Subgroups — Q₈×#6887 #14517832

Order 1944.

414 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6887	C₂	1	3	yes	maximal
2	C₃²×Q₈×C₉	C₃	1	1	yes	maximal, characteristic
3	Q₈×C₉⋊C₉	C₃	1	6	yes	maximal, complemented
4	Q₈×C₃²⋊C₉	C₃	1	3	yes	maximal
5	Q₈×C₉²	C₃	1	3	yes	maximal, complemented
6	C₂×#6887	C₂²	1	1	yes	characteristic
7	C₃²×C₄×C₉	C₂×C₃	1	3	yes	—
8	C₄×C₉⋊C₉	C₂×C₃	1	18	yes	—
9	C₄×C₃²⋊C₉	C₂×C₃	1	9	yes	—
10	C₄×C₉²	C₂×C₃	1	9	yes	—
11	#6887	Q₈	1	1	yes	characteristic, direct product factor, Sylow
12	C₃³×Q₈	C₃²	1	1	yes	characteristic
13	C₃×Q₈×C₉	C₉	1	9	yes	complemented
14	C₃×Q₈×C₉	C₃²	1	9	yes	—
15	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
16	C₃×Q₈×C₉	C₃²	1	2	yes	—
17	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
18	C₂×C₉⋊C₉	C₂²×C₃	1	6	yes	—
19	C₂×C₃²⋊C₉	C₂²×C₃	1	3	yes	—
20	C₂×C₉²	C₂²×C₃	1	3	yes	—
21	C₃³×C₄	C₂×C₃²	1	3	yes	—
22	C₃×C₄×C₉	C₂×C₉	1	27	yes	—
23	C₃×C₄×C₉	C₂×C₃²	1	27	yes	—
24	C₃×C₄×C₉	C₂×C₃²	1	6	yes	—
25	C₃×C₄×C₉	C₂×C₃²	1	3	yes	—
26	C₃²×C₉	C₃×Q₈	1	1	yes	characteristic
27	C₉⋊C₉	C₃×Q₈	1	6	yes	complemented
28	C₃²⋊C₉	C₃×Q₈	1	3	yes	—
29	C₉²	C₃×Q₈	1	3	yes	complemented
30	C₃²×Q₈	—	3	3	no	—
31	C₃²×Q₈	C₃×C₉	1	3	yes	—
32	C₃²×Q₈	C₃³	1	1	yes	characteristic
33	Q₈×C₉	—	3	9	no	—
34	Q₈×C₉	C₃×C₉	1	3	yes	—
35	Q₈×C₉	C₃×C₉	1	6	yes	—
36	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
37	C₂×C₃×C₉	C₂²×C₉	1	9	yes	—
38	C₂×C₃×C₉	C₂²×C₃²	1	9	yes	—
39	C₂×C₃×C₉	C₂²×C₃²	1	2	yes	—
40	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
41	C₃²×C₄	—	3	9	no	—
42	C₃²×C₄	C₂×C₃×C₉	1	9	yes	—
43	C₃²×C₄	C₂×C₃³	1	3	yes	—
44	C₄×C₉	—	3	27	no	—
45	C₄×C₉	C₂×C₃×C₉	1	18	yes	—
46	C₄×C₉	C₂×C₃×C₉	1	9	yes	—
47	C₃³	C₃²×Q₈	1	1	yes	characteristic
48	C₃×C₉	Q₈×C₉	1	9	yes	complemented
49	C₃×C₉	C₃²×Q₈	1	9	yes	—
50	C₃×C₉	C₃²×Q₈	1	2	yes	—
51	C₃×C₉	C₃²×Q₈	1	1	yes	characteristic
52	C₃×Q₈	—	3	3	no	—
53	C₃×Q₈	C₃²×C₉	1	1	yes	characteristic
54	C₃×Q₈	#737	1	3	yes	—
55	C₂×C₃²	—	3	3	no	—
56	C₂×C₃²	C₂²×C₃×C₉	1	3	yes	—
57	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
58	C₂×C₉	—	3	9	no	—
59	C₂×C₉	C₂²×C₃×C₉	1	6	yes	—
60	C₂×C₉	C₂²×C₃×C₉	1	3	yes	—
61	C₃×C₄	—	3	9	no	—
62	C₃×C₄	C₂×C₃²×C₉	1	3	yes	—
63	C₃×C₄	C₂×#737	1	9	yes	—
64	C₃²	—	3	3	no	—
65	C₃²	C₃×Q₈×C₉	1	3	yes	—
66	C₃²	C₃³×Q₈	1	1	yes	characteristic
67	C₉	—	3	9	no	—
68	C₉	C₃×Q₈×C₉	1	6	yes	—
69	C₉	C₃×Q₈×C₉	1	3	yes	—
70	Q₈	#6887	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×C₃	—	3	3	no	—
72	C₂×C₃	C₂²×C₃²×C₉	1	1	yes	characteristic
73	C₂×C₃	C₂²×#737	1	3	yes	—
74	C₄	C₂×#6887	1	3	yes	—
75	C₃	—	3	3	no	—
76	C₃	C₃²×Q₈×C₉	1	1	yes	characteristic
77	C₃	Q₈×#737	1	3	yes	—
78	C₂	C₂²×#6887	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₉	C₂²×C₃²
commutator	C₂×C₃	C₂²×C₃²×C₉
Frattini, socle, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₂×C₃³	C₂²×C₃²
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. Q₈×C₉⋊C₉	C₃	semidirect
5. Q₈×C₉²	C₃	semidirect
11. #6887	Q₈	direct (N × H)
13. C₃×Q₈×C₉	C₉	semidirect
27. C₉⋊C₉	C₃×Q₈	semidirect
29. C₉²	C₃×Q₈	semidirect
48. C₃×C₉	Q₈×C₉	semidirect
70. Q₈	#6887	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×#6887	2
C₃²×Q₈×C₉	3
Q₈×C₉⋊C₉	3
Q₈×C₃²⋊C₉	3
Q₈×C₉²	3

Subgroups — Q8×#6887 #14517832

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#6887 #14517832