Subgroups — D₁₅⋊Q₈ #6770

Order 240.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₅×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Dic₁₀	C₂	1	1	yes	maximal, characteristic, complemented
4	#1204	C₂	1	2	yes	maximal, complemented
5	#1200	C₂	1	2	yes	maximal
6	Q₈×D₅	—	3	1	no	maximal, Hall
7	C₂×D₁₅	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅	C₂²	1	1	yes	characteristic
9	Dic₁₅	C₂²	1	1	yes	characteristic
10	C₃×Dic₅	C₂²	1	2	yes	—
11	C₅×Dic₃	C₂²	1	2	yes	—
12	D₃×Q₈	—	5	1	no	maximal, Hall
13	C₅×Q₈	—	3	1	no	—
14	C₄×D₅	—	3	1	no	—
15	C₄×D₅	—	3	2	no	—
16	Dic₁₀	—	3	2	no	—
17	Dic₁₀	D₃	1	1	yes	characteristic, complemented
18	C₂×C₃×C₅	C₂³	1	1	yes	characteristic
19	D₁₅	Q₈	1	2	yes	complemented
20	C₃×Q₈	—	5	1	no	—
21	C₄×D₃	—	5	1	no	—
22	C₄×D₃	—	5	2	no	—
23	Dic₆	—	5	2	no	—
24	Dic₆	D₅	1	1	yes	characteristic, complemented
25	C₂×D₅	—	3	1	no	—
26	C₄×C₅	—	3	2	no	—
27	C₄×C₅	C₂×D₃	1	1	yes	characteristic
28	Dic₅	—	3	1	no	—
29	Dic₅	C₂×D₃	1	2	yes	—
30	C₂×Q₈	—	15	1	no	Sylow
31	C₃×C₅	C₂×Q₈	1	1	yes	characteristic, complemented, Hall
32	C₂×D₃	—	5	1	no	—
33	C₃×C₄	—	5	2	no	—
34	C₃×C₄	C₂×D₅	1	1	yes	characteristic
35	Dic₃	—	5	1	no	—
36	Dic₃	C₂×D₅	1	2	yes	—
37	C₂×C₅	C₂²×D₃	1	1	yes	characteristic
38	D₅	—	3	2	no	—
39	Q₈	—	3	1	no	—
40	Q₈	—	5	1	no	—
41	Q₈	—	15	2	no	—
42	C₂×C₄	—	15	2	no	—
43	C₂×C₄	—	15	1	no	—
44	C₂×C₃	C₂²×D₅	1	1	yes	characteristic
45	D₃	—	5	2	no	—
46	C₅	D₃×Q₈	1	1	yes	characteristic, complemented, Sylow
47	C₂²	—	15	1	no	—
48	C₄	—	3	2	no	—
49	C₄	—	5	2	no	—
50	C₄	—	15	1	no	—
51	C₄	D₃×D₅	1	1	yes	characteristic
52	C₃	Q₈×D₅	1	1	yes	characteristic, complemented, Sylow
53	C₂	—	15	2	no	—
54	C₂	C₂×D₃×D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×D₃×D₅
commutator, socle, cosocle	C₂×C₃×C₅	C₂³
Fitting	C₃×C₄×C₅	C₂²
prime radical	C₂×D₁₅	C₂²
hypercenter	C₄	D₃×D₅
hypocenter	C₃×C₅	C₂×Q₈
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₅×Dic₆	C₂	semidirect
3. C₃×Dic₁₀	C₂	semidirect
4. #1204	C₂	semidirect
17. Dic₁₀	D₃	semidirect
19. D₁₅	Q₈	semidirect
24. Dic₆	D₅	semidirect
31. C₃×C₅	C₂×Q₈	semidirect
46. C₅	D₃×Q₈	semidirect
52. C₃	Q₈×D₅	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₅	2
C₅×Dic₆	2
C₃×Dic₁₀	2
#1204	2
#1200	2
Q₈×D₅	3
D₃×Q₈	5

Subgroups — D15⋊Q8 #6770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₅⋊Q₈ #6770