Subgroups — C₃²×C₅×Q₈ #67187

Order 360.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₄×C₅	C₂	1	3	yes	maximal
2	C₃×C₅×Q₈	C₃	1	4	yes	maximal, direct product factor
3	C₂×C₃²×C₅	C₂²	1	1	yes	characteristic
4	C₃²×Q₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₃×C₄×C₅	C₂×C₃	1	12	yes	—
6	C₃²×C₅	Q₈	1	1	yes	characteristic, direct product factor, Hall
7	C₅×Q₈	C₃²	1	1	yes	characteristic, direct product factor, Hall
8	C₃²×C₄	C₂×C₅	1	3	yes	—
9	C₂×C₃×C₅	C₂²×C₃	1	4	yes	—
10	C₃×Q₈	C₃×C₅	1	4	yes	direct product factor
11	C₄×C₅	C₂×C₃²	1	3	yes	—
12	C₂×C₃²	C₂²×C₅	1	1	yes	characteristic
13	C₃×C₅	C₃×Q₈	1	4	yes	direct product factor
14	C₃×C₄	C₂×C₃×C₅	1	12	yes	—
15	C₂×C₅	C₂²×C₃²	1	1	yes	characteristic
16	C₃²	C₅×Q₈	1	1	yes	characteristic, direct product factor, Sylow
17	Q₈	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
18	C₂×C₃	C₂²×C₃×C₅	1	4	yes	—
19	C₅	C₃²×Q₈	1	1	yes	characteristic, direct product factor, Sylow
20	C₄	C₂×C₃²×C₅	1	3	yes	—
21	C₃	C₃×C₅×Q₈	1	4	yes	direct product factor
22	C₂	C₂²×C₃²×C₅	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂×C₃²×C₅	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₃²×C₅
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₅ ◁ G

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

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