Subgroups — #11722198

Order 864.

364 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃³×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#88852	C₂	1	1	yes	maximal, characteristic
3	C₃⋊(C₃⋊(C₃⋊C₁₆))	C₂	1	1	yes	maximal, characteristic, complemented
4	#63655	—	3	13	no	maximal
5	C₃³×Q₈	—	2	1	no	—
6	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	1	no	—
7	C₃³×C₈	C₂²	1	1	yes	characteristic
8	C₃²×Q₁₆	D₃	1	13	yes	complemented
9	#3745	—	3	13	no	—
10	C₃⋊(C₃⋊C₁₆)	—	3	13	no	—
11	C₂×C₃⋊(C₃⋊D₃)	—	4	1	no	—
12	C₃³×C₄	—	4	1	no	—
13	C₃³×C₄	D₄	1	1	yes	characteristic
14	#828	—	9	13	no	—
15	C₃²×Q₈	—	2	13	no	—
16	C₄⋊(C₃⋊D₃)	—	6	13	no	—
17	C₃²×C₈	C₂×D₃	1	13	yes	—
18	C₂×C₃³	D₈	1	1	yes	characteristic
19	C₃⋊(C₃⋊D₃)	—	8	1	no	—
20	C₃×Q₁₆	C₃⋊D₃	1	13	yes	complemented
21	D₂₄	—	9	13	no	—
22	C₃⋊C₁₆	—	9	13	no	—
23	C₂×C₃⋊D₃	—	12	13	no	—
24	C₃²×C₄	—	4	13	no	—
25	C₃²×C₄	#67	1	13	yes	—
26	SD₃₂	—	27	1	no	Sylow
27	C₃³	SD₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	2	13	no	—
29	D₁₂	—	18	13	no	—
30	C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
31	C₂×C₃²	#213	1	13	yes	—
32	C₃⋊D₃	—	24	13	no	—
33	Q₁₆	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
34	D₈	—	27	1	no	—
35	C₁₆	—	27	1	no	—
36	C₂×D₃	—	36	13	no	—
37	C₃×C₄	—	4	13	no	—
38	C₃×C₄	#638	1	13	yes	—
39	C₃²	#828	1	13	yes	complemented
40	Q₈	—	2	1	no	—
41	D₄	—	54	1	no	—
42	C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
43	C₂×C₃	#3753	1	13	yes	—
44	D₃	—	72	13	no	—
45	C₂²	—	108	1	no	—
46	C₄	—	4	1	no	—
47	C₄	#6312	1	1	yes	characteristic
48	C₃	#63655	1	13	yes	complemented
49	C₂	—	216	1	no	—
50	C₂	#88860	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#88860
commutator, cosocle	C₃³×C₈	C₂²
Frattini	C₈	C₂×C₃⋊(C₃⋊D₃)
Fitting	C₃³×Q₁₆	C₂
socle	C₂×C₃³	D₈
prime radical	#88852	C₂
hypercenter	Q₁₆	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	SD₃₂
LCS	G ▷ C₃³×C₈ ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₃³×Q₁₆	C₂	semidirect
3. C₃⋊(C₃⋊(C₃⋊C₁₆))	C₂	semidirect
8. C₃²×Q₁₆	D₃	semidirect
20. C₃×Q₁₆	C₃⋊D₃	semidirect
27. C₃³	SD₃₂	semidirect
33. Q₁₆	C₃⋊(C₃⋊D₃)	semidirect
39. C₃²	#828	semidirect
48. C₃	#63655	semidirect

Maximal subgroups

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

subgroup	index
C₃³×Q₁₆	2
#88852	2
C₃⋊(C₃⋊(C₃⋊C₁₆))	2
#63655	3