Subgroups — #13047374

Order 1296.

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

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
C₄×D₂₇⋊C₃	2
C₃×#3701	3
#88403	3