Subgroups — #67599

Order 384.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×M₆₄	C₂	1	1	yes	maximal, characteristic
2	Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#4499	C₂	1	1	yes	maximal, characteristic
4	#1422	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
6	C₈×Dic₃	C₄	1	1	yes	characteristic
7	C₂×C₃⋊C₁₆	C₄	1	1	yes	characteristic
8	C₃×C₃₂	—	2	1	no	—
9	C₃⋊C₃₂	—	2	1	no	—
10	M₆₄	—	3	1	no	—
11	M₆₄	D₃	1	1	yes	characteristic
12	C₄×C₁₆	—	3	1	no	—
13	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
14	C₄×Dic₃	C₈	1	1	yes	characteristic
15	C₂×C₃⋊C₈	C₈	1	1	yes	characteristic
16	C₃×C₁₆	D₄	1	1	yes	characteristic
17	C₃×C₁₆	Q₈	1	1	yes	characteristic
18	C₃⋊C₁₆	—	2	1	no	—
19	C₂×C₁₆	—	3	1	no	—
20	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
21	C₄×C₈	—	3	1	no	—
22	C₃₂	—	2	1	no	—
23	C₃₂	—	6	1	no	—
24	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
25	C₂×Dic₃	—	2	1	no	—
26	C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
27	C₃×C₈	M₁₆	1	1	yes	characteristic
28	C₃⋊C₈	C₁₆	1	2	yes	—
29	C₂×C₈	—	3	1	no	—
30	C₂×C₈	C₄×D₃	1	1	yes	characteristic
31	C₄²	—	3	1	no	—
32	C₁₆	—	6	1	no	—
33	C₁₆	Dic₆	1	1	yes	characteristic
34	C₁₆	#67	1	1	yes	characteristic
35	C₂²×C₃	M₃₂	1	1	yes	characteristic
36	C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
37	C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
38	Dic₃	—	2	2	no	—
39	C₂×C₄	—	6	1	no	—
40	C₂×C₄	D₃×C₈	1	1	yes	characteristic
41	C₈	—	3	2	no	—
42	C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
43	C₈	Dic₃⋊C₄	1	1	yes	characteristic
44	C₂×C₃	—	2	1	no	—
45	C₂×C₃	C₄⋊C₁₆	1	1	yes	characteristic
46	C₂²	#798	1	1	yes	characteristic
47	C₄	—	6	2	no	—
48	C₄	D₃×C₁₆	1	1	yes	characteristic
49	C₄	Dic₃⋊C₈	1	1	yes	characteristic
50	C₃	#1422	1	1	yes	characteristic, complemented, Sylow
51	C₂	—	2	1	no	—
52	C₂	Dic₃⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#67
commutator	C₃×C₄	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₃×M₆₄	C₂
socle	C₂×C₃	C₄⋊C₁₆
prime radical	C₂²×C₃	M₃₂
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	M₆₄	D₃
hypocenter	C₃	#1422
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
50. C₃	#1422	semidirect

Maximal subgroups

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

subgroup	index
C₃×M₆₄	2
Dic₃×C₁₆	2
#4499	2
#1422	3