Subgroups — C₃×Q₈×Dic₃ #64068

Order 288.

159 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊Dic₃	C₂	1	3	yes	maximal
3	C₃×C₄×Dic₃	C₂	1	3	yes	maximal
4	C₃×C₄×Q₈	—	3	1	no	maximal
5	Q₈×Dic₃	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₃²×Q₈	C₄	1	4	yes	complemented
7	C₂×C₃²×C₄	C₂²	1	3	yes	—
8	C₂×C₃×Dic₃	C₂²	1	3	yes	—
9	C₂×C₃×Dic₃	C₂²	1	1	yes	characteristic
10	C₂×C₃×Q₈	—	2	1	no	—
11	C₂×C₃×Q₈	D₃	1	1	yes	characteristic
12	C₂×C₃×Q₈	C₂×C₃	1	1	yes	characteristic
13	C₃×C₄⋊C₄	—	3	3	no	—
14	C₃×C₄²	—	3	3	no	—
15	C₄⋊Dic₃	C₂×C₃	1	3	yes	—
16	C₄×Dic₃	C₂×C₃	1	3	yes	—
17	C₂²×C₃²	C₂³	1	1	yes	characteristic
18	C₃²×C₄	C₂×C₄	1	6	yes	—
19	C₃×Dic₃	—	2	3	no	—
20	C₃×Dic₃	Q₈	1	2	yes	direct product factor
21	C₄×Q₈	—	3	1	no	Sylow
22	C₃×Q₈	—	2	4	no	—
23	C₃×Q₈	Dic₃	1	4	yes	direct product factor
24	C₃×Q₈	C₃×C₄	1	4	yes	complemented
25	C₂×C₃×C₄	—	2	3	no	—
26	C₂×C₃×C₄	—	3	1	no	—
27	C₂×C₃×C₄	—	3	3	no	—
28	C₂×C₃×C₄	C₂×D₃	1	3	yes	—
29	C₂×C₃×C₄	C₂²×C₃	1	3	yes	—
30	C₂×Dic₃	C₂²×C₃	1	3	yes	—
31	C₂×Dic₃	C₂²×C₃	1	1	yes	characteristic
32	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₃²	#41	1	1	yes	characteristic
35	C₂×Q₈	C₃×D₃	1	1	yes	characteristic
36	C₄⋊C₄	—	3	3	no	—
37	C₄²	—	3	3	no	—
38	C₂²×C₃	—	2	1	no	—
39	C₂²×C₃	C₂²×D₃	1	1	yes	characteristic
40	C₂²×C₃	C₂³×C₃	1	1	yes	characteristic
41	C₃×C₄	—	2	6	no	—
42	C₃×C₄	—	3	2	no	—
43	C₃×C₄	—	6	3	no	—
44	C₃×C₄	C₂×Dic₃	1	6	yes	—
45	C₃×C₄	C₂×C₃×C₄	1	6	yes	—
46	Dic₃	—	2	3	no	—
47	Dic₃	C₃×Q₈	1	2	yes	direct product factor
48	C₃²	C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
49	Q₈	C₃×Dic₃	1	4	yes	direct product factor
50	C₂×C₄	—	3	1	no	—
51	C₂×C₄	—	3	3	no	—
52	C₂×C₄	C₂×C₃×D₃	1	3	yes	—
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	—	2	1	no	—
56	C₂×C₃	D₃×Q₈	1	1	yes	characteristic
57	C₂×C₃	#239	1	1	yes	characteristic
58	C₂×C₃	C₂²×Dic₃	1	1	yes	characteristic
59	C₂×C₃	C₂²×C₃×C₄	1	1	yes	characteristic
60	C₂×C₃	C₂×C₃×Q₈	1	1	yes	characteristic
61	C₂×C₃	C₃×#41	1	1	yes	characteristic
62	C₂²	C₂²×C₃×D₃	1	1	yes	characteristic
63	C₄	—	3	2	no	—
64	C₄	—	6	3	no	—
65	C₄	C₂×C₃×Dic₃	1	6	yes	—
66	C₃	—	2	1	no	—
67	C₃	Q₈×Dic₃	1	1	yes	characteristic, direct product factor
68	C₃	C₃×C₄×Q₈	1	1	yes	characteristic, complemented
69	C₂	C₃×D₃×Q₈	1	1	yes	characteristic
70	C₂	C₃×#239	1	1	yes	characteristic
71	C₂	C₂²×C₃×Dic₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂²×D₃
commutator	C₂×C₃	C₂²×C₃×C₄
Frattini	C₂²	C₂²×C₃×D₃
Fitting	C₂×C₃²×Q₈	C₂
socle, prime radical	C₂²×C₃²	C₂³
cosocle	C₂²×C₃	C₂³×C₃
hypercenter	C₂×C₃×Q₈	D₃
hypocenter	C₃	C₃×C₄×Q₈
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×Q₈

Complements

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

N	H ≅ G/N	type
5. Q₈×Dic₃	C₃	direct (N × H)
6. C₃²×Q₈	C₄	semidirect
20. C₃×Dic₃	Q₈	direct (N × H)
23. C₃×Q₈	Dic₃	direct (N × H)
24. C₃×Q₈	C₃×C₄	semidirect
47. Dic₃	C₃×Q₈	direct (N × H)
48. C₃²	C₄×Q₈	semidirect
49. Q₈	C₃×Dic₃	direct (N × H)
67. C₃	Q₈×Dic₃	direct (N × H)
68. C₃	C₃×C₄×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃²×Q₈	2
C₃×C₄⋊Dic₃	2
C₃×C₄×Dic₃	2
C₃×C₄×Q₈	3
Q₈×Dic₃	3

Subgroups — C3×Q8×Dic3 #64068

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×Dic₃ #64068