Subgroups — #13049153

Order 1296.

2120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⁴×C₄	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	1	2	yes	maximal, complemented
3	#88855	—	3	40	no	maximal
4	C₂²×C₃⁴	C₄	1	1	yes	characteristic
5	C₃⁴×C₄	C₄	1	1	yes	characteristic
6	C₃⁴×C₄	C₂²	1	1	yes	characteristic
7	C₂×C₃³×C₄	D₃	1	40	yes	—
8	C₃⋊(C₃⋊(C₃⋊C₈))	—	3	80	no	—
9	C₂×C₃⁴	—	2	1	no	—
10	C₂×C₃⁴	C₂×C₄	1	1	yes	characteristic
11	#3748	—	9	130	no	—
12	C₂²×C₃³	Dic₃	1	40	yes	—
13	C₃³×C₄	Dic₃	1	40	yes	—
14	C₃³×C₄	C₂×D₃	1	40	yes	—
15	C₃⁴	M₁₆	1	1	yes	characteristic, complemented, Sylow
16	C₂×C₃²×C₄	C₃⋊D₃	1	130	yes	—
17	C₃⋊(C₃⋊C₈)	—	9	260	no	—
18	C₂×C₃³	—	2	40	no	—
19	C₂×C₃³	C₂×Dic₃	1	40	yes	—
20	#208	—	27	40	no	—
21	C₂²×C₃²	C₃⋊Dic₃	1	130	yes	—
22	C₃²×C₄	C₃⋊Dic₃	1	130	yes	—
23	C₃²×C₄	C₂×C₃⋊D₃	1	130	yes	—
24	C₃³	#208	1	40	yes	complemented
25	C₂×C₃×C₄	C₃⋊(C₃⋊D₃)	1	40	yes	—
26	C₃⋊C₈	—	27	80	no	—
27	C₂×C₃²	—	2	130	no	—
28	C₂×C₃²	C₂×C₃⋊Dic₃	1	130	yes	—
29	M₁₆	—	81	1	no	Sylow
30	C₂²×C₃	C₃⋊(C₃⋊Dic₃)	1	40	yes	—
31	C₃×C₄	C₃⋊(C₃⋊Dic₃)	1	40	yes	—
32	C₃×C₄	C₂×C₃⋊(C₃⋊D₃)	1	40	yes	—
33	C₃²	#3748	1	130	yes	complemented
34	C₂×C₄	C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
35	C₈	—	81	2	no	—
36	C₂×C₃	—	2	40	no	—
37	C₂×C₃	C₂×C₃⋊(C₃⋊Dic₃)	1	40	yes	—
38	C₂²	C₃⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic
39	C₄	C₃⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic
40	C₄	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
41	C₃	#88855	1	40	yes	complemented
42	C₂	—	2	1	no	—
43	C₂	C₂×C₃⋊(C₃⋊(C₃⋊Dic₃))	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂×C₃⋊(C₃⋊(C₃⋊D₃))
commutator, socle	C₂×C₃⁴	C₂×C₄
Fitting	C₂×C₃⁴×C₄	C₂
prime radical	C₂²×C₃⁴	C₄
cosocle	C₃⁴×C₄	C₂²
hypercenter	C₂×C₄	C₃⋊(C₃⋊(C₃⋊D₃))
hypocenter	C₃⁴	M₁₆
LCS	G ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	semidirect
15. C₃⁴	M₁₆	semidirect
24. C₃³	#208	semidirect
33. C₃²	#3748	semidirect
41. C₃	#88855	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⁴×C₄	2
C₃⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	2
#88855	3