Subgroups — C₃×#870 #63993

Order 288.

127 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄²	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	C₃×Dic₃⋊C₄	C₂	1	4	yes	maximal
4	C₃×#125	—	3	1	no	maximal
5	#870	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×C₄	C₂²	1	2	yes	—
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×Dic₃	C₂²	1	4	yes	—
9	C₃×C₄⋊C₄	—	3	4	no	—
10	C₃×C₄⋊C₄	—	3	2	no	—
11	C₃×C₄²	—	2	1	no	—
12	C₃×C₄²	D₃	1	1	yes	characteristic
13	C₃×C₄²	C₂×C₃	1	1	yes	characteristic
14	C₄⋊Dic₃	C₂×C₃	1	2	yes	—
15	Dic₃⋊C₄	C₂×C₃	1	4	yes	—
16	C₂²×C₃²	C₂³	1	1	yes	characteristic
17	C₃²×C₄	—	2	2	no	—
18	C₃²×C₄	Q₈	1	2	yes	—
19	C₃×Dic₃	—	2	4	no	—
20	#125	—	3	1	no	Sylow
21	C₂×C₃×C₄	—	2	2	no	—
22	C₂×C₃×C₄	—	2	1	no	—
23	C₂×C₃×C₄	—	3	4	no	—
24	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
25	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
26	C₂×C₃×C₄	C₂²×C₃	1	2	yes	—
27	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
28	C₂×Dic₃	C₂²×C₃	1	4	yes	—
29	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₃²	#41	1	2	yes	—
31	C₄⋊C₄	—	3	4	no	—
32	C₄⋊C₄	—	3	2	no	—
33	C₄²	C₃×D₃	1	1	yes	characteristic
34	C₂²×C₃	—	2	1	no	—
35	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
36	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic
37	C₃×C₄	—	2	4	no	—
38	C₃×C₄	—	2	2	no	—
39	C₃×C₄	—	2	2	no	—
40	C₃×C₄	—	2	2	no	—
41	C₃×C₄	—	6	4	no	—
42	C₃×C₄	Dic₆	1	2	yes	—
43	C₃×C₄	C₃×Q₈	1	2	yes	—
44	Dic₃	—	2	4	no	—
45	C₃²	#125	1	1	yes	characteristic, complemented, Sylow
46	C₂×C₄	—	3	4	no	—
47	C₂×C₄	C₂×C₃×D₃	1	2	yes	—
48	C₂×C₄	C₂×C₃×D₃	1	1	yes	characteristic
49	C₂×C₃	—	2	1	no	—
50	C₂×C₃	—	2	2	no	—
51	C₂×C₃	C₂×Dic₆	1	1	yes	characteristic
52	C₂×C₃	#235	1	2	yes	—
53	C₂×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
54	C₂×C₃	C₃×#41	1	2	yes	—
55	C₂²	C₂²×C₃×D₃	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	6	4	no	—
58	C₄	C₃×Dic₆	1	2	yes	—
59	C₃	—	2	1	no	—
60	C₃	#870	1	1	yes	characteristic, direct product factor
61	C₃	C₃×#125	1	1	yes	characteristic, complemented
62	C₂	C₂×C₃×Dic₆	1	1	yes	characteristic
63	C₂	C₃×#235	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂³×C₃
Frattini	C₂²	C₂²×C₃×D₃
Fitting	C₃²×C₄²	C₂
socle, prime radical	C₂²×C₃²	C₂³
hypercenter	C₃×C₄²	D₃
hypocenter	C₃	C₃×#125
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄²

Complements

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

N	H ≅ G/N	type
5. #870	C₃	direct (N × H)
45. C₃²	#125	semidirect
60. C₃	#870	direct (N × H)
61. C₃	C₃×#125	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄²	2
C₃×C₄⋊Dic₃	2
C₃×Dic₃⋊C₄	2
C₃×#125	3
#870	3

Subgroups — C3×#870 #63993

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#870 #63993