Subgroups — C₃×#875 #63998

Order 288.

167 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×(C₂×D₃)⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₃×#124	—	3	1	no	maximal
6	#875	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂²×C₃×D₃	C₂²	1	2	yes	—
8	C₂×C₃²×C₄	C₂²	1	2	yes	—
9	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×Dic₃	C₂²	1	2	yes	—
11	C₃×D₁₂	—	2	2	no	—
12	C₃×Dic₆	—	2	2	no	—
13	C₂×C₃×Q₈	—	3	1	no	—
14	C₂×C₃×D₄	—	3	1	no	—
15	C₂×D₁₂	C₂×C₃	1	1	yes	characteristic
16	C₂×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
17	C₃×C₂²⋊C₄	—	3	4	no	—
18	C₃×C₄²	—	2	1	no	—
19	C₃×C₄²	D₃	1	1	yes	characteristic, complemented
20	C₃×C₄²	C₂×C₃	1	1	yes	characteristic, complemented
21	(C₂×D₃)⋊C₄	C₂×C₃	1	4	yes	complemented
22	C₂²×C₃²	C₂³	1	1	yes	characteristic
23	C₂×C₃×D₃	—	2	2	no	—
24	C₂×C₃×D₃	—	2	4	no	—
25	C₃²×C₄	—	2	2	no	—
26	C₃²×C₄	D₄	1	2	yes	—
27	C₃×Dic₃	—	2	2	no	—
28	#124	—	3	1	no	Sylow
29	C₂³×C₃	—	3	2	no	—
30	C₂²×D₃	C₂²×C₃	1	2	yes	—
31	C₃×Q₈	—	6	2	no	—
32	C₃×D₄	—	6	2	no	—
33	C₂×C₃×C₄	—	2	2	no	—
34	C₂×C₃×C₄	—	2	1	no	—
35	C₂×C₃×C₄	—	3	2	no	—
36	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
37	C₂×C₃×C₄	C₂×D₃	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂²×C₃	1	2	yes	—
39	C₂×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
40	C₂×Dic₃	C₂²×C₃	1	2	yes	—
41	D₁₂	—	2	2	no	—
42	Dic₆	—	2	2	no	—
43	C₂×C₃²	C₂×D₄	1	1	yes	characteristic
44	C₂×C₃²	#41	1	2	yes	—
45	C₃×D₃	—	4	2	no	—
46	C₂×Q₈	—	3	1	no	—
47	C₂×D₄	—	3	1	no	—
48	C₂²⋊C₄	—	3	4	no	—
49	C₄²	C₃×D₃	1	1	yes	characteristic, complemented
50	C₂²×C₃	—	2	1	no	—
51	C₂²×C₃	—	6	2	no	—
52	C₂²×C₃	—	6	4	no	—
53	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
54	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic
55	C₂×D₃	—	2	2	no	—
56	C₂×D₃	—	2	4	no	—
57	C₃×C₄	—	2	4	no	—
58	C₃×C₄	—	2	2	no	—
59	C₃×C₄	—	2	2	no	—
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	—	6	2	no	—
62	C₃×C₄	D₁₂	1	2	yes	—
63	C₃×C₄	C₃×D₄	1	2	yes	—
64	Dic₃	—	2	2	no	—
65	C₃²	#124	1	1	yes	characteristic, complemented, Sylow
66	C₂³	—	3	2	no	—
67	Q₈	—	6	2	no	—
68	D₄	—	6	2	no	—
69	C₂×C₄	—	3	2	no	—
70	C₂×C₄	C₂×C₃×D₃	1	2	yes	—
71	C₂×C₄	C₂×C₃×D₃	1	1	yes	characteristic
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	2	2	no	—
74	C₂×C₃	—	12	2	no	—
75	C₂×C₃	C₂×D₁₂	1	1	yes	characteristic
76	C₂×C₃	#235	1	2	yes	—
77	C₂×C₃	C₂×C₃×D₄	1	1	yes	characteristic
78	C₂×C₃	C₃×#41	1	2	yes	—
79	D₃	—	4	2	no	—
80	C₂²	—	6	2	no	—
81	C₂²	—	6	4	no	—
82	C₂²	C₂²×C₃×D₃	1	1	yes	characteristic
83	C₄	—	2	2	no	—
84	C₄	—	6	2	no	—
85	C₄	C₃×D₁₂	1	2	yes	—
86	C₃	—	2	1	no	—
87	C₃	#875	1	1	yes	characteristic, direct product factor
88	C₃	C₃×#124	1	1	yes	characteristic, complemented
89	C₂	—	12	2	no	—
90	C₂	C₂×C₃×D₁₂	1	1	yes	characteristic
91	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	C₂²×C₃²	C₂³
prime radical	C₂×C₃×D₁₂	C₂
hypercenter	C₃×C₄²	D₃
hypocenter	C₃	C₃×#124
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
2. C₂×C₃×Dic₆	C₂	semidirect
3. C₃²×C₄²	C₂	semidirect
4. C₃×(C₂×D₃)⋊C₄	C₂	semidirect
6. #875	C₃	direct (N × H)
16. C₂×Dic₆	C₂×C₃	semidirect
19. C₃×C₄²	D₃	semidirect
20. C₃×C₄²	C₂×C₃	semidirect
21. (C₂×D₃)⋊C₄	C₂×C₃	semidirect
49. C₄²	C₃×D₃	semidirect
65. C₃²	#124	semidirect
87. C₃	#875	direct (N × H)
88. C₃	C₃×#124	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₁₂	2
C₂×C₃×Dic₆	2
C₃²×C₄²	2
C₃×(C₂×D₃)⋊C₄	2
C₃×#124	3
#875	3

Subgroups — C3×#875 #63998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#875 #63998