Subgroups — #67442

Order 384.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#322	C₂	1	1	yes	maximal, characteristic
2	#4461	C₂	1	1	yes	maximal, characteristic
3	#4461	C₂	1	1	yes	maximal, characteristic
4	#1313	—	3	1	no	maximal, Sylow
5	C₃×C₄×C₈	C₄	1	2	yes	—
6	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊C₁₆	C₄	1	2	yes	—
8	C₂×C₃⋊C₁₆	C₄	1	2	yes	—
9	#346	—	3	1	no	—
10	#346	—	3	1	no	—
11	#322	D₃	1	1	yes	characteristic
12	C₂×C₃×C₈	—	2	2	no	—
13	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
14	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
15	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
16	C₃⋊C₁₆	—	2	2	no	—
17	C₃⋊C₁₆	C₈	1	4	yes	complemented
18	C₂×C₁₆	—	3	2	no	—
19	C₂×C₁₆	—	3	2	no	—
20	C₄×C₈	Dic₃	1	2	yes	—
21	C₄×C₈	C₂×D₃	1	1	yes	characteristic
22	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
23	C₂×C₃×C₄	M₁₆	1	2	yes	—
24	C₃×C₈	—	4	2	no	—
25	C₃×C₈	C₂×C₈	1	2	yes	—
26	C₃×C₈	M₁₆	1	2	yes	—
27	C₂×C₈	—	2	2	no	—
28	C₂×C₈	C₄×D₃	1	1	yes	characteristic
29	C₂×C₈	C₄×D₃	1	1	yes	characteristic
30	C₄²	C₂×Dic₃	1	1	yes	characteristic
31	C₁₆	—	3	4	no	—
32	C₁₆	—	6	2	no	—
33	C₂²×C₃	#97	1	1	yes	characteristic
34	C₃×C₄	—	2	2	no	—
35	C₃×C₄	C₄×C₈	1	1	yes	characteristic
36	C₃×C₄	#97	1	1	yes	characteristic
37	C₂×C₄	#208	1	2	yes	—
38	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
39	C₈	—	4	2	no	—
40	C₈	D₃×C₈	1	2	yes	—
41	C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
42	C₂×C₃	#322	1	1	yes	characteristic
43	C₂×C₃	#347	1	2	yes	—
44	C₂²	#803	1	1	yes	characteristic
45	C₄	—	2	2	no	—
46	C₄	C₈×Dic₃	1	1	yes	characteristic
47	C₄	#815	1	1	yes	characteristic
48	C₃	#1313	1	1	yes	characteristic, complemented, Sylow
49	C₂	#4454	1	1	yes	characteristic
50	C₂	#4466	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₃
commutator	C₃×C₄	C₄×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#322	C₂
socle, prime radical	C₂²×C₃	#97
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#322	D₃
hypocenter	C₃	#1313
LCS	G ▷ C₃×C₄ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ #322

Complements

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

N	H ≅ G/N	type
17. C₃⋊C₁₆	C₈	semidirect
48. C₃	#1313	semidirect

Maximal subgroups

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

subgroup	index
C₃×#322	2
#4461	2
#1313	3