Subgroups — #3711

Order 144.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×D₃	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#208	—	3	1	no	maximal
5	ΓL(1,𝔽₂₅)	—	3	1	no	maximal
6	C₂×C₃×D₃	C₄	1	1	yes	characteristic
7	C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₃×Dic₃	C₄	1	1	yes	characteristic
9	C₂×C₃×C₄	—	3	1	no	—
10	C₄×D₃	D₃	1	1	yes	characteristic
11	C₃×C₈	—	3	1	no	—
12	C₃⋊C₈	—	3	1	no	—
13	C₃⋊C₈	—	3	1	no	—
14	C₃⋊C₈	—	6	1	no	—
15	C₃⋊C₈	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃²	C₂×C₄	1	1	yes	characteristic
17	C₃×D₃	—	2	1	no	—
18	M₁₆	—	9	1	no	Sylow
19	C₂²×C₃	—	3	1	no	—
20	C₂×D₃	Dic₃	1	1	yes	characteristic
21	C₃×C₄	—	2	1	no	—
22	C₃×C₄	—	3	1	no	—
23	C₃×C₄	C₂×D₃	1	1	yes	characteristic
24	C₃×C₄	C₂×D₃	1	1	yes	characteristic
25	Dic₃	Dic₃	1	1	yes	characteristic
26	C₃²	M₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₂×C₄	—	3	1	no	—
28	C₈	—	3	1	no	—
29	C₈	—	9	1	no	—
30	C₂×C₃	—	2	1	no	—
31	C₂×C₃	—	6	1	no	—
32	C₂×C₃	C₄×D₃	1	1	yes	characteristic
33	C₂×C₃	C₂×Dic₃	1	1	yes	characteristic
34	D₃	—	2	1	no	—
35	C₂²	—	3	1	no	—
36	C₄	—	3	1	no	—
37	C₄	D₃²	1	1	yes	characteristic
38	C₃	—	2	1	no	—
39	C₃	ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented
40	C₃	#208	1	1	yes	characteristic, complemented
41	C₂	—	6	1	no	—
42	C₂	D₃×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃²
commutator, socle	C₂×C₃²	C₂×C₄
Fitting, cosocle	C₃²×C₄	C₂²
prime radical	C₂×C₃×D₃	C₄
hypocenter	C₃²	M₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄

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.