Subgroups — C₃×C₅×Q₈² #11756647

Order 960.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₅×C₄⋊Q₈	C₂	1	9	yes	maximal
2	C₃×C₄×C₅×Q₈	C₂	1	6	yes	maximal
3	C₅×Q₈²	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
4	C₂×C₃×C₅×Q₈	C₂²	1	6	yes	—
5	C₂×C₃×C₅×Q₈	C₂²	1	2	yes	—
6	C₃×C₅×C₄⋊C₄	C₂²	1	18	yes	—
7	C₃×C₄²×C₅	C₂²	1	9	yes	—
8	C₃×Q₈²	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₅×C₄⋊Q₈	C₂×C₃	1	9	yes	—
10	C₄×C₅×Q₈	C₂×C₃	1	6	yes	—
11	C₃×C₅×Q₈	—	4	6	no	—
12	C₃×C₅×Q₈	Q₈	1	8	yes	direct product factor
13	C₂×C₃×C₄×C₅	C₂³	1	9	yes	—
14	C₂×C₃×C₄×C₅	C₂³	1	6	yes	—
15	C₃×C₄⋊Q₈	C₂×C₅	1	9	yes	—
16	C₃×C₄×Q₈	C₂×C₅	1	6	yes	—
17	C₂×C₅×Q₈	C₂²×C₃	1	6	yes	—
18	C₂×C₅×Q₈	C₂²×C₃	1	2	yes	—
19	C₅×C₄⋊C₄	C₂²×C₃	1	18	yes	—
20	C₄²×C₅	C₂²×C₃	1	9	yes	—
21	Q₈²	C₃×C₅	1	1	yes	characteristic, direct product factor, Sylow
22	C₂²×C₃×C₅	C₂⁴	1	1	yes	characteristic
23	C₃×C₄×C₅	—	2	9	no	—
24	C₃×C₄×C₅	C₂×Q₈	1	12	yes	—
25	C₂×C₃×Q₈	C₂²×C₅	1	6	yes	—
26	C₂×C₃×Q₈	C₂²×C₅	1	2	yes	—
27	C₃×C₄⋊C₄	C₂²×C₅	1	18	yes	—
28	C₃×C₄²	C₂²×C₅	1	9	yes	—
29	C₅×Q₈	—	4	6	no	—
30	C₅×Q₈	C₃×Q₈	1	8	yes	direct product factor
31	C₂×C₄×C₅	C₂³×C₃	1	9	yes	—
32	C₂×C₄×C₅	C₂³×C₃	1	6	yes	—
33	C₄⋊Q₈	C₂×C₃×C₅	1	9	yes	—
34	C₄×Q₈	C₂×C₃×C₅	1	6	yes	—
35	C₂×C₃×C₅	C₂²×Q₈	1	2	yes	—
36	C₂×C₃×C₅	H(5,𝔽₂)	1	1	yes	characteristic
37	C₃×Q₈	—	4	6	no	—
38	C₃×Q₈	C₅×Q₈	1	8	yes	direct product factor
39	C₂×C₃×C₄	C₂³×C₅	1	9	yes	—
40	C₂×C₃×C₄	C₂³×C₅	1	6	yes	—
41	C₂²×C₅	C₂⁴×C₃	1	1	yes	characteristic
42	C₄×C₅	—	2	9	no	—
43	C₄×C₅	C₂×C₃×Q₈	1	12	yes	—
44	C₂×Q₈	C₂²×C₃×C₅	1	6	yes	—
45	C₂×Q₈	C₂²×C₃×C₅	1	2	yes	—
46	C₄⋊C₄	C₂²×C₃×C₅	1	18	yes	—
47	C₄²	C₂²×C₃×C₅	1	9	yes	—
48	C₃×C₅	Q₈²	1	1	yes	characteristic, direct product factor, Hall
49	C₂²×C₃	C₂⁴×C₅	1	1	yes	characteristic
50	C₃×C₄	—	2	9	no	—
51	C₃×C₄	C₂×C₅×Q₈	1	12	yes	—
52	C₂×C₅	C₂²×C₃×Q₈	1	2	yes	—
53	C₂×C₅	C₃×H(5,𝔽₂)	1	1	yes	characteristic
54	Q₈	—	4	6	no	—
55	Q₈	C₃×C₅×Q₈	1	8	yes	direct product factor
56	C₂×C₄	C₂³×C₃×C₅	1	9	yes	—
57	C₂×C₄	C₂³×C₃×C₅	1	6	yes	—
58	C₂×C₃	C₂²×C₅×Q₈	1	2	yes	—
59	C₂×C₃	C₅×H(5,𝔽₂)	1	1	yes	characteristic
60	C₅	C₃×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
61	C₂²	C₂⁴×C₃×C₅	1	1	yes	characteristic
62	C₄	—	2	9	no	—
63	C₄	C₂×C₃×C₅×Q₈	1	12	yes	—
64	C₃	C₅×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
65	C₂	C₂²×C₃×C₅×Q₈	1	2	yes	—
66	C₂	C₃×C₅×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×Q₈²	C₃	direct (N × H)
8. C₃×Q₈²	C₅	direct (N × H)
12. C₃×C₅×Q₈	Q₈	direct (N × H)
21. Q₈²	C₃×C₅	direct (N × H)
30. C₅×Q₈	C₃×Q₈	direct (N × H)
38. C₃×Q₈	C₅×Q₈	direct (N × H)
48. C₃×C₅	Q₈²	direct (N × H)
55. Q₈	C₃×C₅×Q₈	direct (N × H)
60. C₅	C₃×Q₈²	direct (N × H)
64. C₃	C₅×Q₈²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₅×C₄⋊Q₈	2
C₃×C₄×C₅×Q₈	2
C₅×Q₈²	3
C₃×Q₈²	5

Subgroups — C3×C5×Q82 #11756647

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₅×Q₈² #11756647