Subgroups — C₂⁴×C₃²×Q₈ #11770638

Order 1152.

18792 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃²×Q₈	C₂	1	60	yes	maximal, direct product factor
2	C₂⁴×C₃²×C₄	C₂	1	3	yes	maximal
3	C₂⁴×C₃×Q₈	C₃	1	4	yes	maximal, direct product factor
4	C₂⁵×C₃²	C₂²	1	1	yes	characteristic
5	C₂²×C₃²×Q₈	C₂²	1	560	yes	direct product factor
6	C₂³×C₃²×C₄	C₂²	1	90	yes	—
7	C₂³×C₃×Q₈	C₂×C₃	1	240	yes	direct product factor
8	C₂⁴×C₃×C₄	C₂×C₃	1	12	yes	—
9	C₂⁴×C₃²	Q₈	1	16	yes	direct product factor
10	C₂⁴×C₃²	C₂³	1	15	yes	—
11	C₂×C₃²×Q₈	C₂³	1	960	yes	direct product factor
12	C₂²×C₃²×C₄	C₂³	1	420	yes	—
13	C₂⁴×Q₈	C₃²	1	1	yes	characteristic, direct product factor, Sylow
14	C₂⁵×C₃	C₂²×C₃	1	4	yes	—
15	C₂²×C₃×Q₈	C₂²×C₃	1	2240	yes	direct product factor
16	C₂³×C₃×C₄	C₂²×C₃	1	360	yes	—
17	C₂³×C₃²	C₂×Q₈	1	120	yes	direct product factor
18	C₂³×C₃²	C₂⁴	1	35	yes	—
19	C₃²×Q₈	C₂⁴	1	256	yes	direct product factor
20	C₂×C₃²×C₄	C₂⁴	1	360	yes	—
21	C₂³×Q₈	C₂×C₃²	1	60	yes	direct product factor
22	C₂⁴×C₄	C₂×C₃²	1	3	yes	—
23	C₂⁴×C₃	C₃×Q₈	1	64	yes	direct product factor
24	C₂⁴×C₃	C₂³×C₃	1	60	yes	—
25	C₂×C₃×Q₈	C₂³×C₃	1	3840	yes	direct product factor
26	C₂²×C₃×C₄	C₂³×C₃	1	1680	yes	—
27	C₂²×C₃²	C₂²×Q₈	1	140	yes	direct product factor
28	C₂²×C₃²	C₂⁵	1	15	yes	—
29	C₃²×C₄	C₂⁵	1	48	yes	—
30	C₂⁵	C₂²×C₃²	1	1	yes	characteristic
31	C₂²×Q₈	C₂²×C₃²	1	560	yes	direct product factor
32	C₂³×C₄	C₂²×C₃²	1	90	yes	—
33	C₂³×C₃	C₂×C₃×Q₈	1	480	yes	direct product factor
34	C₂³×C₃	C₂⁴×C₃	1	140	yes	—
35	C₃×Q₈	C₂⁴×C₃	1	1024	yes	direct product factor
36	C₂×C₃×C₄	C₂⁴×C₃	1	1440	yes	—
37	C₂×C₃²	C₂³×Q₈	1	30	yes	direct product factor
38	C₂×C₃²	C₂⁶	1	1	yes	characteristic
39	C₂⁴	C₃²×Q₈	1	16	yes	direct product factor
40	C₂⁴	C₂³×C₃²	1	15	yes	—
41	C₂×Q₈	C₂³×C₃²	1	960	yes	direct product factor
42	C₂²×C₄	C₂³×C₃²	1	420	yes	—
43	C₂²×C₃	C₂²×C₃×Q₈	1	560	yes	direct product factor
44	C₂²×C₃	C₂⁵×C₃	1	60	yes	—
45	C₃×C₄	C₂⁵×C₃	1	192	yes	—
46	C₃²	C₂⁴×Q₈	1	1	yes	characteristic, direct product factor, Sylow
47	C₂³	C₂×C₃²×Q₈	1	120	yes	direct product factor
48	C₂³	C₂⁴×C₃²	1	35	yes	—
49	Q₈	C₂⁴×C₃²	1	256	yes	direct product factor
50	C₂×C₄	C₂⁴×C₃²	1	360	yes	—
51	C₂×C₃	C₂³×C₃×Q₈	1	120	yes	direct product factor
52	C₂×C₃	C₂⁶×C₃	1	4	yes	—
53	C₂²	C₂²×C₃²×Q₈	1	140	yes	direct product factor
54	C₂²	C₂⁵×C₃²	1	15	yes	—
55	C₄	C₂⁵×C₃²	1	48	yes	—
56	C₃	C₂⁴×C₃×Q₈	1	4	yes	direct product factor
57	C₂	C₂³×C₃²×Q₈	1	30	yes	direct product factor
58	C₂	C₂⁶×C₃²	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵×C₃²	C₂²
commutator, Frattini, cosocle	C₂	C₂⁶×C₃²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₃²×Q₈	C₂	direct (N × H)
3. C₂⁴×C₃×Q₈	C₃	direct (N × H)
5. C₂²×C₃²×Q₈	C₂²	direct (N × H)
7. C₂³×C₃×Q₈	C₂×C₃	direct (N × H)
9. C₂⁴×C₃²	Q₈	direct (N × H)
11. C₂×C₃²×Q₈	C₂³	direct (N × H)
13. C₂⁴×Q₈	C₃²	direct (N × H)
15. C₂²×C₃×Q₈	C₂²×C₃	direct (N × H)
17. C₂³×C₃²	C₂×Q₈	direct (N × H)
19. C₃²×Q₈	C₂⁴	direct (N × H)
21. C₂³×Q₈	C₂×C₃²	direct (N × H)
23. C₂⁴×C₃	C₃×Q₈	direct (N × H)
25. C₂×C₃×Q₈	C₂³×C₃	direct (N × H)
27. C₂²×C₃²	C₂²×Q₈	direct (N × H)
31. C₂²×Q₈	C₂²×C₃²	direct (N × H)
33. C₂³×C₃	C₂×C₃×Q₈	direct (N × H)
35. C₃×Q₈	C₂⁴×C₃	direct (N × H)
37. C₂×C₃²	C₂³×Q₈	direct (N × H)
39. C₂⁴	C₃²×Q₈	direct (N × H)
41. C₂×Q₈	C₂³×C₃²	direct (N × H)
43. C₂²×C₃	C₂²×C₃×Q₈	direct (N × H)
46. C₃²	C₂⁴×Q₈	direct (N × H)
47. C₂³	C₂×C₃²×Q₈	direct (N × H)
49. Q₈	C₂⁴×C₃²	direct (N × H)
51. C₂×C₃	C₂³×C₃×Q₈	direct (N × H)
53. C₂²	C₂²×C₃²×Q₈	direct (N × H)
56. C₃	C₂⁴×C₃×Q₈	direct (N × H)
57. C₂	C₂³×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
C₂³×C₃²×Q₈	2
C₂⁴×C₃²×C₄	2
C₂⁴×C₃×Q₈	3

Subgroups — C24×C32×Q8 #11770638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₃²×Q₈ #11770638