Subgroups — #13049162

Order 1296.

3604 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃⁴	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊(C₃⋊(C₃⋊Dic₃))	C₂	1	2	yes	maximal, complemented
3	#88864	—	3	40	no	maximal
4	C₂²×C₃⁴	—	2	2	no	—
5	C₂²×C₃⁴	C₄	1	2	yes	complemented
6	C₂²×C₃⁴	C₂²	1	1	yes	characteristic
7	C₃⋊(C₃⋊(C₃⋊Dic₃))	—	2	2	no	—
8	C₂³×C₃³	D₃	1	40	yes	—
9	C₂×C₃⋊(C₃⋊Dic₃)	—	3	80	no	—
10	C₂×C₃⁴	—	2	2	no	—
11	C₂×C₃⁴	C₂×C₄	1	1	yes	characteristic
12	C₂×C₃⁴	D₄	1	2	yes	—
13	#3757	—	9	130	no	—
14	C₂²×C₃³	—	2	80	no	—
15	C₂²×C₃³	Dic₃	1	80	yes	complemented
16	C₂²×C₃³	C₂×D₃	1	40	yes	—
17	C₃⋊(C₃⋊Dic₃)	—	6	80	no	—
18	C₃⁴	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
19	C₂³×C₃²	C₃⋊D₃	1	130	yes	—
20	C₂×C₃⋊Dic₃	—	9	260	no	—
21	C₂×C₃³	—	2	80	no	—
22	C₂×C₃³	C₂×Dic₃	1	40	yes	—
23	C₂×C₃³	#67	1	80	yes	—
24	#217	—	27	40	no	—
25	C₂²×C₃²	—	2	260	no	—
26	C₂²×C₃²	C₃⋊Dic₃	1	260	yes	complemented
27	C₂²×C₃²	C₂×C₃⋊D₃	1	130	yes	—
28	C₃⋊Dic₃	—	18	260	no	—
29	C₃³	#217	1	40	yes	complemented
30	C₂³×C₃	C₃⋊(C₃⋊D₃)	1	40	yes	—
31	C₂×Dic₃	—	27	80	no	—
32	C₂×C₃²	—	2	260	no	—
33	C₂×C₃²	C₂×C₃⋊Dic₃	1	130	yes	—
34	C₂×C₃²	#638	1	260	yes	—
35	C₂²⋊C₄	—	81	1	no	Sylow
36	C₂²×C₃	—	2	80	no	—
37	C₂²×C₃	C₃⋊(C₃⋊Dic₃)	1	80	yes	complemented
38	C₂²×C₃	C₂×C₃⋊(C₃⋊D₃)	1	40	yes	—
39	Dic₃	—	54	80	no	—
40	C₃²	#3757	1	130	yes	complemented
41	C₂³	C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
42	C₂×C₄	—	81	2	no	—
43	C₂×C₃	—	2	80	no	—
44	C₂×C₃	C₂×C₃⋊(C₃⋊Dic₃)	1	40	yes	—
45	C₂×C₃	#6312	1	80	yes	—
46	C₂²	—	2	2	no	—
47	C₂²	C₃⋊(C₃⋊(C₃⋊Dic₃))	1	2	yes	complemented
48	C₂²	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
49	C₄	—	162	2	no	—
50	C₃	#88864	1	40	yes	complemented
51	C₂	—	2	2	no	—
52	C₂	C₂×C₃⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic
53	C₂	#10620179	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂×C₃⋊(C₃⋊(C₃⋊D₃))
commutator	C₂×C₃⁴	C₂×C₄
Fitting, prime radical	C₂³×C₃⁴	C₂
socle, cosocle	C₂²×C₃⁴	C₂²
hypercenter	C₂³	C₃⋊(C₃⋊(C₃⋊D₃))
hypocenter	C₃⁴	C₂²⋊C₄
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×C₃⋊(C₃⋊(C₃⋊Dic₃))	C₂	semidirect
5. C₂²×C₃⁴	C₄	semidirect
15. C₂²×C₃³	Dic₃	semidirect
18. C₃⁴	C₂²⋊C₄	semidirect
26. C₂²×C₃²	C₃⋊Dic₃	semidirect
29. C₃³	#217	semidirect
37. C₂²×C₃	C₃⋊(C₃⋊Dic₃)	semidirect
40. C₃²	#3757	semidirect
47. C₂²	C₃⋊(C₃⋊(C₃⋊Dic₃))	semidirect
50. C₃	#88864	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₃⁴	2
C₂×C₃⋊(C₃⋊(C₃⋊Dic₃))	2
#88864	3