Subgroups — C₃³×#126 #11724007

Order 864.

840 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×C₄⋊C₄	C₂	1	3	yes	maximal, complemented
2	C₃³×C₂²⋊C₄	C₂	1	3	yes	maximal
3	C₃³×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#126	C₃	1	13	yes	maximal, direct product factor
5	C₂³×C₃³	C₂²	1	1	yes	characteristic
6	C₂×C₃³×C₄	C₂²	1	3	yes	—
7	C₂×C₃³×C₄	C₂²	1	3	yes	—
8	C₃²×C₄⋊C₄	C₂×C₃	1	39	yes	complemented
9	C₃²×C₂²⋊C₄	C₂×C₃	1	39	yes	—
10	C₃²×C₄²	C₂×C₃	1	13	yes	complemented
11	C₂²×C₃³	—	2	3	no	—
12	C₂²×C₃³	C₂³	1	1	yes	characteristic
13	C₃³×C₄	—	2	3	no	—
14	C₃³×C₄	—	2	3	no	—
15	C₃×#126	C₃²	1	13	yes	direct product factor
16	C₂³×C₃²	C₂²×C₃	1	13	yes	—
17	C₂×C₃²×C₄	C₂²×C₃	1	39	yes	—
18	C₂×C₃²×C₄	C₂²×C₃	1	39	yes	—
19	C₂×C₃³	—	4	1	no	—
20	C₂×C₃³	#41	1	3	yes	—
21	C₃×C₄⋊C₄	C₂×C₃²	1	39	yes	complemented
22	C₃×C₂²⋊C₄	C₂×C₃²	1	39	yes	—
23	C₃×C₄²	C₂×C₃²	1	13	yes	complemented
24	C₂²×C₃²	—	2	39	no	—
25	C₂²×C₃²	C₂³×C₃	1	13	yes	—
26	C₃²×C₄	—	2	39	no	—
27	C₃²×C₄	—	2	39	no	—
28	#126	C₃³	1	1	yes	characteristic, direct product factor, Sylow
29	C₃³	#126	1	1	yes	characteristic, direct product factor, Sylow
30	C₂³×C₃	C₂²×C₃²	1	13	yes	—
31	C₂×C₃×C₄	C₂²×C₃²	1	39	yes	—
32	C₂×C₃×C₄	C₂²×C₃²	1	39	yes	—
33	C₂×C₃²	—	4	13	no	—
34	C₂×C₃²	C₃×#41	1	39	yes	—
35	C₄⋊C₄	C₂×C₃³	1	3	yes	complemented
36	C₂²⋊C₄	C₂×C₃³	1	3	yes	—
37	C₄²	C₂×C₃³	1	1	yes	characteristic, complemented
38	C₂²×C₃	—	2	39	no	—
39	C₂²×C₃	C₂³×C₃²	1	13	yes	—
40	C₃×C₄	—	2	39	no	—
41	C₃×C₄	—	2	39	no	—
42	C₃²	C₃×#126	1	13	yes	direct product factor
43	C₂³	C₂²×C₃³	1	1	yes	characteristic
44	C₂×C₄	C₂²×C₃³	1	3	yes	—
45	C₂×C₄	C₂²×C₃³	1	3	yes	—
46	C₂×C₃	—	4	13	no	—
47	C₂×C₃	C₃²×#41	1	39	yes	—
48	C₂²	—	2	3	no	—
49	C₂²	C₂³×C₃³	1	1	yes	characteristic
50	C₄	—	2	3	no	—
51	C₄	—	2	3	no	—
52	C₃	C₃²×#126	1	13	yes	direct product factor
53	C₂	—	4	1	no	—
54	C₂	C₃³×#41	1	3	yes	—

	subgroup	G/H
center, socle	C₂²×C₃³	C₂³
commutator, Frattini, cosocle	C₂²	C₂³×C₃³
prime radical	C₂³×C₃³	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃³ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃³×C₄⋊C₄	C₂	semidirect
3. C₃³×C₄²	C₂	semidirect
4. C₃²×#126	C₃	direct (N × H)
8. C₃²×C₄⋊C₄	C₂×C₃	semidirect
10. C₃²×C₄²	C₂×C₃	semidirect
15. C₃×#126	C₃²	direct (N × H)
21. C₃×C₄⋊C₄	C₂×C₃²	semidirect
23. C₃×C₄²	C₂×C₃²	semidirect
28. #126	C₃³	direct (N × H)
29. C₃³	#126	direct (N × H)
35. C₄⋊C₄	C₂×C₃³	semidirect
37. C₄²	C₂×C₃³	semidirect
42. C₃²	C₃×#126	direct (N × H)
52. C₃	C₃²×#126	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×C₄⋊C₄	2
C₃³×C₂²⋊C₄	2
C₃³×C₄²	2
C₃²×#126	3

Subgroups — C33×#126 #11724007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃³×#126 #11724007