Subgroups — #63576

Order 288.

83 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄⋊Dic₃	C₂	1	2	yes	maximal
3	#807	—	3	2	no	maximal
4	C₂×C₃²×C₄	C₂²	1	1	yes	characteristic
5	C₂×C₃×Dic₃	—	2	2	no	—
6	C₃⋊(C₃⋊C₈)	C₄	1	2	yes	complemented
7	C₃×C₄⋊C₄	—	3	2	no	—
8	C₄⋊Dic₃	D₃	1	2	yes	—
9	C₂×C₃⋊C₈	—	3	2	no	—
10	C₂×C₃⋊C₈	—	6	1	no	—
11	C₂²×C₃²	D₄	1	1	yes	characteristic
12	C₃²×C₄	C₂×C₄	1	1	yes	characteristic
13	C₃²×C₄	Q₈	1	1	yes	characteristic
14	C₃×Dic₃	—	4	2	no	—
15	#107	—	9	1	no	Sylow
16	C₂×C₃×C₄	—	2	1	no	—
17	C₂×C₃×C₄	—	6	2	no	—
18	C₂×C₃×C₄	C₂×D₃	1	2	yes	—
19	C₂×Dic₃	—	2	2	no	—
20	C₃⋊C₈	—	3	4	no	—
21	C₃⋊C₈	—	6	2	no	—
22	C₂×C₃²	C₄⋊C₄	1	1	yes	characteristic
23	C₂×C₃²	D₈	1	1	yes	characteristic
24	C₂×C₃²	Q₁₆	1	1	yes	characteristic
25	C₂×C₈	—	9	1	no	—
26	C₄⋊C₄	—	3	2	no	—
27	C₂²×C₃	—	2	1	no	—
28	C₂²×C₃	#67	1	2	yes	—
29	C₃×C₄	—	2	1	no	—
30	C₃×C₄	—	2	1	no	—
31	C₃×C₄	—	12	2	no	—
32	C₃×C₄	Dic₆	1	2	yes	—
33	C₃×C₄	C₄×D₃	1	2	yes	—
34	Dic₃	—	4	2	no	—
35	C₃²	#107	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₄	—	6	2	no	—
37	C₂×C₄	D₃²	1	1	yes	characteristic
38	C₈	—	9	2	no	—
39	C₂×C₃	—	2	1	no	—
40	C₂×C₃	—	2	1	no	—
41	C₂×C₃	—	2	1	no	—
42	C₂×C₃	Dic₃⋊C₄	1	2	yes	—
43	C₂×C₃	#213	1	2	yes	—
44	C₂×C₃	#216	1	2	yes	—
45	C₂²	#625	1	1	yes	characteristic
46	C₄	—	12	2	no	—
47	C₄	#624	1	1	yes	characteristic
48	C₄	#627	1	1	yes	characteristic
49	C₃	—	2	1	no	—
50	C₃	#807	1	2	yes	complemented
51	C₂	#3713	1	1	yes	characteristic
52	C₂	#3718	1	1	yes	characteristic
53	C₂	#3724	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#625
commutator	C₃²×C₄	C₂×C₄
Frattini, hypercenter	C₂×C₄	D₃²
Fitting, cosocle	C₂×C₃²×C₄	C₂²
socle, prime radical	C₂²×C₃²	D₄
hypocenter	C₃²	#107
LCS	G ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
6. C₃⋊(C₃⋊C₈)	C₄	semidirect
35. C₃²	#107	semidirect
50. C₃	#807	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃⋊(C₃⋊C₈)	2
C₃×C₄⋊Dic₃	2
#807	3