃	—	16	3	no	—
31	C₂×C₃	C₂²⋊A₄	1	1	yes	characteristic
32	C₂²	—	3	1	no	—
33	C₂²	—	3	1	no	—
34	C₂²	C₃×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
35	C₄	—	3	4	no	—
36	C₃	—	16	3	no	—
37	C₃	C₂²⋊SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
38	C₂	—	3	1	no	—
39	C₂	—	3	1	no	—
40	C₂	C₃×C₂²⋊A₄	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	C₂²⋊A₄
commutator, cosocle, hypocenter	C₂²×Q₈	C₃²
Frattini	C₂	C₃×C₂²⋊A₄
Fitting	C₂²×C₃×Q₈	C₃
socle	C₂³×C₃	A₄
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂×C₃

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.