Subgroups — C₃×D₃×Q₈ #3821

Order 144.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₄×D₃	C₂	1	3	yes	maximal
3	C₃×Dic₆	C₂	1	3	yes	maximal, complemented
4	C₂×C₃×Q₈	—	3	1	no	maximal
5	D₃×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor
6	C₂×C₃×D₃	C₂²	1	1	yes	characteristic
7	C₃²×C₄	C₂²	1	3	yes	—
8	C₃×Dic₃	C₂²	1	3	yes	—
9	C₃×Q₈	—	2	1	no	—
10	C₃×Q₈	—	3	3	no	—
11	C₃×Q₈	D₃	1	1	yes	characteristic, direct product factor
12	C₃×Q₈	C₂×C₃	1	1	yes	characteristic, complemented
13	C₂×C₃×C₄	—	3	3	no	—
14	C₄×D₃	C₂×C₃	1	3	yes	—
15	Dic₆	C₂×C₃	1	3	yes	complemented
16	C₂×C₃²	C₂³	1	1	yes	characteristic
17	C₃×D₃	Q₈	1	2	yes	direct product factor
18	C₂×Q₈	—	3	1	no	Sylow
19	C₂²×C₃	—	3	1	no	—
20	C₂×D₃	C₂²×C₃	1	1	yes	characteristic
21	C₃×C₄	—	2	3	no	—
22	C₃×C₄	—	3	3	no	—
23	C₃×C₄	C₂×D₃	1	3	yes	—
24	C₃×C₄	C₂²×C₃	1	3	yes	—
25	Dic₃	C₂²×C₃	1	3	yes	—
26	C₃²	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
27	Q₈	—	3	3	no	—
28	Q₈	C₃×D₃	1	1	yes	characteristic, direct product factor
29	C₂×C₄	—	3	3	no	—
30	C₂×C₃	—	2	1	no	—
31	C₂×C₃	—	3	2	no	—
32	C₂×C₃	C₂²×D₃	1	1	yes	characteristic
33	C₂×C₃	C₂³×C₃	1	1	yes	characteristic
34	D₃	C₃×Q₈	1	2	yes	direct product factor
35	C₂²	—	3	1	no	—
36	C₄	—	3	3	no	—
37	C₄	C₂×C₃×D₃	1	3	yes	—
38	C₃	—	2	1	no	—
39	C₃	D₃×Q₈	1	1	yes	characteristic, direct product factor
40	C₃	C₂×C₃×Q₈	1	1	yes	characteristic, complemented
41	C₂	—	3	2	no	—
42	C₂	C₂²×C₃×D₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃²×Q₈	C₂	semidirect
3. C₃×Dic₆	C₂	semidirect
5. D₃×Q₈	C₃	direct (N × H)
11. C₃×Q₈	D₃	direct (N × H)
12. C₃×Q₈	C₂×C₃	semidirect
15. Dic₆	C₂×C₃	semidirect
17. C₃×D₃	Q₈	direct (N × H)
26. C₃²	C₂×Q₈	semidirect
28. Q₈	C₃×D₃	direct (N × H)
34. D₃	C₃×Q₈	direct (N × H)
39. C₃	D₃×Q₈	direct (N × H)
40. C₃	C₂×C₃×Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₃²×Q₈	2
C₃×C₄×D₃	2
C₃×Dic₆	2
C₂×C₃×Q₈	3
D₃×Q₈	3

Subgroups — C3×D3×Q8 #3821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×D₃×Q₈ #3821