Subgroups — C₂²×C₃²×Q₈ #64372

Order 288.

468 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×Q₈	C₂	1	12	yes	maximal, direct product factor
2	C₂²×C₃²×C₄	C₂	1	3	yes	maximal
3	C₂²×C₃×Q₈	C₃	1	4	yes	maximal, direct product factor
4	C₂³×C₃²	C₂²	1	1	yes	characteristic
5	C₃²×Q₈	C₂²	1	16	yes	direct product factor
6	C₂×C₃²×C₄	C₂²	1	18	yes	—
7	C₂×C₃×Q₈	C₂×C₃	1	48	yes	direct product factor
8	C₂²×C₃×C₄	C₂×C₃	1	12	yes	—
9	C₂²×C₃²	Q₈	1	4	yes	direct product factor
10	C₂²×C₃²	C₂³	1	3	yes	—
11	C₃²×C₄	C₂³	1	12	yes	—
12	C₂²×Q₈	C₃²	1	1	yes	characteristic, direct product factor, Sylow
13	C₂³×C₃	C₂²×C₃	1	4	yes	—
14	C₃×Q₈	C₂²×C₃	1	64	yes	direct product factor
15	C₂×C₃×C₄	C₂²×C₃	1	72	yes	—
16	C₂×C₃²	C₂×Q₈	1	6	yes	direct product factor
17	C₂×C₃²	C₂⁴	1	1	yes	characteristic
18	C₂×Q₈	C₂×C₃²	1	12	yes	direct product factor
19	C₂²×C₄	C₂×C₃²	1	3	yes	—
20	C₂²×C₃	C₃×Q₈	1	16	yes	direct product factor
21	C₂²×C₃	C₂³×C₃	1	12	yes	—
22	C₃×C₄	C₂³×C₃	1	48	yes	—
23	C₃²	C₂²×Q₈	1	1	yes	characteristic, direct product factor, Sylow
24	C₂³	C₂²×C₃²	1	1	yes	characteristic
25	Q₈	C₂²×C₃²	1	16	yes	direct product factor
26	C₂×C₄	C₂²×C₃²	1	18	yes	—
27	C₂×C₃	C₂×C₃×Q₈	1	24	yes	direct product factor
28	C₂×C₃	C₂⁴×C₃	1	4	yes	—
29	C₂²	C₃²×Q₈	1	4	yes	direct product factor
30	C₂²	C₂³×C₃²	1	3	yes	—
31	C₄	C₂³×C₃²	1	12	yes	—
32	C₃	C₂²×C₃×Q₈	1	4	yes	direct product factor
33	C₂	C₂×C₃²×Q₈	1	6	yes	direct product factor
34	C₂	C₂⁴×C₃²	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃²	C₂²
commutator, Frattini, cosocle	C₂	C₂⁴×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃²×Q₈	C₂	direct (N × H)
3. C₂²×C₃×Q₈	C₃	direct (N × H)
5. C₃²×Q₈	C₂²	direct (N × H)
7. C₂×C₃×Q₈	C₂×C₃	direct (N × H)
9. C₂²×C₃²	Q₈	direct (N × H)
12. C₂²×Q₈	C₃²	direct (N × H)
14. C₃×Q₈	C₂²×C₃	direct (N × H)
16. C₂×C₃²	C₂×Q₈	direct (N × H)
18. C₂×Q₈	C₂×C₃²	direct (N × H)
20. C₂²×C₃	C₃×Q₈	direct (N × H)
23. C₃²	C₂²×Q₈	direct (N × H)
25. Q₈	C₂²×C₃²	direct (N × H)
27. C₂×C₃	C₂×C₃×Q₈	direct (N × H)
29. C₂²	C₃²×Q₈	direct (N × H)
32. C₃	C₂²×C₃×Q₈	direct (N × H)
33. C₂	C₂×C₃²×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
C₂²×C₃²×C₄	2
C₂²×C₃×Q₈	3

Subgroups — C22×C32×Q8 #64372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃²×Q₈ #64372