Subgroups — C₂²×#732 #66396

Order 324.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#732	C₂	1	3	yes	maximal, direct product factor
2	C₂²×H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	C₂²×C₃×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	#732	C₂²	1	1	yes	characteristic, direct product factor, Sylow
5	C₂×H(3,𝔽₃)	C₂×C₃	1	9	yes	complemented
6	C₂×C₃×C₉	C₂×C₃	1	3	yes	complemented
7	C₂²×C₃²	—	3	3	no	—
8	C₂²×C₃²	C₃²	1	1	yes	characteristic
9	C₂²×C₉	—	3	1	no	—
10	H(3,𝔽₃)	C₂²×C₃	1	3	yes	complemented
11	C₃×C₉	C₂²×C₃	1	1	yes	characteristic, complemented
12	C₂×C₃²	—	3	9	no	—
13	C₂×C₃²	C₂×C₃²	1	3	yes	—
14	C₂×C₉	—	3	3	no	—
15	C₂²×C₃	—	3	1	no	—
16	C₂²×C₃	—	9	3	no	—
17	C₂²×C₃	H(3,𝔽₃)	1	1	yes	characteristic
18	C₃²	—	3	3	no	—
19	C₃²	C₂²×C₃²	1	1	yes	characteristic
20	C₉	—	3	1	no	—
21	C₂×C₃	—	3	3	no	—
22	C₂×C₃	—	9	9	no	—
23	C₂×C₃	C₂×H(3,𝔽₃)	1	3	yes	—
24	C₂²	#732	1	1	yes	characteristic, direct product factor, Sylow
25	C₃	—	3	1	no	—
26	C₃	—	9	3	no	—
27	C₃	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
28	C₂	C₂×#732	1	3	yes	direct product factor

	subgroup	G/H
center, socle	C₂²×C₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂²×C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃² ◁ G

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.