Subgroups — #63543

Order 288.

57 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×D₃×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#812	—	3	1	no	maximal
5	#798	—	3	1	no	maximal
6	C₃×C₄×D₃	C₄	1	1	yes	characteristic
7	C₃²×C₈	C₂²	1	1	yes	characteristic
8	C₃×C₃⋊C₈	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈	—	3	1	no	—
10	D₃×C₈	D₃	1	1	yes	characteristic
11	C₃×C₁₆	—	3	1	no	—
12	C₃⋊C₁₆	—	3	1	no	—
13	C₃⋊C₁₆	—	3	1	no	—
14	C₃⋊C₁₆	—	6	1	no	—
15	C₃⋊C₁₆	D₃	1	1	yes	characteristic, complemented
16	C₂×C₃×D₃	C₈	1	1	yes	characteristic
17	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
18	C₃×Dic₃	C₈	1	1	yes	characteristic
19	M₃₂	—	9	1	no	Sylow
20	C₂×C₃×C₄	—	3	1	no	—
21	C₄×D₃	Dic₃	1	1	yes	characteristic
22	C₃×C₈	—	2	1	no	—
23	C₃×C₈	—	3	1	no	—
24	C₃×C₈	C₂×D₃	1	1	yes	characteristic
25	C₃×C₈	C₂×D₃	1	1	yes	characteristic
26	C₃⋊C₈	Dic₃	1	1	yes	characteristic
27	C₂×C₃²	C₂×C₈	1	1	yes	characteristic
28	C₃×D₃	—	2	1	no	—
29	C₂×C₈	—	3	1	no	—
30	C₁₆	—	3	1	no	—
31	C₁₆	—	9	1	no	—
32	C₂²×C₃	—	3	1	no	—
33	C₂×D₃	C₃⋊C₈	1	1	yes	characteristic
34	C₃×C₄	—	2	1	no	—
35	C₃×C₄	—	3	1	no	—
36	C₃×C₄	C₄×D₃	1	1	yes	characteristic
37	C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
38	Dic₃	C₃⋊C₈	1	1	yes	characteristic
39	C₃²	M₃₂	1	1	yes	characteristic, complemented, Sylow
40	C₂×C₄	—	3	1	no	—
41	C₈	—	3	1	no	—
42	C₈	D₃²	1	1	yes	characteristic
43	C₂×C₃	—	2	1	no	—
44	C₂×C₃	—	6	1	no	—
45	C₂×C₃	D₃×C₈	1	1	yes	characteristic
46	C₂×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
47	D₃	—	2	1	no	—
48	C₂²	—	3	1	no	—
49	C₄	—	3	1	no	—
50	C₄	D₃×Dic₃	1	1	yes	characteristic
51	C₃	—	2	1	no	—
52	C₃	#798	1	1	yes	characteristic, complemented
53	C₃	#812	1	1	yes	characteristic, complemented
54	C₂	—	6	1	no	—
55	C₂	D₃×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₈	D₃²
commutator, socle	C₂×C₃²	C₂×C₈
Fitting, cosocle	C₃²×C₈	C₂²
prime radical	C₂×C₃×D₃	C₈
hypocenter	C₃²	M₃₂
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₈

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₃⋊C₁₆)	C₂	semidirect
3. C₃×C₃⋊C₁₆	C₂	semidirect
15. C₃⋊C₁₆	D₃	semidirect
39. C₃²	M₃₂	semidirect
52. C₃	#798	semidirect
53. C₃	#812	semidirect

Maximal subgroups

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

subgroup	index
C₃×D₃×C₈	2
C₃⋊(C₃⋊C₁₆)	2
C₃×C₃⋊C₁₆	2
#812	3
#798	3