Subgroups — #13048173

Order 1296.

291 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₉	C₃	1	1	yes	maximal, characteristic
2	#88907	C₃	1	3	yes	maximal
3	#66359	—	4	5	no	maximal
4	C₂³×C₃×C₉	—	3	5	no	—
5	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
6	C₂⁴×C₉	—	3	1	no	—
7	C₂²⋊(C₂²⋊C₉)	—	3	3	no	—
8	C₂²×C₃×C₉	—	3	10	no	—
9	C₂²×C₃×C₉	A₄	1	5	yes	—
10	#1094	—	4	15	no	—
11	#733	—	16	1	no	Sylow
12	C₂³×C₃²	—	3	5	no	—
13	C₂³×C₉	—	3	15	no	—
14	C₂×C₃×C₉	—	3	5	no	—
15	C₂⁴×C₃	—	3	1	no	—
16	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
17	C₂²×C₃²	—	3	10	no	—
18	C₂²×C₃²	C₃×A₄	1	5	yes	—
19	C₂²×C₉	—	3	30	no	—
20	C₂²×C₉	—	3	5	no	—
21	C₂²⋊C₉	—	12	15	no	—
22	3₋¹⁺²	—	16	3	no	—
23	C₃×C₉	C₂²⋊A₄	1	1	yes	characteristic
24	C₂³×C₃	—	3	15	no	—
25	C₂³×C₃	—	3	5	no	—
26	C₂×C₃²	—	3	5	no	—
27	C₂×C₉	—	3	15	no	—
28	C₂⁴	#733	1	1	yes	characteristic, complemented, Sylow
29	C₂²×C₃	—	3	10	no	—
30	C₂²×C₃	—	3	30	no	—
31	C₂²×C₃	—	3	5	no	—
32	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
33	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
34	C₉	—	3	1	no	—
35	C₉	—	48	3	no	—
36	C₂³	—	3	5	no	—
37	C₂×C₃	—	3	15	no	—
38	C₂×C₃	—	3	5	no	—
39	C₂²	—	3	10	no	—
40	C₂²	#66359	1	5	yes	complemented
41	C₃	—	3	1	no	—
42	C₃	C₃²⋊(C₂²⋊A₄)	1	1	yes	characteristic
43	C₂	—	3	5	no	—

	subgroup	G/H
center	C₃	C₃²⋊(C₂²⋊A₄)
commutator, prime radical, cosocle	C₂⁴×C₃²	C₃²
Frattini	C₃²	C₃×C₂²⋊A₄
Fitting	C₂⁴×C₃×C₉	C₃
socle	C₂⁴×C₃	H(3,𝔽₃)
hypercenter	C₃×C₉	C₂²⋊A₄
hypocenter	C₂⁴	#733
LCS	G ▷ C₂⁴×C₃² ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉

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

N	H ≅ G/N	type
28. C₂⁴	#733	semidirect
40. C₂²	#66359	semidirect

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