Subgroups — Q₈×D₂₇⋊C₃ #13047373

Order 1296.

164 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₂₇⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₂₇⋊C₃	C₂	1	3	yes	maximal
3	(C₂₇⋊C₃)⋊Q₈	C₂	1	3	yes	maximal, complemented
4	C₃×Q₈×D₉	—	3	1	no	maximal
5	Q₈×D₂₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×D₂₇⋊C₃	C₂²	1	1	yes	characteristic
7	C₄×C₂₇⋊C₃	C₂²	1	3	yes	—
8	(C₂₇⋊C₃)⋊C₄	C₂²	1	3	yes	—
9	C₃×Q₈×C₉	D₃	1	1	yes	characteristic
10	C₃×C₄×D₉	—	3	3	no	—
11	C₃×Dic₁₈	—	3	3	no	—
12	Q₈×C₂₇	—	2	1	no	—
13	Q₈×C₂₇	C₂×C₃	1	1	yes	characteristic, complemented
14	C₄×D₂₇	C₂×C₃	1	3	yes	—
15	Dic₅₄	C₂×C₃	1	3	yes	complemented
16	C₂×C₂₇⋊C₃	C₂³	1	1	yes	characteristic
17	D₂₇⋊C₃	Q₈	1	2	yes	direct product factor
18	C₃×D₃×Q₈	—	9	1	no	—
19	Q₈×D₉	—	3	1	no	—
20	C₂×C₃×D₉	—	3	1	no	—
21	C₃×C₄×C₉	C₂×D₃	1	3	yes	—
22	C₃×Dic₉	—	3	3	no	—
23	C₂×D₂₇	C₂²×C₃	1	1	yes	characteristic
24	C₄×C₂₇	—	2	3	no	—
25	C₄×C₂₇	C₂²×C₃	1	3	yes	—
26	Dic₂₇	C₂²×C₃	1	3	yes	—
27	C₂₇⋊C₃	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
28	C₃²×Q₈	D₉	1	1	yes	characteristic
29	C₃×C₄×D₃	—	9	3	no	—
30	C₃×Dic₆	—	9	3	no	—
31	Q₈×C₉	—	2	1	no	—
32	Q₈×C₉	C₃×D₃	1	1	yes	characteristic
33	C₄×D₉	—	3	3	no	—
34	Dic₁₈	—	3	3	no	—
35	C₂×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
36	C₃×D₉	—	3	2	no	—
37	C₂×C₂₇	—	2	1	no	—
38	C₂×C₂₇	C₂³×C₃	1	1	yes	characteristic
39	D₂₇	C₃×Q₈	1	2	yes	complemented
40	C₂×C₃×Q₈	—	27	1	no	—
41	D₃×Q₈	—	9	1	no	—
42	C₂×C₃×D₃	—	9	1	no	—
43	C₃²×C₄	C₂×D₉	1	3	yes	—
44	C₃×Dic₃	—	9	3	no	—
45	C₂×D₉	—	3	1	no	—
46	C₄×C₉	—	2	3	no	—
47	C₄×C₉	C₂×C₃×D₃	1	3	yes	—
48	Dic₉	—	3	3	no	—
49	C₃×C₉	D₃×Q₈	1	1	yes	characteristic
50	C₂₇	—	2	1	no	—
51	C₂₇	C₂×C₃×Q₈	1	1	yes	characteristic, complemented
52	C₃×Q₈	—	3	1	no	—
53	C₃×Q₈	—	27	3	no	—
54	C₃×Q₈	C₃×D₉	1	1	yes	characteristic
55	C₂×C₃×C₄	—	27	3	no	—
56	C₄×D₃	—	9	3	no	—
57	Dic₆	—	9	3	no	—
58	C₂×C₃²	C₂²×D₉	1	1	yes	characteristic
59	C₃×D₃	—	9	2	no	—
60	C₂×C₉	—	2	1	no	—
61	C₂×C₉	C₂²×C₃×D₃	1	1	yes	characteristic
62	D₉	—	3	2	no	—
63	C₂×Q₈	—	27	1	no	Sylow
64	C₂²×C₃	—	27	1	no	—
65	C₂×D₃	—	9	1	no	—
66	C₃×C₄	—	3	3	no	—
67	C₃×C₄	—	27	3	no	—
68	C₃×C₄	C₂×C₃×D₉	1	3	yes	—
69	Dic₃	—	9	3	no	—
70	C₃²	Q₈×D₉	1	1	yes	characteristic
71	C₉	—	2	1	no	—
72	C₉	C₃×D₃×Q₈	1	1	yes	characteristic
73	Q₈	—	27	3	no	—
74	Q₈	D₂₇⋊C₃	1	1	yes	characteristic, direct product factor
75	C₂×C₄	—	27	3	no	—
76	C₂×C₃	—	3	1	no	—
77	C₂×C₃	—	27	2	no	—
78	C₂×C₃	C₂²×C₃×D₉	1	1	yes	characteristic
79	D₃	—	9	2	no	—
80	C₂²	—	27	1	no	—
81	C₄	—	27	3	no	—
82	C₄	C₂×D₂₇⋊C₃	1	3	yes	—
83	C₃	—	3	1	no	—
84	C₃	C₃×Q₈×D₉	1	1	yes	characteristic
85	C₂	—	27	2	no	—
86	C₂	C₂²×D₂₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₂₇⋊C₃
commutator, cosocle	C₂×C₂₇	C₂³×C₃
Frattini	C₂×C₉	C₂²×C₃×D₃
Fitting	Q₈×C₂₇⋊C₃	C₂
socle	C₂×C₃	C₂²×C₃×D₉
prime radical	C₂×D₂₇⋊C₃	C₂²
hypercenter	Q₈	D₂₇⋊C₃
hypocenter	C₂₇	C₂×C₃×Q₈
LCS	G ▷ C₂×C₂₇ ▷ C₂₇
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×C₂₇⋊C₃	C₂	semidirect
3. (C₂₇⋊C₃)⋊Q₈	C₂	semidirect
5. Q₈×D₂₇	C₃	semidirect
13. Q₈×C₂₇	C₂×C₃	semidirect
15. Dic₅₄	C₂×C₃	semidirect
17. D₂₇⋊C₃	Q₈	direct (N × H)
27. C₂₇⋊C₃	C₂×Q₈	semidirect
39. D₂₇	C₃×Q₈	semidirect
51. C₂₇	C₂×C₃×Q₈	semidirect
74. Q₈	D₂₇⋊C₃	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×C₂₇⋊C₃	2
C₄×D₂₇⋊C₃	2
(C₂₇⋊C₃)⋊Q₈	2
C₃×Q₈×D₉	3
Q₈×D₂₇	3

Subgroups — Q8×D27⋊C3 #13047373

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₂₇⋊C₃ #13047373