Subgroups — C₂³×#4185 #13048201

Order 1296.

1072 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#732	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#4185	C₂	1	14	yes	maximal, direct product factor
3	C₂³×(C₃⋊D₃)⋊C₃	—	3	3	no	maximal
4	C₂³×C₃×D₉	—	3	1	no	maximal
5	C₂²×#732	C₂²	1	7	yes	complemented
6	C₂×#4185	C₂²	1	28	yes	direct product factor
7	C₂³×H(3,𝔽₃)	D₃	1	3	yes	complemented
8	C₂³×C₃×C₉	D₃	1	1	yes	characteristic, complemented
9	C₂²×(C₃⋊D₃)⋊C₃	—	3	42	no	—
10	C₂²×C₃×D₉	—	3	14	no	—
11	C₂×#732	C₂³	1	7	yes	complemented
12	#4185	C₂³	1	8	yes	direct product factor
13	C₂³×C₃⋊D₃	—	3	3	no	—
14	C₂³×C₃×D₃	—	9	1	no	—
15	C₂³×D₉	—	3	1	no	—
16	C₂²×H(3,𝔽₃)	C₂×D₃	1	21	yes	complemented
17	C₂²×C₃×C₉	C₂×D₃	1	7	yes	complemented
18	C₂×(C₃⋊D₃)⋊C₃	—	3	84	no	—
19	C₂×C₃×D₉	—	3	28	no	—
20	#732	C₂⁴	1	1	yes	characteristic, complemented, Sylow
21	C₂³×C₃²	—	3	3	no	—
22	C₂³×C₃²	C₃⋊D₃	1	1	yes	characteristic
23	C₂²×C₃⋊D₃	—	3	42	no	—
24	C₂²×C₃×D₃	—	9	14	no	—
25	C₂³×C₉	—	3	1	no	—
26	C₂²×D₉	—	3	14	no	—
27	C₂×H(3,𝔽₃)	C₂²×D₃	1	21	yes	complemented
28	C₂×C₃×C₉	C₂²×D₃	1	7	yes	complemented
29	(C₃⋊D₃)⋊C₃	—	3	24	no	—
30	C₃×D₉	—	3	8	no	—
31	C₂⁴×C₃	—	27	1	no	—
32	C₂³×D₃	—	9	1	no	—
33	C₂³×D₃	—	27	3	no	—
34	C₂²×C₃²	—	3	21	no	—
35	C₂²×C₃²	C₂×C₃⋊D₃	1	7	yes	—
36	C₂×C₃⋊D₃	—	3	84	no	—
37	C₂×C₃×D₃	—	9	28	no	—
38	C₂²×C₉	—	3	7	no	—
39	C₂×D₉	—	3	28	no	—
40	H(3,𝔽₃)	C₂³×D₃	1	3	yes	complemented
41	C₃×C₉	C₂³×D₃	1	1	yes	characteristic, complemented
42	C₂³×C₃	—	3	1	no	—
43	C₂³×C₃	—	9	3	no	—
44	C₂³×C₃	—	27	14	no	—
45	C₂³×C₃	#272	1	1	yes	characteristic
46	C₂²×D₃	—	9	14	no	—
47	C₂²×D₃	—	27	42	no	—
48	C₂×C₃²	—	3	21	no	—
49	C₂×C₃²	C₂²×C₃⋊D₃	1	7	yes	—
50	C₃⋊D₃	—	3	24	no	—
51	C₃×D₃	—	9	8	no	—
52	C₂×C₉	—	3	7	no	—
53	D₉	—	3	8	no	—
54	C₂⁴	—	27	1	no	Sylow
55	C₂²×C₃	—	3	7	no	—
56	C₂²×C₃	—	9	21	no	—
57	C₂²×C₃	—	27	28	no	—
58	C₂²×C₃	C₂×#272	1	7	yes	—
59	C₂×D₃	—	9	28	no	—
60	C₂×D₃	—	27	84	no	—
61	C₃²	—	3	3	no	—
62	C₃²	C₂³×C₃⋊D₃	1	1	yes	characteristic
63	C₉	—	3	1	no	—
64	C₂³	—	27	14	no	—
65	C₂³	#4185	1	1	yes	characteristic, direct product factor
66	C₂×C₃	—	3	7	no	—
67	C₂×C₃	—	9	21	no	—
68	C₂×C₃	—	27	8	no	—
69	C₂×C₃	C₂²×#272	1	7	yes	—
70	D₃	—	9	8	no	—
71	D₃	—	27	24	no	—
72	C₂²	—	27	28	no	—
73	C₂²	C₂×#4185	1	7	yes	direct product factor
74	C₃	—	3	1	no	—
75	C₃	—	9	3	no	—
76	C₃	C₂³×#272	1	1	yes	characteristic
77	C₂	—	27	8	no	—
78	C₂	C₂²×#4185	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	#4185
commutator, cosocle, hypocenter	#732	C₂⁴
Frattini	C₃²	C₂³×C₃⋊D₃
Fitting	C₂³×#732	C₂
socle	C₂³×C₃	#272
LCS	G ▷ #732
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×#732	C₂	semidirect
2. C₂²×#4185	C₂	direct (N × H)
5. C₂²×#732	C₂²	semidirect
6. C₂×#4185	C₂²	direct (N × H)
7. C₂³×H(3,𝔽₃)	D₃	semidirect
8. C₂³×C₃×C₉	D₃	semidirect
11. C₂×#732	C₂³	semidirect
12. #4185	C₂³	direct (N × H)
16. C₂²×H(3,𝔽₃)	C₂×D₃	semidirect
17. C₂²×C₃×C₉	C₂×D₃	semidirect
20. #732	C₂⁴	semidirect
27. C₂×H(3,𝔽₃)	C₂²×D₃	semidirect
28. C₂×C₃×C₉	C₂²×D₃	semidirect
40. H(3,𝔽₃)	C₂³×D₃	semidirect
41. C₃×C₉	C₂³×D₃	semidirect
65. C₂³	#4185	direct (N × H)
73. C₂²	C₂×#4185	direct (N × H)
78. C₂	C₂²×#4185	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#732	2
C₂²×#4185	2
C₂³×(C₃⋊D₃)⋊C₃	3
C₂³×C₃×D₉	3

Subgroups — C23×#4185 #13048201

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#4185 #13048201