Subgroups — Q₈×C₃⋊D₂₇ #13047459

Order 1296.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×C₂₇	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₃⋊D₂₇	C₂	1	3	yes	maximal
3	#10619657	C₂	1	3	yes	maximal, complemented
4	Q₈×C₃⋊D₉	—	3	1	no	maximal
5	Q₈×D₂₇	—	3	3	no	maximal
6	C₂×C₃⋊D₂₇	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₂₇	C₂²	1	3	yes	—
8	C₃⋊Dic₂₇	C₂²	1	3	yes	—
9	C₃×Q₈×C₉	D₃	1	1	yes	characteristic
10	C₄×C₃⋊D₉	—	3	3	no	—
11	#6226	—	3	3	no	—
12	Q₈×C₂₇	D₃	1	3	yes	complemented
13	C₄×D₂₇	—	3	9	no	—
14	Dic₅₄	—	3	9	no	—
15	C₂×C₃×C₂₇	C₂³	1	1	yes	characteristic
16	C₃⋊D₂₇	Q₈	1	2	yes	direct product factor
17	Q₈×C₃⋊D₃	—	9	1	no	—
18	Q₈×D₉	—	9	2	no	—
19	Q₈×D₉	—	9	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	—	3	3	no	—
24	C₄×C₂₇	C₂×D₃	1	9	yes	—
25	Dic₂₇	—	3	9	no	—
26	C₃×C₂₇	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
27	C₃²×Q₈	D₉	1	1	yes	characteristic
28	C₄×C₃⋊D₃	—	9	3	no	—
29	#634	—	9	3	no	—
30	Q₈×C₉	D₉	1	2	yes	—
31	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic
32	C₄×D₉	—	9	6	no	—
33	C₄×D₉	—	9	3	no	—
34	Dic₁₈	—	9	6	no	—
35	Dic₁₈	—	9	3	no	—
36	C₂×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
37	C₃⋊D₉	—	3	2	no	—
38	C₂×C₂₇	C₂²×D₃	1	3	yes	—
39	D₂₇	—	3	6	no	—
40	D₃×Q₈	—	27	3	no	—
41	D₃×Q₈	—	27	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	—	9	2	no	—
46	C₂×D₉	—	9	1	no	—
47	C₄×C₉	C₂×D₉	1	6	yes	—
48	C₄×C₉	C₂×C₃⋊D₃	1	3	yes	—
49	Dic₉	—	9	6	no	—
50	Dic₉	—	9	3	no	—
51	C₃×C₉	D₃×Q₈	1	1	yes	characteristic
52	C₂₇	D₃×Q₈	1	3	yes	complemented
53	C₃×Q₈	D₂₇	1	3	yes	complemented
54	C₃×Q₈	C₃⋊D₉	1	1	yes	characteristic
55	C₄×D₃	—	27	9	no	—
56	C₄×D₃	—	27	3	no	—
57	Dic₆	—	27	9	no	—
58	Dic₆	—	27	3	no	—
59	C₂×C₃²	C₂²×D₉	1	1	yes	characteristic
60	C₃⋊D₃	—	9	2	no	—
61	C₂×C₉	C₂²×D₉	1	2	yes	—
62	C₂×C₉	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	D₉	—	9	4	no	—
64	D₉	—	9	2	no	—
65	C₂×Q₈	—	81	1	no	Sylow
66	C₂×D₃	—	27	3	no	—
67	C₂×D₃	—	27	1	no	—
68	C₃×C₄	C₂×D₂₇	1	9	yes	—
69	C₃×C₄	C₂×C₃⋊D₉	1	3	yes	—
70	Dic₃	—	27	9	no	—
71	Dic₃	—	27	3	no	—
72	C₃²	Q₈×D₉	1	1	yes	characteristic
73	C₉	Q₈×D₉	1	2	yes	—
74	C₉	Q₈×C₃⋊D₃	1	1	yes	characteristic
75	Q₈	—	81	3	no	—
76	Q₈	C₃⋊D₂₇	1	1	yes	characteristic, direct product factor
77	C₂×C₄	—	81	3	no	—
78	C₂×C₃	C₂²×D₂₇	1	3	yes	—
79	C₂×C₃	C₂²×C₃⋊D₉	1	1	yes	characteristic
80	D₃	—	27	6	no	—
81	D₃	—	27	2	no	—
82	C₂²	—	81	1	no	—
83	C₄	—	81	3	no	—
84	C₄	C₂×C₃⋊D₂₇	1	3	yes	—
85	C₃	Q₈×D₂₇	1	3	yes	complemented
86	C₃	Q₈×C₃⋊D₉	1	1	yes	characteristic
87	C₂	—	81	2	no	—
88	C₂	C₂²×C₃⋊D₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×C₃⋊D₂₇
commutator, cosocle	C₂×C₃×C₂₇	C₂³
Frattini	C₂×C₉	C₂²×C₃⋊D₃
Fitting	C₃×Q₈×C₂₇	C₂
socle	C₂×C₃²	C₂²×D₉
prime radical	C₂×C₃⋊D₂₇	C₂²
hypercenter	Q₈	C₃⋊D₂₇
hypocenter	C₃×C₂₇	C₂×Q₈
LCS	G ▷ C₂×C₃×C₂₇ ▷ C₃×C₂₇
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈×C₂₇	C₂	semidirect
3. #10619657	C₂	semidirect
12. Q₈×C₂₇	D₃	semidirect
16. C₃⋊D₂₇	Q₈	direct (N × H)
26. C₃×C₂₇	C₂×Q₈	semidirect
52. C₂₇	D₃×Q₈	semidirect
53. C₃×Q₈	D₂₇	semidirect
76. Q₈	C₃⋊D₂₇	direct (N × H)
85. C₃	Q₈×D₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×C₂₇	2
C₄×C₃⋊D₂₇	2
#10619657	2
Q₈×C₃⋊D₉	3
Q₈×D₂₇	3

Subgroups — Q8×C3⋊D27 #13047459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃⋊D₂₇ #13047459