Subgroups — C₂³×#732 #10619829

Order 648.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#732	C₂	1	7	yes	maximal, direct product factor
2	C₂³×H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
3	C₂³×C₃×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂×#732	C₂²	1	7	yes	direct product factor
5	C₂²×H(3,𝔽₃)	C₂×C₃	1	21	yes	complemented
6	C₂²×C₃×C₉	C₂×C₃	1	7	yes	complemented
7	#732	C₂³	1	1	yes	characteristic, direct product factor, Sylow
8	C₂³×C₃²	—	3	3	no	—
9	C₂³×C₃²	C₃²	1	1	yes	characteristic
10	C₂³×C₉	—	3	1	no	—
11	C₂×H(3,𝔽₃)	C₂²×C₃	1	21	yes	complemented
12	C₂×C₃×C₉	C₂²×C₃	1	7	yes	complemented
13	C₂²×C₃²	—	3	21	no	—
14	C₂²×C₃²	C₂×C₃²	1	7	yes	—
15	C₂²×C₉	—	3	7	no	—
16	H(3,𝔽₃)	C₂³×C₃	1	3	yes	complemented
17	C₃×C₉	C₂³×C₃	1	1	yes	characteristic, complemented
18	C₂³×C₃	—	3	1	no	—
19	C₂³×C₃	—	9	3	no	—
20	C₂³×C₃	H(3,𝔽₃)	1	1	yes	characteristic
21	C₂×C₃²	—	3	21	no	—
22	C₂×C₃²	C₂²×C₃²	1	7	yes	—
23	C₂×C₉	—	3	7	no	—
24	C₂²×C₃	—	3	7	no	—
25	C₂²×C₃	—	9	21	no	—
26	C₂²×C₃	C₂×H(3,𝔽₃)	1	7	yes	—
27	C₃²	—	3	3	no	—
28	C₃²	C₂³×C₃²	1	1	yes	characteristic
29	C₉	—	3	1	no	—
30	C₂³	#732	1	1	yes	characteristic, direct product factor, Sylow
31	C₂×C₃	—	3	7	no	—
32	C₂×C₃	—	9	21	no	—
33	C₂×C₃	C₂²×H(3,𝔽₃)	1	7	yes	—
34	C₂²	C₂×#732	1	7	yes	direct product factor
35	C₃	—	3	1	no	—
36	C₃	—	9	3	no	—
37	C₃	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
38	C₂	C₂²×#732	1	7	yes	direct product factor

	subgroup	G/H
center, socle	C₂³×C₃	H(3,𝔽₃)
commutator, Frattini, cosocle	C₃²	C₂³×C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂³×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.