Subgroups — D₉² #66344

Order 324.

59 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉⋊D₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×D₉	C₂	1	2	yes	maximal, complemented
3	D₃×D₉	—	3	2	no	maximal
4	C₉²	C₂²	1	1	yes	characteristic, complemented, Sylow
5	C₃⋊D₉	—	3	2	no	—
6	C₃⋊D₉	—	6	1	no	—
7	D₃×C₉	—	3	2	no	—
8	C₃×D₉	D₃	1	2	yes	—
9	D₃²	—	9	1	no	—
10	C₂×D₉	—	9	2	no	—
11	C₃×C₉	—	2	1	no	—
12	C₃×C₉	C₂×D₃	1	2	yes	—
13	C₃⋊D₃	—	9	1	no	—
14	C₃×D₃	—	3	2	no	—
15	C₂×C₉	—	9	2	no	—
16	D₉	—	9	2	no	—
17	D₉	—	18	2	no	—
18	D₉	—	18	3	no	—
19	D₉	D₉	1	2	yes	direct product factor
20	C₂×D₃	—	27	2	no	—
21	C₃²	D₃²	1	1	yes	characteristic
22	C₉	—	2	3	no	—
23	C₉	—	2	2	no	—
24	C₉	C₂×D₉	1	2	yes	complemented
25	C₂×C₃	—	9	2	no	—
26	D₃	—	3	2	no	—
27	D₃	—	27	2	no	—
28	D₃	—	54	1	no	—
29	C₂²	—	81	1	no	Sylow
30	C₃	—	2	1	no	—
31	C₃	D₃×D₉	1	2	yes	—
32	C₂	—	9	2	no	—
33	C₂	—	81	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉²	C₂²
Frattini, socle	C₃²	D₃²
LCS	G ▷ C₉²
UCS	C₁

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.