Subgroups — Q₈×C₃⋊(C₃⋊Dic₃) #11723986

Order 864.

980 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₃⋊Dic₃))	C₂	1	3	yes	maximal
3	C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	3	yes	maximal
4	Q₈×C₃⋊Dic₃	—	3	13	no	maximal
5	C₃³×Q₈	C₄	1	4	yes	complemented
6	C₂×C₃³×C₄	C₂²	1	3	yes	—
7	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	3	yes	—
8	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	1	yes	characteristic
9	C₂×C₃²×Q₈	D₃	1	13	yes	—
10	C₄⋊(C₃⋊Dic₃)	—	3	39	no	—
11	C₄×C₃⋊Dic₃	—	3	39	no	—
12	C₂²×C₃³	C₂³	1	1	yes	characteristic
13	C₃³×C₄	C₂×C₄	1	6	yes	—
14	C₃⋊(C₃⋊Dic₃)	—	2	3	no	—
15	C₃⋊(C₃⋊Dic₃)	Q₈	1	2	yes	direct product factor
16	Q₈×Dic₃	—	9	13	no	—
17	C₃²×Q₈	Dic₃	1	52	yes	complemented
18	C₂×C₃²×C₄	C₂×D₃	1	39	yes	—
19	C₂×C₃⋊Dic₃	—	3	13	no	—
20	C₂×C₃⋊Dic₃	—	3	39	no	—
21	C₂×C₃³	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₃³	C₂×Q₈	1	1	yes	characteristic
23	C₂×C₃³	#41	1	1	yes	characteristic
24	C₂×C₃×Q₈	C₃⋊D₃	1	13	yes	—
25	C₄⋊Dic₃	—	9	39	no	—
26	C₄×Dic₃	—	9	39	no	—
27	C₂²×C₃²	C₂²×D₃	1	13	yes	—
28	C₃²×C₄	C₂×Dic₃	1	78	yes	—
29	C₃⋊Dic₃	—	3	26	no	—
30	C₃⋊Dic₃	—	6	39	no	—
31	C₄×Q₈	—	27	1	no	Sylow
32	C₃³	C₄×Q₈	1	1	yes	characteristic, complemented, Sylow
33	C₃×Q₈	C₃⋊Dic₃	1	52	yes	complemented
34	C₂×C₃×C₄	C₂×C₃⋊D₃	1	39	yes	—
35	C₂×Dic₃	—	9	13	no	—
36	C₂×Dic₃	—	9	39	no	—
37	C₂×C₃²	D₃×Q₈	1	13	yes	—
38	C₂×C₃²	#239	1	13	yes	—
39	C₂×C₃²	C₂²×Dic₃	1	13	yes	—
40	C₂×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
41	C₄⋊C₄	—	27	3	no	—
42	C₄²	—	27	3	no	—
43	C₂²×C₃	C₂²×C₃⋊D₃	1	13	yes	—
44	C₃×C₄	C₂×C₃⋊Dic₃	1	78	yes	—
45	Dic₃	—	9	26	no	—
46	Dic₃	—	18	39	no	—
47	C₃²	Q₈×Dic₃	1	13	yes	complemented
48	Q₈	C₃⋊(C₃⋊Dic₃)	1	4	yes	direct product factor
49	C₂×C₄	—	27	1	no	—
50	C₂×C₄	—	27	3	no	—
51	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
52	C₂×C₃	Q₈×C₃⋊D₃	1	13	yes	—
53	C₂×C₃	#3832	1	13	yes	—
54	C₂×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
55	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
56	C₄	—	27	2	no	—
57	C₄	—	54	3	no	—
58	C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	6	yes	—
59	C₃	Q₈×C₃⋊Dic₃	1	13	yes	complemented
60	C₂	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
61	C₂	#89080	1	1	yes	characteristic
62	C₂	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₂²×C₄
Fitting	C₂×C₃³×Q₈	C₂
socle, prime radical, cosocle	C₂²×C₃³	C₂³
hypercenter	C₂×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₄×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
5. C₃³×Q₈	C₄	semidirect
15. C₃⋊(C₃⋊Dic₃)	Q₈	direct (N × H)
17. C₃²×Q₈	Dic₃	semidirect
32. C₃³	C₄×Q₈	semidirect
33. C₃×Q₈	C₃⋊Dic₃	semidirect
47. C₃²	Q₈×Dic₃	semidirect
48. Q₈	C₃⋊(C₃⋊Dic₃)	direct (N × H)
59. C₃	Q₈×C₃⋊Dic₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×Q₈	2
C₄⋊(C₃⋊(C₃⋊Dic₃))	2
C₄×C₃⋊(C₃⋊Dic₃)	2
Q₈×C₃⋊Dic₃	3

Subgroups — Q8×C3⋊(C3⋊Dic3) #11723986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃⋊(C₃⋊Dic₃) #11723986