Subgroups — C₃×Q₈×A₄ #64334

Order 288.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×A₄	C₂	1	3	yes	maximal
2	C₂²×C₃×Q₈	C₃	1	1	yes	maximal, characteristic, complemented
3	Q₈×A₄	C₃	1	3	yes	maximal, direct product factor
4	C₂×C₃×A₄	C₂²	1	1	yes	characteristic
5	C₃²×Q₈	—	4	1	no	maximal
6	C₂×C₃×Q₈	—	3	3	no	—
7	C₂×C₃×Q₈	—	3	1	no	—
8	C₂²×C₃×C₄	C₂×C₃	1	3	yes	—
9	C₄×A₄	C₂×C₃	1	9	yes	—
10	C₃×A₄	Q₈	1	1	yes	characteristic, direct product factor
11	C₃²×C₄	—	4	3	no	—
12	C₂²×Q₈	C₃²	1	1	yes	characteristic, complemented, Sylow
13	C₂³×C₃	C₂²×C₃	1	1	yes	characteristic
14	C₂×A₄	C₂²×C₃	1	3	yes	—
15	C₃×Q₈	—	3	2	no	—
16	C₃×Q₈	—	3	3	no	—
17	C₃×Q₈	—	4	3	no	—
18	C₃×Q₈	A₄	1	1	yes	characteristic, direct product factor
19	C₂×C₃×C₄	—	3	3	no	—
20	C₂×C₃×C₄	—	3	3	no	—
21	C₂×C₃²	—	4	1	no	—
22	C₂×Q₈	—	3	3	no	—
23	C₂×Q₈	—	3	1	no	—
24	C₂²×C₄	C₂×C₃²	1	3	yes	—
25	C₂²×C₃	—	3	1	no	—
26	C₂²×C₃	—	3	1	no	—
27	C₂²×C₃	C₃×Q₈	1	1	yes	characteristic, complemented
28	A₄	C₃×Q₈	1	3	yes	direct product factor
29	C₃×C₄	—	3	3	no	—
30	C₃×C₄	—	4	9	no	—
31	C₃×C₄	C₂×A₄	1	3	yes	—
32	C₃²	—	4	1	no	Sylow
33	C₂³	C₂²×C₃²	1	1	yes	characteristic
34	Q₈	—	3	2	no	—
35	Q₈	—	3	3	no	—
36	Q₈	C₃×A₄	1	1	yes	characteristic, direct product factor
37	C₂×C₄	—	3	3	no	—
38	C₂×C₄	—	3	3	no	—
39	C₂×C₃	—	3	1	no	—
40	C₂×C₃	—	3	1	no	—
41	C₂×C₃	—	4	3	no	—
42	C₂×C₃	C₂²×A₄	1	1	yes	characteristic
43	C₂²	—	3	1	no	—
44	C₂²	—	3	1	no	—
45	C₂²	C₃²×Q₈	1	1	yes	characteristic, complemented
46	C₄	—	3	3	no	—
47	C₄	C₂×C₃×A₄	1	3	yes	—
48	C₃	—	4	3	no	—
49	C₃	Q₈×A₄	1	1	yes	characteristic, direct product factor
50	C₂	—	3	1	no	—
51	C₂	—	3	1	no	—
52	C₂	C₂²×C₃×A₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂²×A₄
commutator, cosocle	C₂³	C₂²×C₃²
Frattini	C₂	C₂²×C₃×A₄
Fitting	C₂²×C₃×Q₈	C₃
socle	C₂³×C₃	C₂²×C₃
prime radical	C₂×C₃×A₄	C₂²
hypercenter	C₃×Q₈	A₄
hypocenter	C₂²	C₃²×Q₈
LCS	G ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂×C₃ ◁ C₃×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Q₈	C₃	semidirect
3. Q₈×A₄	C₃	direct (N × H)
10. C₃×A₄	Q₈	direct (N × H)
12. C₂²×Q₈	C₃²	semidirect
18. C₃×Q₈	A₄	direct (N × H)
27. C₂²×C₃	C₃×Q₈	semidirect
28. A₄	C₃×Q₈	direct (N × H)
36. Q₈	C₃×A₄	direct (N × H)
45. C₂²	C₃²×Q₈	semidirect
49. C₃	Q₈×A₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×A₄	2
C₂²×C₃×Q₈	3
Q₈×A₄	3
C₃²×Q₈	4

Subgroups — C3×Q8×A4 #64334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×Q₈×A₄ #64334