Subgroups — C₄×#733 #66342

Order 324.

39 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#733	C₂	1	1	yes	maximal, characteristic
2	C₄×3₋¹⁺²	C₃	1	3	yes	maximal
3	C₃×C₄×C₉	C₃	1	1	yes	maximal, characteristic
4	#733	C₄	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×3₋¹⁺²	C₂×C₃	1	3	yes	—
6	C₂×C₃×C₉	C₂×C₃	1	1	yes	characteristic
7	C₃²×C₄	C₃²	1	1	yes	characteristic
8	C₄×C₉	—	3	1	no	—
9	C₄×C₉	—	3	3	no	—
10	3₋¹⁺²	C₃×C₄	1	3	yes	—
11	C₃×C₉	C₃×C₄	1	1	yes	characteristic
12	C₂×C₃²	C₂×C₃²	1	1	yes	characteristic
13	C₂×C₉	—	3	1	no	—
14	C₂×C₉	—	3	3	no	—
15	C₃×C₄	—	3	1	no	—
16	C₃×C₄	H(3,𝔽₃)	1	1	yes	characteristic
17	C₃²	C₃²×C₄	1	1	yes	characteristic
18	C₉	—	3	1	no	—
19	C₉	—	3	3	no	—
20	C₂×C₃	—	3	1	no	—
21	C₂×C₃	C₂×H(3,𝔽₃)	1	1	yes	characteristic
22	C₄	#733	1	1	yes	characteristic, direct product factor, Sylow
23	C₃	—	3	1	no	—
24	C₃	C₄×H(3,𝔽₃)	1	1	yes	characteristic
25	C₂	C₂×#733	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄	H(3,𝔽₃)
commutator	C₃²	C₃²×C₄
Frattini, prime radical, cosocle	C₂×C₃²	C₂×C₃²
socle	C₂×C₃	C₂×H(3,𝔽₃)
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₃²×C₄ ◁ G

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #733	C₄	direct (N × H)
22. C₄	#733	direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.