Subgroups — C₂⁴×C₃×Q₈ #87579

Order 384.

6264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×Q₈	C₂	1	60	yes	maximal, direct product factor
2	C₂⁴×C₃×C₄	C₂	1	3	yes	maximal
3	C₂⁴×Q₈	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂⁵×C₃	C₂²	1	1	yes	characteristic
5	C₂²×C₃×Q₈	C₂²	1	560	yes	direct product factor
6	C₂³×C₃×C₄	C₂²	1	90	yes	—
7	C₂³×Q₈	C₂×C₃	1	60	yes	direct product factor
8	C₂⁴×C₄	C₂×C₃	1	3	yes	—
9	C₂⁴×C₃	Q₈	1	16	yes	direct product factor
10	C₂⁴×C₃	C₂³	1	15	yes	—
11	C₂×C₃×Q₈	C₂³	1	960	yes	direct product factor
12	C₂²×C₃×C₄	C₂³	1	420	yes	—
13	C₂⁵	C₂²×C₃	1	1	yes	characteristic
14	C₂²×Q₈	C₂²×C₃	1	560	yes	direct product factor
15	C₂³×C₄	C₂²×C₃	1	90	yes	—
16	C₂³×C₃	C₂×Q₈	1	120	yes	direct product factor
17	C₂³×C₃	C₂⁴	1	35	yes	—
18	C₃×Q₈	C₂⁴	1	256	yes	direct product factor
19	C₂×C₃×C₄	C₂⁴	1	360	yes	—
20	C₂⁴	C₃×Q₈	1	16	yes	direct product factor
21	C₂⁴	C₂³×C₃	1	15	yes	—
22	C₂×Q₈	C₂³×C₃	1	960	yes	direct product factor
23	C₂²×C₄	C₂³×C₃	1	420	yes	—
24	C₂²×C₃	C₂²×Q₈	1	140	yes	direct product factor
25	C₂²×C₃	C₂⁵	1	15	yes	—
26	C₃×C₄	C₂⁵	1	48	yes	—
27	C₂³	C₂×C₃×Q₈	1	120	yes	direct product factor
28	C₂³	C₂⁴×C₃	1	35	yes	—
29	Q₈	C₂⁴×C₃	1	256	yes	direct product factor
30	C₂×C₄	C₂⁴×C₃	1	360	yes	—
31	C₂×C₃	C₂³×Q₈	1	30	yes	direct product factor
32	C₂×C₃	C₂⁶	1	1	yes	characteristic
33	C₂²	C₂²×C₃×Q₈	1	140	yes	direct product factor
34	C₂²	C₂⁵×C₃	1	15	yes	—
35	C₄	C₂⁵×C₃	1	48	yes	—
36	C₃	C₂⁴×Q₈	1	1	yes	characteristic, direct product factor, Sylow
37	C₂	C₂³×C₃×Q₈	1	30	yes	direct product factor
38	C₂	C₂⁶×C₃	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃	C₂²
commutator, Frattini, cosocle	C₂	C₂⁶×C₃
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃×Q₈	C₂	direct (N × H)
3. C₂⁴×Q₈	C₃	direct (N × H)
5. C₂²×C₃×Q₈	C₂²	direct (N × H)
7. C₂³×Q₈	C₂×C₃	direct (N × H)
9. C₂⁴×C₃	Q₈	direct (N × H)
11. C₂×C₃×Q₈	C₂³	direct (N × H)
14. C₂²×Q₈	C₂²×C₃	direct (N × H)
16. C₂³×C₃	C₂×Q₈	direct (N × H)
18. C₃×Q₈	C₂⁴	direct (N × H)
20. C₂⁴	C₃×Q₈	direct (N × H)
22. C₂×Q₈	C₂³×C₃	direct (N × H)
24. C₂²×C₃	C₂²×Q₈	direct (N × H)
27. C₂³	C₂×C₃×Q₈	direct (N × H)
29. Q₈	C₂⁴×C₃	direct (N × H)
31. C₂×C₃	C₂³×Q₈	direct (N × H)
33. C₂²	C₂²×C₃×Q₈	direct (N × H)
36. C₃	C₂⁴×Q₈	direct (N × H)
37. C₂	C₂³×C₃×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
C₂³×C₃×Q₈	2
C₂⁴×C₃×C₄	2
C₂⁴×Q₈	3

Subgroups — C24×C3×Q8 #87579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃×Q₈ #87579