Subgroups — C₃²×C₄×Q₈ #64168

Order 288.

210 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	C₃²×C₄⋊C₄	C₂	1	3	yes	maximal
3	C₃²×C₄²	C₂	1	3	yes	maximal
4	C₃×C₄×Q₈	C₃	1	4	yes	maximal, direct product factor
5	C₃²×Q₈	C₄	1	4	yes	direct product factor
6	C₂×C₃²×C₄	C₂²	1	3	yes	—
7	C₂×C₃²×C₄	C₂²	1	3	yes	—
8	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×Q₈	C₂×C₃	1	4	yes	—
10	C₃×C₄⋊C₄	C₂×C₃	1	12	yes	—
11	C₃×C₄²	C₂×C₃	1	12	yes	—
12	C₂²×C₃²	C₂³	1	1	yes	characteristic
13	C₃²×C₄	—	2	3	no	—
14	C₃²×C₄	C₂×C₄	1	6	yes	—
15	C₃²×C₄	Q₈	1	2	yes	direct product factor
16	C₄×Q₈	C₃²	1	1	yes	characteristic, direct product factor, Sylow
17	C₃×Q₈	C₃×C₄	1	16	yes	direct product factor
18	C₂×C₃×C₄	C₂²×C₃	1	12	yes	—
19	C₂×C₃×C₄	C₂²×C₃	1	12	yes	—
20	C₂×C₃×C₄	C₂²×C₃	1	4	yes	—
21	C₂×C₃²	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₃²	#41	1	1	yes	characteristic
24	C₂×Q₈	C₂×C₃²	1	1	yes	characteristic
25	C₄⋊C₄	C₂×C₃²	1	3	yes	—
26	C₄²	C₂×C₃²	1	3	yes	—
27	C₂²×C₃	C₂³×C₃	1	4	yes	—
28	C₃×C₄	—	2	12	no	—
29	C₃×C₄	C₂×C₃×C₄	1	24	yes	—
30	C₃×C₄	C₃×Q₈	1	8	yes	direct product factor
31	C₃²	C₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
32	Q₈	C₃²×C₄	1	4	yes	direct product factor
33	C₂×C₄	C₂²×C₃²	1	3	yes	—
34	C₂×C₄	C₂²×C₃²	1	3	yes	—
35	C₂×C₄	C₂²×C₃²	1	1	yes	characteristic
36	C₂×C₃	C₂²×C₃×C₄	1	4	yes	—
37	C₂×C₃	C₂×C₃×Q₈	1	4	yes	—
38	C₂×C₃	C₃×#41	1	4	yes	—
39	C₂²	C₂³×C₃²	1	1	yes	characteristic
40	C₄	—	2	3	no	—
41	C₄	C₂×C₃²×C₄	1	6	yes	—
42	C₄	C₃²×Q₈	1	2	yes	direct product factor
43	C₃	C₃×C₄×Q₈	1	4	yes	direct product factor
44	C₂	C₂²×C₃²×C₄	1	1	yes	characteristic
45	C₂	C₂×C₃²×Q₈	1	1	yes	characteristic
46	C₂	C₃²×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃²×C₄	C₂²
commutator	C₂	C₂²×C₃²×C₄
Frattini, cosocle	C₂²	C₂³×C₃²
socle, prime radical	C₂²×C₃²	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₄×Q₈	C₃	direct (N × H)
5. C₃²×Q₈	C₄	direct (N × H)
15. C₃²×C₄	Q₈	direct (N × H)
16. C₄×Q₈	C₃²	direct (N × H)
17. C₃×Q₈	C₃×C₄	direct (N × H)
30. C₃×C₄	C₃×Q₈	direct (N × H)
31. C₃²	C₄×Q₈	direct (N × H)
32. Q₈	C₃²×C₄	direct (N × H)
42. C₄	C₃²×Q₈	direct (N × H)
43. C₃	C₃×C₄×Q₈	direct (N × H)

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
C₃²×C₄⋊C₄	2
C₃²×C₄²	2
C₃×C₄×Q₈	3

Subgroups — C32×C4×Q8 #64168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₄×Q₈ #64168