Subgroups — #6258

Order 216.

57 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²⋊H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×S₄	—	3	3	no	maximal
3	C₃×#67	—	3	1	no	maximal
4	#272	—	4	1	no	maximal
5	C₂²×C₃²	D₃	1	1	yes	characteristic, complemented
6	C₂×C₃×D₃	—	3	1	no	—
7	C₃×A₄	D₃	1	3	yes	complemented
8	C₃×Dic₃	—	3	1	no	—
9	H(3,𝔽₃)	—	4	1	no	Sylow
10	S₄	—	3	3	no	—
11	C₃×D₄	—	9	1	no	—
12	#67	—	3	1	no	—
13	C₂×C₃²	—	3	1	no	—
14	C₃×D₃	—	6	1	no	—
15	C₃×D₃	—	12	3	no	—
16	C₂²×C₃	—	3	1	no	—
17	C₂²×C₃	—	9	1	no	—
18	C₂²×C₃	C₃⋊D₃	1	1	yes	characteristic
19	C₂×D₃	—	3	1	no	—
20	A₄	—	3	3	no	—
21	C₃×C₄	—	9	1	no	—
22	Dic₃	—	3	1	no	—
23	C₃²	—	4	3	no	—
24	C₃²	S₄	1	1	yes	characteristic, complemented
25	D₄	—	9	1	no	Sylow
26	C₂×C₃	—	3	1	no	—
27	C₂×C₃	—	3	1	no	—
28	C₂×C₃	—	6	1	no	—
29	C₂×C₃	—	18	1	no	—
30	D₃	—	6	1	no	—
31	D₃	—	12	3	no	—
32	C₂²	—	9	1	no	—
33	C₂²	#272	1	1	yes	characteristic, complemented
34	C₄	—	9	1	no	—
35	C₃	—	3	1	no	—
36	C₃	—	12	3	no	—
37	C₃	C₃⋊S₄	1	1	yes	characteristic
38	C₂	—	3	1	no	—
39	C₂	—	18	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₃	C₃⋊S₄
commutator, cosocle, hypocenter	C₂²⋊H(3,𝔽₃)	C₂
Fitting	C₂²×C₃²	D₃
socle	C₂²×C₃	C₃⋊D₃
LCS	G ▷ C₂²⋊H(3,𝔽₃)
UCS	C₁ ◁ 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.