Subgroups — #71451

Order 384.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#504	C₂	1	1	yes	maximal, characteristic
2	#4499	C₂	1	3	yes	maximal, complemented
3	C₂×C₃⋊C₃₂	C₂	1	3	yes	maximal, complemented
4	#2258	—	3	1	no	maximal, Sylow
5	C₃×#131	C₄	1	1	yes	characteristic
6	C₃×M₃₂	C₄	1	3	yes	—
7	C₂×C₃×C₁₆	C₂²	1	3	yes	—
8	C₃⋊C₃₂	C₂²	1	1	yes	characteristic
9	C₃⋊C₃₂	C₂²	1	3	yes	—
10	#504	D₃	1	1	yes	characteristic
11	M₆₄	—	3	3	no	—
12	C₂×C₃₂	—	3	3	no	—
13	C₃×#41	C₈	1	1	yes	characteristic
14	C₃×M₁₆	C₈	1	3	yes	—
15	C₂×C₃×C₈	C₂×C₄	1	3	yes	—
16	C₃×C₁₆	C₂×C₄	1	3	yes	—
17	C₃×C₁₆	C₂³	1	1	yes	characteristic
18	#131	Dic₃	1	1	yes	characteristic
19	M₃₂	Dic₃	1	3	yes	—
20	C₂×C₁₆	C₂×D₃	1	3	yes	—
21	C₃₂	—	3	1	no	—
22	C₃₂	—	3	3	no	—
23	C₃×Q₈	C₁₆	1	1	yes	characteristic
24	C₃×D₄	C₁₆	1	3	yes	—
25	C₂×C₃×C₄	C₂×C₈	1	3	yes	—
26	C₃×C₈	C₂×C₈	1	3	yes	—
27	C₃×C₈	C₂²×C₄	1	1	yes	characteristic
28	#41	C₃⋊C₈	1	1	yes	characteristic
29	M₁₆	C₃⋊C₈	1	3	yes	—
30	C₂×C₈	C₂×Dic₃	1	3	yes	—
31	C₁₆	C₂×Dic₃	1	3	yes	—
32	C₁₆	C₂²×D₃	1	1	yes	characteristic
33	C₂²×C₃	C₂×C₁₆	1	3	yes	—
34	C₃×C₄	C₂×C₁₆	1	3	yes	—
35	C₃×C₄	C₂²×C₈	1	1	yes	characteristic
36	Q₈	C₃⋊C₁₆	1	1	yes	characteristic
37	D₄	C₃⋊C₁₆	1	3	yes	—
38	C₂×C₄	C₂×C₃⋊C₈	1	3	yes	—
39	C₈	C₂×C₃⋊C₈	1	3	yes	—
40	C₈	C₂²×Dic₃	1	1	yes	characteristic
41	C₂×C₃	—	2	3	no	—
42	C₂×C₃	C₂²×C₁₆	1	1	yes	characteristic
43	C₂²	C₂×C₃⋊C₁₆	1	3	yes	—
44	C₄	C₂×C₃⋊C₁₆	1	3	yes	—
45	C₄	C₂²×C₃⋊C₈	1	1	yes	characteristic
46	C₃	#2258	1	1	yes	characteristic, complemented, Sylow
47	C₂	—	2	3	no	—
48	C₂	C₂²×C₃⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₃
commutator, socle	C₂×C₃	C₂²×C₁₆
Fitting	C₃×#504	C₂
prime radical	C₃×#41	C₈
cosocle	C₃×C₁₆	C₂³
hypercenter	#504	D₃
hypocenter	C₃	#2258
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₁₆ ◁ #504

Complements

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

N	H ≅ G/N	type
2. #4499	C₂	semidirect
3. C₂×C₃⋊C₃₂	C₂	semidirect
46. C₃	#2258	semidirect

Maximal subgroups

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

subgroup	index
C₃×#504	2
#4499	2
C₂×C₃⋊C₃₂	2
#2258	3