Subgroups — #6293

Order 216.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₃²×Dic₃	C₂	1	1	yes	maximal, characteristic
4	#634	—	3	1	no	maximal
5	#627	—	3	4	no	maximal
6	C₂×C₃³	C₂²	1	1	yes	characteristic
7	C₃²×C₄	—	3	1	no	—
8	C₃⋊Dic₃	—	3	1	no	—
9	C₃⋊Dic₃	—	3	4	no	—
10	C₃⋊Dic₃	—	6	4	no	—
11	C₃⋊Dic₃	D₃	1	1	yes	characteristic
12	C₃×Dic₃	—	3	4	no	—
13	C₃×Dic₃	D₃	1	4	yes	—
14	C₃³	Q₈	1	1	yes	characteristic, complemented, Sylow
15	Dic₆	—	9	4	no	—
16	Dic₆	—	9	1	no	—
17	C₂×C₃²	—	2	4	no	—
18	C₂×C₃²	C₂×D₃	1	4	yes	—
19	C₂×C₃²	C₂×D₃	1	1	yes	characteristic
20	C₃×C₄	—	3	4	no	—
21	C₃×C₄	—	9	1	no	—
22	Dic₃	—	3	4	no	—
23	Dic₃	—	9	4	no	—
24	Dic₃	—	9	1	no	—
25	Dic₃	—	18	4	no	—
26	Dic₃	C₃⋊D₃	1	1	yes	characteristic
27	C₃²	—	2	4	no	—
28	C₃²	Dic₆	1	4	yes	complemented
29	C₃²	Dic₆	1	1	yes	characteristic, complemented
30	Q₈	—	27	1	no	Sylow
31	C₂×C₃	—	2	4	no	—
32	C₂×C₃	D₃²	1	4	yes	—
33	C₂×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
34	C₄	—	3	1	no	—
35	C₄	—	9	1	no	—
36	C₄	—	27	1	no	—
37	C₃	—	2	4	no	—
38	C₃	#627	1	4	yes	complemented
39	C₃	#634	1	1	yes	characteristic, complemented
40	C₂	D₃×C₃⋊D₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃×C₃⋊D₃
commutator, Fitting, socle, prime radical, cosocle	C₂×C₃³	C₂²
hypocenter	C₃³	Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂

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.