Subgroups — Q₈⋊#6855 #14515883

Order 1944.

257 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×Q₈×C₉	C₃	1	1	yes	maximal, characteristic
2	C₃²×Q₈⋊C₉	C₃	1	3	yes	maximal
3	C₂×#6855	—	4	1	no	maximal
4	C₃²×C₄×C₉	—	3	1	no	—
5	#6855	—	4	1	no	Sylow
6	C₃³×Q₈	C₃²	1	1	yes	characteristic
7	C₃×Q₈×C₉	—	3	1	no	—
8	C₃×Q₈×C₉	—	3	2	no	—
9	C₃×Q₈×C₉	C₉	1	3	yes	complemented
10	C₃×Q₈⋊C₉	—	3	3	no	—
11	C₃×Q₈⋊C₉	—	3	6	no	—
12	C₃×Q₈⋊C₉	C₉	1	9	yes	complemented
13	C₂×C₃²×C₉	—	4	3	no	—
14	C₂×C₃²×C₉	A₄	1	1	yes	characteristic
15	C₃³×C₄	—	3	1	no	—
16	C₃×C₄×C₉	—	3	6	no	—
17	C₃×C₄×C₉	—	3	3	no	—
18	C₃×C₄×C₉	—	3	3	no	—
19	C₃²×C₉	—	4	3	no	—
20	C₃²×C₉	SL(2,𝔽₃)	1	1	yes	characteristic
21	C₃²×Q₈	C₃×C₉	1	3	yes	—
22	C₃²×Q₈	C₃×C₉	1	1	yes	characteristic
23	C₃²×Q₈	H(3,𝔽₃)	1	1	yes	characteristic
24	C₃²×Q₈	3₋¹⁺²	1	6	yes	—
25	C₃²×Q₈	3₋¹⁺²	1	2	yes	—
26	Q₈×C₉	—	3	3	no	—
27	Q₈⋊C₉	—	3	9	no	—
28	C₂×C₃³	C₃×A₄	1	1	yes	characteristic
29	C₂×C₃×C₉	—	3	2	no	—
30	C₂×C₃×C₉	—	3	1	no	—
31	C₂×C₃×C₉	—	4	9	no	—
32	C₂×C₃×C₉	—	12	6	no	—
33	C₂×C₃×C₉	—	12	3	no	—
34	C₂×C₃×C₉	C₂²⋊C₉	1	3	yes	—
35	C₃²×C₄	—	3	3	no	—
36	C₃²×C₄	—	3	6	no	—
37	C₃²×C₄	—	3	1	no	—
38	C₃²×C₄	—	3	2	no	—
39	C₃²×C₄	—	3	1	no	—
40	C₄×C₉	—	3	9	no	—
41	C₃³	C₃×SL(2,𝔽₃)	1	1	yes	characteristic
42	C₃×C₉	—	3	2	no	—
43	C₃×C₉	—	3	1	no	—
44	C₃×C₉	—	4	9	no	—
45	C₃×C₉	—	12	6	no	—
46	C₃×C₉	—	12	3	no	—
47	C₃×C₉	Q₈⋊C₉	1	3	yes	complemented
48	C₃×Q₈	C₉²	1	1	yes	characteristic
49	C₃×Q₈	C₃²⋊C₉	1	3	yes	—
50	C₃×Q₈	C₃²⋊C₉	1	1	yes	characteristic
51	C₃×Q₈	C₉⋊C₉	1	6	yes	—
52	C₃×Q₈	C₉⋊C₉	1	2	yes	—
53	C₂×C₃²	C₉×A₄	1	3	yes	—
54	C₂×C₃²	C₉⋊A₄	1	6	yes	—
55	C₂×C₃²	C₃×C₂²⋊C₉	1	1	yes	characteristic
56	C₂×C₃²	#1094	1	2	yes	—
57	C₂×C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
58	C₂×C₉	—	3	3	no	—
59	C₂×C₉	—	12	9	no	—
60	C₃×C₄	—	3	1	no	—
61	C₃×C₄	—	3	6	no	—
62	C₃×C₄	—	3	2	no	—
63	C₃×C₄	—	3	3	no	—
64	C₃×C₄	—	3	1	no	—
65	C₃²	C₉×SL(2,𝔽₃)	1	3	yes	—
66	C₃²	C₉⋊SL(2,𝔽₃)	1	6	yes	—
67	C₃²	C₃×Q₈⋊C₉	1	1	yes	characteristic
68	C₃²	#6204	1	2	yes	—
69	C₃²	Q₈⋊H(3,𝔽₃)	1	1	yes	characteristic
70	C₉	—	3	3	no	—
71	C₉	—	12	9	no	—
72	Q₈	#6855	1	1	yes	characteristic, complemented, Sylow
73	C₂×C₃	C₉×C₂²⋊C₉	1	1	yes	characteristic
74	C₂×C₃	(C₂²⋊C₉)⋊C₉	1	6	yes	—
75	C₂×C₃	C₉⋊(C₂²⋊C₉)	1	2	yes	—
76	C₂×C₃	(C₃×A₄)⋊C₉	1	3	yes	—
77	C₂×C₃	C₃²⋊(C₂²⋊C₉)	1	1	yes	characteristic
78	C₄	—	3	1	no	—
79	C₃	C₉×Q₈⋊C₉	1	1	yes	characteristic
80	C₃	(Q₈⋊C₉)⋊C₉	1	6	yes	—
81	C₃	C₉⋊(Q₈⋊C₉)	1	2	yes	—
82	C₃	(C₃×SL(2,𝔽₃))⋊C₉	1	3	yes	—
83	C₃	#10619584	1	1	yes	characteristic
84	C₂	C₂²⋊#6855	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, socle, prime radical	C₂×C₃³	C₃×A₄
commutator	C₃×Q₈	C₉²
Fitting	C₃²×Q₈×C₉	C₃
cosocle	C₃³×Q₈	C₃²
hypercenter	C₂×C₃²×C₉	A₄
hypocenter	Q₈	#6855
LCS	G ▷ C₃×Q₈ ▷ Q₈
UCS	C₁ ◁ C₂×C₃³ ◁ C₂×C₃²×C₉

Complements

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

N	H ≅ G/N	type
9. C₃×Q₈×C₉	C₉	semidirect
12. C₃×Q₈⋊C₉	C₉	semidirect
47. C₃×C₉	Q₈⋊C₉	semidirect
72. Q₈	#6855	semidirect

Maximal subgroups

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

subgroup	index
C₃²×Q₈×C₉	3
C₃²×Q₈⋊C₉	3
C₂×#6855	4

Subgroups — Q8⋊#6855 #14515883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈⋊#6855 #14515883