Subgroups — C₃×#812 #63598

Order 288.

63 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₃⋊C₁₆	C₂	1	2	yes	maximal, complemented
3	C₃×M₃₂	—	3	1	no	maximal
4	#812	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃²×C₄	C₄	1	1	yes	characteristic
6	C₃²×C₈	C₄	1	1	yes	characteristic
7	C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₈	—	2	1	no	—
9	C₂×C₃×C₈	D₃	1	1	yes	characteristic
10	C₂×C₃×C₈	C₂×C₃	1	1	yes	characteristic
11	C₃×C₁₆	—	3	2	no	—
12	C₃⋊C₁₆	C₂×C₃	1	2	yes	complemented
13	C₂²×C₃²	C₈	1	1	yes	characteristic
14	C₃²×C₄	C₈	1	1	yes	characteristic
15	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
16	M₃₂	—	3	1	no	Sylow
17	C₂×C₃×C₄	—	2	1	no	—
18	C₂×C₃×C₄	Dic₃	1	1	yes	characteristic
19	C₂×C₃×C₄	C₃×C₄	1	1	yes	characteristic
20	C₃×C₈	—	2	1	no	—
21	C₃×C₈	—	2	1	no	—
22	C₃×C₈	Dic₃	1	1	yes	characteristic
23	C₃×C₈	C₃×C₄	1	1	yes	characteristic
24	C₃×C₈	C₂×D₃	1	1	yes	characteristic
25	C₃×C₈	C₂²×C₃	1	1	yes	characteristic
26	C₂×C₃²	—	2	1	no	—
27	C₂×C₃²	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈	C₃×D₃	1	1	yes	characteristic
29	C₁₆	—	3	2	no	—
30	C₂²×C₃	—	2	1	no	—
31	C₂²×C₃	C₃⋊C₈	1	1	yes	characteristic
32	C₂²×C₃	C₃×C₈	1	1	yes	characteristic
33	C₃×C₄	—	2	1	no	—
34	C₃×C₄	—	2	1	no	—
35	C₃×C₄	C₃⋊C₈	1	1	yes	characteristic
36	C₃×C₄	C₃×C₈	1	1	yes	characteristic
37	C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
38	C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
39	C₃²	M₃₂	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₄	C₃×Dic₃	1	1	yes	characteristic
41	C₈	C₃×Dic₃	1	1	yes	characteristic
42	C₈	C₂×C₃×D₃	1	1	yes	characteristic
43	C₂×C₃	—	2	2	no	—
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	—	2	1	no	—
46	C₂×C₃	—	2	1	no	—
47	C₂×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
48	C₂×C₃	C₂×C₃×C₈	1	1	yes	characteristic
49	C₂²	C₃×C₃⋊C₈	1	1	yes	characteristic
50	C₄	C₃×C₃⋊C₈	1	1	yes	characteristic
51	C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
52	C₃	—	2	1	no	—
53	C₃	#812	1	1	yes	characteristic, direct product factor
54	C₃	C₃×M₃₂	1	1	yes	characteristic, complemented
55	C₂	—	2	1	no	—
56	C₂	C₂×C₃×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₈	C₂×D₃
commutator	C₂×C₃	C₂×C₃×C₈
Frattini	C₈	C₂×C₃×D₃
Fitting	C₂×C₃²×C₈	C₂
socle	C₂×C₃²	C₂×C₈
prime radical	C₂²×C₃²	C₈
cosocle	C₃×C₈	C₂²×C₃
hypercenter	C₂×C₃×C₈	D₃
hypocenter	C₃	C₃×M₃₂
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₃⋊C₁₆	C₂	semidirect
4. #812	C₃	direct (N × H)
12. C₃⋊C₁₆	C₂×C₃	semidirect
39. C₃²	M₃₂	semidirect
53. C₃	#812	direct (N × H)
54. C₃	C₃×M₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×C₈	2
C₃×C₃⋊C₁₆	2
C₃×M₃₂	3
#812	3

Subgroups — C3×#812 #63598

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#812 #63598