Subgroups — C₃×#808 #63594

Order 288.

85 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#106	—	3	1	no	maximal
5	#808	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃²×C₄	—	2	1	no	—
7	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
8	C₂×C₃×Dic₃	—	2	1	no	—
9	C₃×C₃⋊C₈	C₄	1	2	yes	complemented
10	C₂×C₃×C₈	—	3	1	no	—
11	C₃×C₄⋊C₄	—	2	1	no	—
12	C₃×C₄⋊C₄	—	3	1	no	—
13	C₃×C₄⋊C₄	D₃	1	1	yes	characteristic
14	C₃×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
15	C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
16	C₂×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₃²	D₄	1	1	yes	characteristic
18	C₃²×C₄	—	4	1	no	—
19	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
20	C₃²×C₄	Q₈	1	1	yes	characteristic
21	C₃×Dic₃	—	4	1	no	—
22	#106	—	3	1	no	Sylow
23	C₂×C₃×C₄	—	2	2	no	—
24	C₂×C₃×C₄	—	2	1	no	—
25	C₂×C₃×C₄	—	2	1	no	—
26	C₂×C₃×C₄	—	2	1	no	—
27	C₂×C₃×C₄	—	6	1	no	—
28	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
29	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
30	C₂×Dic₃	—	2	1	no	—
31	C₃×C₈	—	3	2	no	—
32	C₃⋊C₈	C₃×C₄	1	2	yes	complemented
33	C₂×C₃²	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₃²	SD₁₆	1	1	yes	characteristic
35	C₂×C₃²	SD₁₆	1	1	yes	characteristic
36	C₂×C₈	—	3	1	no	—
37	C₄⋊C₄	—	3	1	no	—
38	C₄⋊C₄	C₃×D₃	1	1	yes	characteristic
39	C₂²×C₃	—	2	1	no	—
40	C₂²×C₃	#67	1	1	yes	characteristic
41	C₂²×C₃	C₃×D₄	1	1	yes	characteristic
42	C₃×C₄	—	2	1	no	—
43	C₃×C₄	—	2	1	no	—
44	C₃×C₄	—	4	2	no	—
45	C₃×C₄	—	4	1	no	—
46	C₃×C₄	—	4	1	no	—
47	C₃×C₄	—	12	1	no	—
48	C₃×C₄	Dic₆	1	1	yes	characteristic
49	C₃×C₄	C₄×D₃	1	1	yes	characteristic
50	C₃×C₄	C₂×C₃×C₄	1	1	yes	characteristic
51	C₃×C₄	C₃×Q₈	1	1	yes	characteristic
52	Dic₃	—	4	1	no	—
53	C₃²	#106	1	1	yes	characteristic, complemented, Sylow
54	C₂×C₄	—	2	1	no	—
55	C₂×C₄	—	6	1	no	—
56	C₂×C₄	C₂×C₃×D₃	1	1	yes	characteristic
57	C₈	—	3	2	no	—
58	C₂×C₃	—	2	1	no	—
59	C₂×C₃	—	2	1	no	—
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	Dic₃⋊C₄	1	1	yes	characteristic
62	C₂×C₃	#214	1	1	yes	characteristic
63	C₂×C₃	#215	1	1	yes	characteristic
64	C₂×C₃	C₃×C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃	C₃×SD₁₆	1	1	yes	characteristic
66	C₂×C₃	C₃×SD₁₆	1	1	yes	characteristic
67	C₂²	C₃×#67	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	—	12	1	no	—
70	C₄	C₃×Dic₆	1	1	yes	characteristic
71	C₄	C₃×C₄×D₃	1	1	yes	characteristic
72	C₃	—	2	1	no	—
73	C₃	#808	1	1	yes	characteristic, direct product factor
74	C₃	C₃×#106	1	1	yes	characteristic, complemented
75	C₂	C₃×Dic₃⋊C₄	1	1	yes	characteristic
76	C₂	C₃×#214	1	1	yes	characteristic
77	C₂	C₃×#215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#67
commutator	C₃×C₄	C₂×C₃×C₄
Frattini	C₂×C₄	C₂×C₃×D₃
Fitting	C₃²×C₄⋊C₄	C₂
socle, prime radical	C₂²×C₃²	D₄
cosocle	C₂×C₃×C₄	C₂²×C₃
hypercenter	C₃×C₄⋊C₄	D₃
hypocenter	C₃	C₃×#106
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₃×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
5. #808	C₃	direct (N × H)
9. C₃×C₃⋊C₈	C₄	semidirect
32. C₃⋊C₈	C₃×C₄	semidirect
53. C₃²	#106	semidirect
73. C₃	#808	direct (N × H)
74. C₃	C₃×#106	semidirect

Maximal subgroups

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

subgroup	index
C₃²×C₄⋊C₄	2
C₃×C₄⋊Dic₃	2
C₂×C₃×C₃⋊C₈	2
C₃×#106	3
#808	3

Subgroups — C3×#808 #63594

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#808 #63594