Subgroups — C₂³×#4177 #13048193

Order 1296.

702 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#731	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#4177	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃⋊D₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₂³×(C₃⋊D₃)⋊C₃	—	3	1	no	maximal
5	C₂²×#731	C₂²	1	7	yes	complemented
6	C₂×#4177	C₂²	1	28	yes	direct product factor
7	C₂³×3₋¹⁺²	—	2	1	no	—
8	C₂³×H(3,𝔽₃)	D₃	1	1	yes	characteristic
9	C₂³×C₃×C₉	C₂×C₃	1	1	yes	characteristic, complemented
10	C₂²×C₃⋊D₉	C₂×C₃	1	14	yes	complemented
11	C₂²×(C₃⋊D₃)⋊C₃	—	3	14	no	—
12	C₂×#731	C₂³	1	7	yes	complemented
13	#4177	C₂³	1	8	yes	direct product factor
14	C₂³×C₃⋊D₃	—	3	1	no	—
15	C₂³×C₃×D₃	—	9	1	no	—
16	C₂³×D₉	—	9	1	no	—
17	C₂²×3₋¹⁺²	—	2	7	no	—
18	C₂²×H(3,𝔽₃)	C₂×D₃	1	7	yes	—
19	C₂²×C₃×C₉	C₂²×C₃	1	7	yes	complemented
20	C₂×C₃⋊D₉	C₂²×C₃	1	28	yes	complemented
21	C₂×(C₃⋊D₃)⋊C₃	—	3	28	no	—
22	#731	C₂⁴	1	1	yes	characteristic, complemented, Sylow
23	C₂³×C₃²	—	3	1	no	—
24	C₂³×C₃²	C₃×D₃	1	1	yes	characteristic
25	C₂²×C₃⋊D₃	—	3	14	no	—
26	C₂²×C₃×D₃	—	9	14	no	—
27	C₂³×C₉	—	3	1	no	—
28	C₂³×C₉	—	6	1	no	—
29	C₂²×D₉	—	9	14	no	—
30	C₂×3₋¹⁺²	—	2	7	no	—
31	C₂×H(3,𝔽₃)	C₂²×D₃	1	7	yes	—
32	C₂×C₃×C₉	C₂³×C₃	1	7	yes	complemented
33	C₃⋊D₉	C₂³×C₃	1	8	yes	complemented
34	(C₃⋊D₃)⋊C₃	—	3	8	no	—
35	C₂⁴×C₃	—	27	1	no	—
36	C₂³×D₃	—	9	1	no	—
37	C₂³×D₃	—	27	1	no	—
38	C₂²×C₃²	—	3	7	no	—
39	C₂²×C₃²	C₂×C₃×D₃	1	7	yes	—
40	C₂×C₃⋊D₃	—	3	28	no	—
41	C₂×C₃×D₃	—	9	28	no	—
42	C₂²×C₉	—	3	7	no	—
43	C₂²×C₉	—	6	7	no	—
44	C₂×D₉	—	9	28	no	—
45	3₋¹⁺²	—	2	1	no	—
46	H(3,𝔽₃)	C₂³×D₃	1	1	yes	characteristic
47	C₃×C₉	C₂⁴×C₃	1	1	yes	characteristic, complemented
48	C₂³×C₃	—	3	1	no	—
49	C₂³×C₃	—	9	1	no	—
50	C₂³×C₃	—	27	14	no	—
51	C₂³×C₃	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
52	C₂²×D₃	—	9	14	no	—
53	C₂²×D₃	—	27	14	no	—
54	C₂×C₃²	—	3	7	no	—
55	C₂×C₃²	C₂²×C₃×D₃	1	7	yes	—
56	C₃⋊D₃	—	3	8	no	—
57	C₃×D₃	—	9	8	no	—
58	C₂×C₉	—	3	7	no	—
59	C₂×C₉	—	6	7	no	—
60	D₉	—	9	8	no	—
61	C₂⁴	—	27	1	no	Sylow
62	C₂²×C₃	—	3	7	no	—
63	C₂²×C₃	—	9	7	no	—
64	C₂²×C₃	—	27	28	no	—
65	C₂²×C₃	C₂×(C₃⋊D₃)⋊C₃	1	7	yes	—
66	C₂×D₃	—	9	28	no	—
67	C₂×D₃	—	27	28	no	—
68	C₃²	—	3	1	no	—
69	C₃²	C₂³×C₃×D₃	1	1	yes	characteristic
70	C₉	—	3	1	no	—
71	C₉	—	6	1	no	—
72	C₂³	—	27	14	no	—
73	C₂³	#4177	1	1	yes	characteristic, direct product factor
74	C₂×C₃	—	3	7	no	—
75	C₂×C₃	—	9	7	no	—
76	C₂×C₃	—	27	8	no	—
77	C₂×C₃	C₂²×(C₃⋊D₃)⋊C₃	1	7	yes	—
78	D₃	—	9	8	no	—
79	D₃	—	27	8	no	—
80	C₂²	—	27	28	no	—
81	C₂²	C₂×#4177	1	7	yes	direct product factor
82	C₃	—	3	1	no	—
83	C₃	—	9	1	no	—
84	C₃	C₂³×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
85	C₂	—	27	8	no	—
86	C₂	C₂²×#4177	1	7	yes	direct product factor

	subgroup	G/H
center, hypercenter	C₂³	#4177
commutator, cosocle, hypocenter	C₃×C₉	C₂⁴×C₃
Frattini	C₃²	C₂³×C₃×D₃
Fitting	C₂³×#731	C₂
socle	C₂³×C₃	(C₃⋊D₃)⋊C₃
LCS	G ▷ C₃×C₉
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
1. C₂³×#731	C₂	semidirect
2. C₂²×#4177	C₂	direct (N × H)
3. C₂³×C₃⋊D₉	C₃	semidirect
5. C₂²×#731	C₂²	semidirect
6. C₂×#4177	C₂²	direct (N × H)
9. C₂³×C₃×C₉	C₂×C₃	semidirect
10. C₂²×C₃⋊D₉	C₂×C₃	semidirect
12. C₂×#731	C₂³	semidirect
13. #4177	C₂³	direct (N × H)
19. C₂²×C₃×C₉	C₂²×C₃	semidirect
20. C₂×C₃⋊D₉	C₂²×C₃	semidirect
22. #731	C₂⁴	semidirect
32. C₂×C₃×C₉	C₂³×C₃	semidirect
33. C₃⋊D₉	C₂³×C₃	semidirect
47. C₃×C₉	C₂⁴×C₃	semidirect
73. C₂³	#4177	direct (N × H)
81. C₂²	C₂×#4177	direct (N × H)
86. C₂	C₂²×#4177	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×#731	2
C₂²×#4177	2
C₂³×C₃⋊D₉	3
C₂³×(C₃⋊D₃)⋊C₃	3

Subgroups — C23×#4177 #13048193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#4177 #13048193