Subgroups — #13049160

Order 1296.

2120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄⋊(C₃⋊(C₃⋊(C₃⋊D₃)))	C₂	1	1	yes	maximal, characteristic
3	C₃⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	1	1	yes	maximal, characteristic, complemented
4	#88862	—	3	40	no	maximal
5	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	—	2	1	no	—
6	C₃⁴×C₄	—	2	1	no	—
7	C₃⁴×C₄	C₂²	1	1	yes	characteristic
8	C₃³×Q₈	D₃	1	40	yes	complemented
9	C₄⋊(C₃⋊(C₃⋊D₃))	—	3	40	no	—
10	C₃⋊(C₃⋊(C₃⋊C₈))	—	3	40	no	—
11	C₂×C₃⁴	D₄	1	1	yes	characteristic
12	C₃⋊(C₃⋊(C₃⋊D₃))	—	4	1	no	—
13	#3755	—	9	130	no	—
14	C₂×C₃⋊(C₃⋊D₃)	—	6	40	no	—
15	C₃³×C₄	—	2	40	no	—
16	C₃³×C₄	C₂×D₃	1	40	yes	—
17	C₃⁴	SD₁₆	1	1	yes	characteristic, complemented, Sylow
18	C₃²×Q₈	C₃⋊D₃	1	130	yes	complemented
19	C₄⋊(C₃⋊D₃)	—	9	130	no	—
20	C₃⋊(C₃⋊C₈)	—	9	130	no	—
21	C₂×C₃³	#67	1	40	yes	—
22	C₃⋊(C₃⋊D₃)	—	12	40	no	—
23	#215	—	27	40	no	—
24	C₂×C₃⋊D₃	—	18	130	no	—
25	C₃²×C₄	—	2	130	no	—
26	C₃²×C₄	C₂×C₃⋊D₃	1	130	yes	—
27	C₃³	#215	1	40	yes	complemented
28	C₃×Q₈	C₃⋊(C₃⋊D₃)	1	40	yes	complemented
29	D₁₂	—	27	40	no	—
30	C₃⋊C₈	—	27	40	no	—
31	C₂×C₃²	#638	1	130	yes	—
32	C₃⋊D₃	—	36	130	no	—
33	SD₁₆	—	81	1	no	Sylow
34	C₂×D₃	—	54	40	no	—
35	C₃×C₄	—	2	40	no	—
36	C₃×C₄	C₂×C₃⋊(C₃⋊D₃)	1	40	yes	—
37	C₃²	#3755	1	130	yes	complemented
38	Q₈	C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic, complemented
39	D₄	—	81	1	no	—
40	C₈	—	81	1	no	—
41	C₂×C₃	#6312	1	40	yes	—
42	D₃	—	108	40	no	—
43	C₂²	—	162	1	no	—
44	C₄	—	2	1	no	—
45	C₄	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
46	C₃	#88862	1	40	yes	complemented
47	C₂	—	324	1	no	—
48	C₂	#10620179	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#10620179
commutator, cosocle	C₃⁴×C₄	C₂²
Frattini	C₄	C₂×C₃⋊(C₃⋊(C₃⋊D₃))
Fitting	C₃⁴×Q₈	C₂
socle	C₂×C₃⁴	D₄
prime radical	C₄⋊(C₃⋊(C₃⋊(C₃⋊D₃)))	C₂
hypercenter	Q₈	C₃⋊(C₃⋊(C₃⋊D₃))
hypocenter	C₃⁴	SD₁₆
LCS	G ▷ C₃⁴×C₄ ▷ C₂×C₃⁴ ▷ C₃⁴
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
1. C₃⁴×Q₈	C₂	semidirect
3. C₃⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	C₂	semidirect
8. C₃³×Q₈	D₃	semidirect
17. C₃⁴	SD₁₆	semidirect
18. C₃²×Q₈	C₃⋊D₃	semidirect
27. C₃³	#215	semidirect
28. C₃×Q₈	C₃⋊(C₃⋊D₃)	semidirect
37. C₃²	#3755	semidirect
38. Q₈	C₃⋊(C₃⋊(C₃⋊D₃))	semidirect
46. C₃	#88862	semidirect

Maximal subgroups

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

subgroup	index
C₃⁴×Q₈	2
C₄⋊(C₃⋊(C₃⋊(C₃⋊D₃)))	2
C₃⋊(C₃⋊(C₃⋊(C₃⋊C₈)))	2
#88862	3