Subgroups — C₃²×#131 #64180

Order 288.

186 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41	C₂	1	1	yes	maximal, characteristic
2	C₃²×M₁₆	C₂	1	3	yes	maximal, complemented
3	C₂×C₃²×C₈	C₂	1	3	yes	maximal, complemented
4	C₃×#131	C₃	1	4	yes	maximal, direct product factor
5	C₃²×Q₈	C₄	1	1	yes	characteristic
6	C₃²×D₄	C₄	1	3	yes	—
7	C₂×C₃²×C₄	C₂²	1	3	yes	—
8	C₃²×C₈	C₂²	1	3	yes	—
9	C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₃×#41	C₂×C₃	1	4	yes	—
11	C₃×M₁₆	C₂×C₃	1	12	yes	complemented
12	C₂×C₃×C₈	C₂×C₃	1	12	yes	complemented
13	C₂²×C₃²	C₂×C₄	1	3	yes	—
14	C₃²×C₄	C₂×C₄	1	3	yes	—
15	C₃²×C₄	C₂³	1	1	yes	characteristic
16	#131	C₃²	1	1	yes	characteristic, direct product factor, Sylow
17	C₃×Q₈	C₃×C₄	1	4	yes	—
18	C₃×D₄	C₃×C₄	1	12	yes	—
19	C₂×C₃×C₄	C₂²×C₃	1	12	yes	—
20	C₃×C₈	C₂²×C₃	1	12	yes	—
21	C₃×C₈	C₂²×C₃	1	4	yes	—
22	C₂×C₃²	—	2	3	no	—
23	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
24	#41	C₂×C₃²	1	1	yes	characteristic
25	M₁₆	C₂×C₃²	1	3	yes	complemented
26	C₂×C₈	C₂×C₃²	1	3	yes	complemented
27	C₂²×C₃	C₂×C₃×C₄	1	12	yes	—
28	C₃×C₄	C₂×C₃×C₄	1	12	yes	—
29	C₃×C₄	C₂³×C₃	1	4	yes	—
30	C₃²	#131	1	1	yes	characteristic, direct product factor, Sylow
31	Q₈	C₃²×C₄	1	1	yes	characteristic
32	D₄	C₃²×C₄	1	3	yes	—
33	C₂×C₄	C₂²×C₃²	1	3	yes	—
34	C₈	C₂²×C₃²	1	1	yes	characteristic
35	C₈	C₂²×C₃²	1	3	yes	—
36	C₂×C₃	—	2	12	no	—
37	C₂×C₃	C₂²×C₃×C₄	1	4	yes	—
38	C₂²	C₂×C₃²×C₄	1	3	yes	—
39	C₄	C₂×C₃²×C₄	1	3	yes	—
40	C₄	C₂³×C₃²	1	1	yes	characteristic
41	C₃	C₃×#131	1	4	yes	direct product factor
42	C₂	—	2	3	no	—
43	C₂	C₂²×C₃²×C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃²×C₈	C₂²
commutator	C₂	C₂²×C₃²×C₄
Frattini, cosocle	C₄	C₂³×C₃²
socle	C₂×C₃²	C₂²×C₄
prime radical	C₃²×#41	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃²×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃²×M₁₆	C₂	semidirect
3. C₂×C₃²×C₈	C₂	semidirect
4. C₃×#131	C₃	direct (N × H)
11. C₃×M₁₆	C₂×C₃	semidirect
12. C₂×C₃×C₈	C₂×C₃	semidirect
16. #131	C₃²	direct (N × H)
25. M₁₆	C₂×C₃²	semidirect
26. C₂×C₈	C₂×C₃²	semidirect
30. C₃²	#131	direct (N × H)
41. C₃	C₃×#131	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×#41	2
C₃²×M₁₆	2
C₂×C₃²×C₈	2
C₃×#131	3

Subgroups — C32×#131 #64180

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#131 #64180