Subgroups — Q₈×C₃⋊(C₃⋊(C₃⋊C₈)) #13163663

Order 1728.

1428 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	1	3	yes	maximal
3	C₄×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	3	yes	maximal
4	Q₈×C₃⋊(C₃⋊C₈)	—	3	13	no	maximal
5	C₂×C₃³×Q₈	C₄	1	1	yes	characteristic
6	C₃³×C₄⋊C₄	C₄	1	3	yes	—
7	C₃³×C₄²	C₂²	1	3	yes	—
8	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	3	yes	—
9	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
10	C₃²×C₄×Q₈	D₃	1	13	yes	—
11	C₄⋊(C₃⋊(C₃⋊C₈))	—	3	39	no	—
12	C₄×C₃⋊(C₃⋊C₈)	—	3	39	no	—
13	C₃³×Q₈	C₈	1	4	yes	complemented
14	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
15	C₂×C₃³×C₄	C₂×C₄	1	3	yes	—
16	C₂×C₃³×C₄	C₂³	1	1	yes	characteristic
17	C₃⋊(C₃⋊(C₃⋊C₈))	—	2	3	no	—
18	C₃⋊(C₃⋊(C₃⋊C₈))	Q₈	1	2	yes	direct product factor
19	Q₈×C₃⋊C₈	—	9	13	no	—
20	C₂×C₃²×Q₈	Dic₃	1	13	yes	—
21	C₃²×C₄⋊C₄	Dic₃	1	39	yes	—
22	C₃²×C₄²	C₂×D₃	1	39	yes	—
23	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
24	C₂×C₃⋊(C₃⋊C₈)	—	3	39	no	—
25	C₂²×C₃³	C₂²×C₄	1	1	yes	characteristic
26	C₃³×C₄	—	2	3	no	—
27	C₃³×C₄	C₂×C₈	1	6	yes	—
28	C₃³×C₄	C₂×Q₈	1	1	yes	characteristic
29	C₃³×C₄	#41	1	1	yes	characteristic
30	C₃×C₄×Q₈	C₃⋊D₃	1	13	yes	—
31	C₄⋊(C₃⋊C₈)	—	9	39	no	—
32	C₄×C₃⋊C₈	—	9	39	no	—
33	C₃²×Q₈	C₃⋊C₈	1	52	yes	complemented
34	C₂×C₃²×C₄	C₂×Dic₃	1	39	yes	—
35	C₂×C₃²×C₄	C₂×Dic₃	1	39	yes	—
36	C₂×C₃²×C₄	C₂²×D₃	1	13	yes	—
37	C₃⋊(C₃⋊C₈)	—	3	26	no	—
38	C₃⋊(C₃⋊C₈)	—	6	39	no	—
39	C₈×Q₈	—	27	1	no	Sylow
40	C₂×C₃³	C₄×Q₈	1	1	yes	characteristic
41	C₂×C₃³	C₂²×C₈	1	1	yes	characteristic
42	C₂×C₃³	#131	1	1	yes	characteristic
43	C₂×C₃×Q₈	C₃⋊Dic₃	1	13	yes	—
44	C₃×C₄⋊C₄	C₃⋊Dic₃	1	39	yes	—
45	C₃×C₄²	C₂×C₃⋊D₃	1	39	yes	—
46	C₂×C₃⋊C₈	—	9	13	no	—
47	C₂×C₃⋊C₈	—	9	39	no	—
48	C₂²×C₃²	C₂²×Dic₃	1	13	yes	—
49	C₃²×C₄	—	2	39	no	—
50	C₃²×C₄	C₂×C₃⋊C₈	1	78	yes	—
51	C₃²×C₄	D₃×Q₈	1	13	yes	—
52	C₃²×C₄	#239	1	13	yes	—
53	C₄×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
54	C₄⋊C₈	—	27	3	no	—
55	C₄×C₈	—	27	3	no	—
56	C₃³	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
57	C₃×Q₈	C₃⋊(C₃⋊C₈)	1	52	yes	complemented
58	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	39	yes	—
59	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	39	yes	—
60	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
61	C₃⋊C₈	—	9	26	no	—
62	C₃⋊C₈	—	18	39	no	—
63	C₂×C₃²	C₂²×C₃⋊C₈	1	13	yes	—
64	C₂×C₃²	Q₈×Dic₃	1	13	yes	—
65	C₂×C₃²	#948	1	13	yes	—
66	C₂×Q₈	C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
67	C₂×C₈	—	27	1	no	—
68	C₂×C₈	—	27	3	no	—
69	C₄⋊C₄	C₃⋊(C₃⋊Dic₃)	1	3	yes	—
70	C₄²	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
71	C₂²×C₃	C₂²×C₃⋊Dic₃	1	13	yes	—
72	C₃×C₄	—	2	39	no	—
73	C₃×C₄	C₂×C₃⋊(C₃⋊C₈)	1	78	yes	—
74	C₃×C₄	Q₈×C₃⋊D₃	1	13	yes	—
75	C₃×C₄	#3832	1	13	yes	—
76	C₃²	Q₈×C₃⋊C₈	1	13	yes	complemented
77	Q₈	C₃⋊(C₃⋊(C₃⋊C₈))	1	4	yes	direct product factor
78	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	3	yes	—
79	C₂×C₄	C₂×C₃⋊(C₃⋊Dic₃)	1	3	yes	—
80	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
81	C₈	—	27	2	no	—
82	C₈	—	54	3	no	—
83	C₂×C₃	C₂²×C₃⋊(C₃⋊C₈)	1	13	yes	—
84	C₂×C₃	Q₈×C₃⋊Dic₃	1	13	yes	—
85	C₂×C₃	#64157	1	13	yes	—
86	C₂²	C₂²×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
87	C₄	—	2	3	no	—
88	C₄	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	1	6	yes	—
89	C₄	Q₈×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
90	C₄	#89080	1	1	yes	characteristic
91	C₃	Q₈×C₃⋊(C₃⋊C₈)	1	13	yes	complemented
92	C₂	C₂²×C₃⋊(C₃⋊(C₃⋊C₈))	1	1	yes	characteristic
93	C₂	Q₈×C₃⋊(C₃⋊Dic₃)	1	1	yes	characteristic
94	C₂	#11723989	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×C₃⋊(C₃⋊D₃)
commutator	C₂×C₃³	C₂²×C₈
Fitting	C₃³×C₄×Q₈	C₂
socle, prime radical	C₂²×C₃³	C₂²×C₄
cosocle	C₂×C₃³×C₄	C₂³
hypercenter	C₄×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₈×Q₈
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
13. C₃³×Q₈	C₈	semidirect
18. C₃⋊(C₃⋊(C₃⋊C₈))	Q₈	direct (N × H)
33. C₃²×Q₈	C₃⋊C₈	semidirect
56. C₃³	C₈×Q₈	semidirect
57. C₃×Q₈	C₃⋊(C₃⋊C₈)	semidirect
76. C₃²	Q₈×C₃⋊C₈	semidirect
77. Q₈	C₃⋊(C₃⋊(C₃⋊C₈))	direct (N × H)
91. C₃	Q₈×C₃⋊(C₃⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
C₃³×C₄×Q₈	2
C₄⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	2
C₄×C₃⋊(C₃⋊(C₃⋊C₈))	2
Q₈×C₃⋊(C₃⋊C₈)	3

Subgroups — Q8×C3⋊(C3⋊(C3⋊C8)) #13163663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃⋊(C₃⋊(C₃⋊C₈)) #13163663