Subgroups — #13047460

Order 1296.

280 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, complemented
3	C₄×C₃⋊D₂₇	C₂	1	3	yes	maximal, complemented
4	#88746	—	3	1	no	maximal
5	#88403	—	3	3	no	maximal
6	C₂×C₃⋊D₂₇	C₂²	1	3	yes	—
7	C₃×C₄×C₂₇	C₂²	1	3	yes	—
8	C₃⋊Dic₂₇	C₂²	1	1	yes	characteristic
9	C₃×Q₈×C₉	D₃	1	1	yes	characteristic
10	C₄⋊(C₃⋊D₉)	—	3	3	no	—
11	C₄×C₃⋊D₉	—	3	3	no	—
12	Q₈×C₂₇	D₃	1	3	yes	complemented
13	D₁₀₈	—	3	9	no	—
14	C₄×D₂₇	—	3	9	no	—
15	C₂×C₃×C₂₇	C₂³	1	1	yes	characteristic
16	C₃⋊D₂₇	—	2	3	no	—
17	#3832	—	9	1	no	—
18	#3701	—	9	2	no	—
19	#3701	—	9	1	no	—
20	C₂×C₃⋊D₉	—	3	3	no	—
21	C₃×C₄×C₉	C₂×D₃	1	3	yes	—
22	C₃⋊Dic₉	—	3	1	no	—
23	C₂×D₂₇	—	3	9	no	—
24	C₄×C₂₇	C₂×D₃	1	9	yes	—
25	Dic₂₇	—	3	3	no	—
26	C₃×C₂₇	#41	1	1	yes	characteristic, complemented, Sylow
27	C₃²×Q₈	D₉	1	1	yes	characteristic
28	C₄⋊(C₃⋊D₃)	—	9	3	no	—
29	C₄×C₃⋊D₃	—	9	3	no	—
30	Q₈×C₉	D₉	1	2	yes	—
31	Q₈×C₉	C₃⋊D₃	1	1	yes	characteristic
32	D₃₆	—	9	6	no	—
33	D₃₆	—	9	3	no	—
34	C₄×D₉	—	9	6	no	—
35	C₄×D₉	—	9	3	no	—
36	C₂×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
37	C₃⋊D₉	—	6	3	no	—
38	C₂×C₂₇	C₂²×D₃	1	3	yes	—
39	D₂₇	—	6	9	no	—
40	#239	—	27	3	no	—
41	#239	—	27	1	no	—
42	C₂×C₃⋊D₃	—	9	3	no	—
43	C₃²×C₄	C₂×D₉	1	3	yes	—
44	C₃⋊Dic₃	—	9	1	no	—
45	C₂×D₉	—	9	6	no	—
46	C₂×D₉	—	9	3	no	—
47	C₄×C₉	C₂×D₉	1	6	yes	—
48	C₄×C₉	C₂×C₃⋊D₃	1	3	yes	—
49	Dic₉	—	9	2	no	—
50	Dic₉	—	9	1	no	—
51	C₃×C₉	#239	1	1	yes	characteristic
52	C₂₇	#239	1	3	yes	complemented
53	C₃×Q₈	D₂₇	1	3	yes	complemented
54	C₃×Q₈	C₃⋊D₉	1	1	yes	characteristic
55	D₁₂	—	27	9	no	—
56	D₁₂	—	27	3	no	—
57	C₄×D₃	—	27	9	no	—
58	C₄×D₃	—	27	3	no	—
59	C₂×C₃²	C₂²×D₉	1	1	yes	characteristic
60	C₃⋊D₃	—	18	3	no	—
61	C₂×C₉	C₂²×D₉	1	2	yes	—
62	C₂×C₉	C₂²×C₃⋊D₃	1	1	yes	characteristic
63	D₉	—	18	6	no	—
64	D₉	—	18	3	no	—
65	#41	—	81	1	no	Sylow
66	C₂×D₃	—	27	9	no	—
67	C₂×D₃	—	27	3	no	—
68	C₃×C₄	C₂×D₂₇	1	9	yes	—
69	C₃×C₄	C₂×C₃⋊D₉	1	3	yes	—
70	Dic₃	—	27	3	no	—
71	Dic₃	—	27	1	no	—
72	C₃²	#3701	1	1	yes	characteristic
73	C₉	#3701	1	2	yes	—
74	C₉	#3832	1	1	yes	characteristic
75	Q₈	C₃⋊D₂₇	1	1	yes	characteristic, complemented
76	D₄	—	81	3	no	—
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	—	54	9	no	—
81	D₃	—	54	3	no	—
82	C₂²	—	81	3	no	—
83	C₄	—	81	1	no	—
84	C₄	C₂×C₃⋊D₂₇	1	3	yes	—
85	C₃	#88403	1	3	yes	complemented
86	C₃	#88746	1	1	yes	characteristic
87	C₂	—	162	3	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
hypercenter	Q₈	C₃⋊D₂₇
hypocenter	C₃×C₂₇	#41
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
2. C₄⋊(C₃⋊D₂₇)	C₂	semidirect
3. C₄×C₃⋊D₂₇	C₂	semidirect
12. Q₈×C₂₇	D₃	semidirect
26. C₃×C₂₇	#41	semidirect
52. C₂₇	#239	semidirect
53. C₃×Q₈	D₂₇	semidirect
75. Q₈	C₃⋊D₂₇	semidirect
85. C₃	#88403	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
C₄×C₃⋊D₂₇	2
#88746	3
#88403	3