Subgroups — Q₈×C₃⋊(C₇⋊C₈) #13056660

Order 1344.

204 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₇⋊C₈))	C₂	1	3	yes	maximal
3	C₄×C₃⋊(C₇⋊C₈)	C₂	1	3	yes	maximal
4	Q₈×C₇⋊C₈	—	3	1	no	maximal, Hall
5	C₂×C₃×C₇×Q₈	C₄	1	1	yes	characteristic
6	C₃×C₇×C₄⋊C₄	C₄	1	3	yes	—
7	C₃×C₄²×C₇	C₂²	1	3	yes	—
8	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	3	yes	—
9	C₂×C₃⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₄×C₇×Q₈	D₃	1	1	yes	characteristic
11	C₄⋊(C₇⋊C₈)	—	3	3	no	—
12	C₄×C₇⋊C₈	—	3	3	no	—
13	Q₈×C₃⋊C₈	—	7	1	no	maximal, Hall
14	C₃×C₇×Q₈	C₈	1	4	yes	complemented
15	C₂×C₃×C₄×C₇	C₂×C₄	1	3	yes	—
16	C₂×C₃×C₄×C₇	C₂×C₄	1	3	yes	—
17	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₃⋊(C₇⋊C₈)	—	2	3	no	—
19	C₃⋊(C₇⋊C₈)	Q₈	1	2	yes	direct product factor
20	C₂×C₇×Q₈	Dic₃	1	1	yes	characteristic
21	C₇×C₄⋊C₄	Dic₃	1	3	yes	—
22	C₄²×C₇	C₂×D₃	1	3	yes	—
23	C₂×C₇⋊C₈	—	3	1	no	—
24	C₂×C₇⋊C₈	—	3	3	no	—
25	C₃×C₄×Q₈	D₇	1	1	yes	characteristic
26	C₄⋊(C₃⋊C₈)	—	7	3	no	—
27	C₄×C₃⋊C₈	—	7	3	no	—
28	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
29	C₃×C₄×C₇	—	2	3	no	—
30	C₃×C₄×C₇	C₂×C₈	1	6	yes	—
31	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
32	C₃×C₄×C₇	#41	1	1	yes	characteristic
33	C₈×Q₈	—	21	1	no	Sylow
34	C₇×Q₈	C₃⋊C₈	1	4	yes	complemented
35	C₂×C₄×C₇	C₂×Dic₃	1	3	yes	—
36	C₂×C₄×C₇	C₂×Dic₃	1	3	yes	—
37	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
38	C₇⋊C₈	—	3	2	no	—
39	C₇⋊C₈	—	6	3	no	—
40	C₂×C₃×Q₈	Dic₇	1	1	yes	characteristic
41	C₃×C₄⋊C₄	Dic₇	1	3	yes	—
42	C₃×C₄²	C₂×D₇	1	3	yes	—
43	C₂×C₃⋊C₈	—	7	1	no	—
44	C₂×C₃⋊C₈	—	7	3	no	—
45	C₂×C₃×C₇	C₄×Q₈	1	1	yes	characteristic
46	C₂×C₃×C₇	C₂²×C₈	1	1	yes	characteristic
47	C₂×C₃×C₇	#131	1	1	yes	characteristic
48	C₄×Q₈	D₂₁	1	1	yes	characteristic
49	C₄⋊C₈	—	21	3	no	—
50	C₄×C₈	—	21	3	no	—
51	C₂²×C₇	C₂²×Dic₃	1	1	yes	characteristic
52	C₄×C₇	—	2	3	no	—
53	C₄×C₇	C₂×C₃⋊C₈	1	6	yes	—
54	C₄×C₇	D₃×Q₈	1	1	yes	characteristic
55	C₄×C₇	#239	1	1	yes	characteristic
56	C₃×Q₈	C₇⋊C₈	1	4	yes	complemented
57	C₂×C₃×C₄	C₂×Dic₇	1	3	yes	—
58	C₂×C₃×C₄	C₂×Dic₇	1	3	yes	—
59	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
60	C₃⋊C₈	—	7	2	no	—
61	C₃⋊C₈	—	14	3	no	—
62	C₃×C₇	C₈×Q₈	1	1	yes	characteristic, complemented, Hall
63	C₂×Q₈	Dic₂₁	1	1	yes	characteristic
64	C₂×C₈	—	21	1	no	—
65	C₂×C₈	—	21	3	no	—
66	C₄⋊C₄	Dic₂₁	1	3	yes	—
67	C₄²	C₂×D₂₁	1	3	yes	—
68	C₂×C₇	C₂²×C₃⋊C₈	1	1	yes	characteristic
69	C₂×C₇	Q₈×Dic₃	1	1	yes	characteristic
70	C₂×C₇	#948	1	1	yes	characteristic
71	C₂²×C₃	C₂²×Dic₇	1	1	yes	characteristic
72	C₃×C₄	—	2	3	no	—
73	C₃×C₄	C₂×C₇⋊C₈	1	6	yes	—
74	C₃×C₄	Q₈×D₇	1	1	yes	characteristic
75	C₃×C₄	#1161	1	1	yes	characteristic
76	Q₈	C₃⋊(C₇⋊C₈)	1	4	yes	direct product factor
77	C₂×C₄	C₂×Dic₂₁	1	3	yes	—
78	C₂×C₄	C₂×Dic₂₁	1	3	yes	—
79	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
80	C₈	—	21	2	no	—
81	C₈	—	42	3	no	—
82	C₇	Q₈×C₃⋊C₈	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	C₂²×C₇⋊C₈	1	1	yes	characteristic
84	C₂×C₃	Q₈×Dic₇	1	1	yes	characteristic
85	C₂×C₃	#6511	1	1	yes	characteristic
86	C₂²	C₂²×Dic₂₁	1	1	yes	characteristic
87	C₄	—	2	3	no	—
88	C₄	C₂×C₃⋊(C₇⋊C₈)	1	6	yes	—
89	C₄	Q₈×D₂₁	1	1	yes	characteristic
90	C₄	#66732	1	1	yes	characteristic
91	C₃	Q₈×C₇⋊C₈	1	1	yes	characteristic, complemented, Sylow
92	C₂	C₂²×C₃⋊(C₇⋊C₈)	1	1	yes	characteristic
93	C₂	Q₈×Dic₂₁	1	1	yes	characteristic
94	C₂	#10621431	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₂₁
commutator	C₂×C₃×C₇	C₂²×C₈
Fitting	C₃×C₄×C₇×Q₈	C₂
socle, prime radical	C₂²×C₃×C₇	C₂²×C₄
cosocle	C₂×C₃×C₄×C₇	C₂³
hypercenter	C₄×Q₈	D₂₁
hypocenter	C₃×C₇	C₈×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
14. C₃×C₇×Q₈	C₈	semidirect
19. C₃⋊(C₇⋊C₈)	Q₈	direct (N × H)
34. C₇×Q₈	C₃⋊C₈	semidirect
56. C₃×Q₈	C₇⋊C₈	semidirect
62. C₃×C₇	C₈×Q₈	semidirect
76. Q₈	C₃⋊(C₇⋊C₈)	direct (N × H)
82. C₇	Q₈×C₃⋊C₈	semidirect
91. C₃	Q₈×C₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₇×Q₈	2
C₄⋊(C₃⋊(C₇⋊C₈))	2
C₄×C₃⋊(C₇⋊C₈)	2
Q₈×C₇⋊C₈	3
Q₈×C₃⋊C₈	7

Subgroups — Q8×C3⋊(C7⋊C8) #13056660

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃⋊(C₇⋊C₈) #13056660