Subgroups — C₃²×C₅×C₄⋊Q₈ #13088237

Order 1440.

408 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃²×C₅×Q₈	C₂	1	2	yes	maximal
2	C₃²×C₅×C₄⋊C₄	C₂	1	4	yes	maximal
3	C₃²×C₄²×C₅	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×C₄⋊Q₈	C₃	1	4	yes	maximal, direct product factor
5	C₃²×C₅×Q₈	—	2	4	no	—
6	C₂×C₃²×C₄×C₅	C₂²	1	4	yes	—
7	C₂×C₃²×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₄×C₅	C₂²	1	2	yes	—
9	C₃²×C₄⋊Q₈	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
10	C₂×C₃×C₅×Q₈	C₂×C₃	1	8	yes	—
11	C₃×C₅×C₄⋊C₄	C₂×C₃	1	16	yes	—
12	C₃×C₄²×C₅	C₂×C₃	1	4	yes	—
13	C₂²×C₃²×C₅	C₂³	1	1	yes	characteristic
14	C₃²×C₄×C₅	—	2	4	no	—
15	C₃²×C₄×C₅	D₄	1	2	yes	—
16	C₃²×C₄×C₅	Q₈	1	4	yes	complemented
17	C₅×C₄⋊Q₈	C₃²	1	1	yes	characteristic, direct product factor, Hall
18	C₂×C₃²×Q₈	C₂×C₅	1	2	yes	—
19	C₃²×C₄⋊C₄	C₂×C₅	1	4	yes	—
20	C₃²×C₄²	C₂×C₅	1	1	yes	characteristic
21	C₃×C₅×Q₈	—	2	16	no	—
22	C₂×C₃×C₄×C₅	C₂²×C₃	1	16	yes	—
23	C₂×C₃×C₄×C₅	C₂²×C₃	1	4	yes	—
24	C₂×C₃×C₄×C₅	C₂²×C₃	1	8	yes	—
25	C₃×C₄⋊Q₈	C₃×C₅	1	4	yes	direct product factor
26	C₂×C₃²×C₅	C₂×D₄	1	1	yes	characteristic
27	C₂×C₃²×C₅	C₂×Q₈	1	2	yes	—
28	C₂×C₅×Q₈	C₂×C₃²	1	2	yes	—
29	C₅×C₄⋊C₄	C₂×C₃²	1	4	yes	—
30	C₄²×C₅	C₂×C₃²	1	1	yes	characteristic
31	C₃²×Q₈	—	2	4	no	—
32	C₂×C₃²×C₄	C₂²×C₅	1	4	yes	—
33	C₂×C₃²×C₄	C₂²×C₅	1	1	yes	characteristic
34	C₂×C₃²×C₄	C₂²×C₅	1	2	yes	—
35	C₂²×C₃×C₅	C₂³×C₃	1	4	yes	—
36	C₃×C₄×C₅	—	2	16	no	—
37	C₃×C₄×C₅	C₃×D₄	1	8	yes	—
38	C₃×C₄×C₅	C₃×Q₈	1	16	yes	complemented
39	C₂×C₃×Q₈	C₂×C₃×C₅	1	8	yes	—
40	C₃×C₄⋊C₄	C₂×C₃×C₅	1	16	yes	—
41	C₃×C₄²	C₂×C₃×C₅	1	4	yes	—
42	C₃²×C₅	C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Hall
43	C₅×Q₈	—	2	4	no	—
44	C₂×C₄×C₅	C₂²×C₃²	1	4	yes	—
45	C₂×C₄×C₅	C₂²×C₃²	1	1	yes	characteristic
46	C₂×C₄×C₅	C₂²×C₃²	1	2	yes	—
47	C₂²×C₃²	C₂³×C₅	1	1	yes	characteristic
48	C₃²×C₄	—	2	4	no	—
49	C₃²×C₄	C₅×D₄	1	2	yes	—
50	C₃²×C₄	C₅×Q₈	1	4	yes	complemented
51	C₄⋊Q₈	C₃²×C₅	1	1	yes	characteristic, direct product factor, Sylow
52	C₂×C₃×C₅	C₂×C₃×D₄	1	4	yes	—
53	C₂×C₃×C₅	C₂×C₃×Q₈	1	8	yes	—
54	C₃×Q₈	—	2	16	no	—
55	C₂×C₃×C₄	C₂²×C₃×C₅	1	16	yes	—
56	C₂×C₃×C₄	C₂²×C₃×C₅	1	4	yes	—
57	C₂×C₃×C₄	C₂²×C₃×C₅	1	8	yes	—
58	C₂²×C₅	C₂³×C₃²	1	1	yes	characteristic
59	C₄×C₅	—	2	4	no	—
60	C₄×C₅	C₃²×D₄	1	2	yes	—
61	C₄×C₅	C₃²×Q₈	1	4	yes	complemented
62	C₂×C₃²	C₂×C₅×D₄	1	1	yes	characteristic
63	C₂×C₃²	C₂×C₅×Q₈	1	2	yes	—
64	C₂×Q₈	C₂×C₃²×C₅	1	2	yes	—
65	C₄⋊C₄	C₂×C₃²×C₅	1	4	yes	—
66	C₄²	C₂×C₃²×C₅	1	1	yes	characteristic
67	C₃×C₅	C₃×C₄⋊Q₈	1	4	yes	direct product factor
68	C₂²×C₃	C₂³×C₃×C₅	1	4	yes	—
69	C₃×C₄	—	2	16	no	—
70	C₃×C₄	C₃×C₅×D₄	1	8	yes	—
71	C₃×C₄	C₃×C₅×Q₈	1	16	yes	complemented
72	C₂×C₅	C₂×C₃²×D₄	1	1	yes	characteristic
73	C₂×C₅	C₂×C₃²×Q₈	1	2	yes	—
74	C₃²	C₅×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
75	Q₈	—	2	4	no	—
76	C₂×C₄	C₂²×C₃²×C₅	1	4	yes	—
77	C₂×C₄	C₂²×C₃²×C₅	1	1	yes	characteristic
78	C₂×C₄	C₂²×C₃²×C₅	1	2	yes	—
79	C₂×C₃	C₂×C₃×C₅×D₄	1	4	yes	—
80	C₂×C₃	C₂×C₃×C₅×Q₈	1	8	yes	—
81	C₅	C₃²×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
82	C₂²	C₂³×C₃²×C₅	1	1	yes	characteristic
83	C₄	—	2	4	no	—
84	C₄	C₃²×C₅×D₄	1	2	yes	—
85	C₄	C₃²×C₅×Q₈	1	4	yes	complemented
86	C₃	C₃×C₅×C₄⋊Q₈	1	4	yes	direct product factor
87	C₂	C₂×C₃²×C₅×D₄	1	1	yes	characteristic
88	C₂	C₂×C₃²×C₅×Q₈	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²×C₅	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃²×C₅
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₅×C₄⋊Q₈	C₃	direct (N × H)
9. C₃²×C₄⋊Q₈	C₅	direct (N × H)
16. C₃²×C₄×C₅	Q₈	semidirect
17. C₅×C₄⋊Q₈	C₃²	direct (N × H)
25. C₃×C₄⋊Q₈	C₃×C₅	direct (N × H)
38. C₃×C₄×C₅	C₃×Q₈	semidirect
42. C₃²×C₅	C₄⋊Q₈	direct (N × H)
50. C₃²×C₄	C₅×Q₈	semidirect
51. C₄⋊Q₈	C₃²×C₅	direct (N × H)
61. C₄×C₅	C₃²×Q₈	semidirect
67. C₃×C₅	C₃×C₄⋊Q₈	direct (N × H)
71. C₃×C₄	C₃×C₅×Q₈	semidirect
74. C₃²	C₅×C₄⋊Q₈	direct (N × H)
81. C₅	C₃²×C₄⋊Q₈	direct (N × H)
85. C₄	C₃²×C₅×Q₈	semidirect
86. C₃	C₃×C₅×C₄⋊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×C₅×Q₈	2
C₃²×C₅×C₄⋊C₄	2
C₃²×C₄²×C₅	2
C₃×C₅×C₄⋊Q₈	3
C₃²×C₄⋊Q₈	5

Subgroups — C32×C5×C4⋊Q8 #13088237

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×C₅×C₄⋊Q₈ #13088237