Subgroups — #67508

Order 384.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃⋊C₃₂	C₂	1	2	yes	maximal
3	#1397	—	3	1	no	maximal, Sylow
4	C₂×C₃×C₁₆	C₄	1	1	yes	characteristic
5	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
6	C₃×C₄×C₈	C₄	1	1	yes	characteristic
7	C₃⋊C₃₂	C₄	1	4	yes	complemented
8	C₂×C₃₂	—	3	2	no	—
9	C₄×C₁₆	D₃	1	1	yes	characteristic
10	C₂×C₃×C₈	C₈	1	1	yes	characteristic
11	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
12	C₃×C₄²	C₈	1	1	yes	characteristic
13	C₃×C₁₆	C₈	1	2	yes	—
14	C₃×C₁₆	C₂×C₄	1	2	yes	—
15	C₂×C₁₆	Dic₃	1	1	yes	characteristic
16	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
17	C₄×C₈	Dic₃	1	1	yes	characteristic
18	C₃₂	—	3	4	no	—
19	C₂×C₃×C₄	C₁₆	1	2	yes	—
20	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
21	C₃×C₈	C₁₆	1	2	yes	—
22	C₃×C₈	C₄²	1	1	yes	characteristic
23	C₃×C₈	C₂×C₈	1	1	yes	characteristic
24	C₂×C₈	C₃⋊C₈	1	1	yes	characteristic
25	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
26	C₄²	C₃⋊C₈	1	1	yes	characteristic
27	C₁₆	C₃⋊C₈	1	2	yes	—
28	C₁₆	C₄×D₃	1	2	yes	—
29	C₂²×C₃	C₂×C₁₆	1	1	yes	characteristic
30	C₃×C₄	—	2	2	no	—
31	C₃×C₄	C₄×C₈	1	1	yes	characteristic
32	C₃×C₄	C₂×C₁₆	1	1	yes	characteristic
33	C₂×C₄	C₃⋊C₁₆	1	2	yes	—
34	C₂×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
35	C₈	C₃⋊C₁₆	1	2	yes	—
36	C₈	C₂×C₃⋊C₈	1	1	yes	characteristic
37	C₈	C₄×Dic₃	1	1	yes	characteristic
38	C₂×C₃	C₄×C₁₆	1	1	yes	characteristic
39	C₂×C₃	M₆₄	1	2	yes	—
40	C₂²	C₂×C₃⋊C₁₆	1	1	yes	characteristic
41	C₄	—	2	2	no	—
42	C₄	C₄×C₃⋊C₈	1	1	yes	characteristic
43	C₄	C₂×C₃⋊C₁₆	1	1	yes	characteristic
44	C₃	#1397	1	1	yes	characteristic, complemented, Sylow
45	C₂	C₄×C₃⋊C₁₆	1	1	yes	characteristic
46	C₂	#4499	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₃
commutator	C₂×C₃	C₄×C₁₆
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×C₁₆
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#1397
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
7. C₃⋊C₃₂	C₄	semidirect
44. C₃	#1397	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₁₆	2
C₂×C₃⋊C₃₂	2
#1397	3