Subgroups — #67498

Order 384.

96 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	C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#1311	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₁₆	C₄	1	2	yes	—
6	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊C₁₆	C₄	1	2	yes	—
8	C₄×C₃⋊C₈	C₄	1	2	yes	—
9	C₄×C₁₆	—	3	1	no	—
10	C₄×C₁₆	D₃	1	1	yes	characteristic
11	C₈²	—	3	1	no	—
12	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
14	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃⋊C₈	C₈	1	4	yes	—
16	C₃×C₁₆	C₈	1	4	yes	complemented
17	C₃⋊C₁₆	C₈	1	4	yes	complemented
18	C₂×C₁₆	—	3	2	no	—
19	C₂×C₁₆	Dic₃	1	2	yes	—
20	C₄×C₈	—	3	2	no	—
21	C₄×C₈	C₂×D₃	1	1	yes	characteristic
22	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
23	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
24	C₃×C₈	C₂×C₈	1	2	yes	—
25	C₃×C₈	C₂×C₈	1	2	yes	—
26	C₃⋊C₈	—	2	4	no	—
27	C₂×C₈	—	3	4	no	—
28	C₂×C₈	C₄×D₃	1	1	yes	characteristic
29	C₂×C₈	C₂×Dic₃	1	1	yes	characteristic
30	C₄²	C₄×D₃	1	1	yes	characteristic
31	C₁₆	—	3	4	no	—
32	C₁₆	C₃⋊C₈	1	4	yes	complemented
33	C₂²×C₃	C₄×C₈	1	1	yes	characteristic
34	C₃×C₄	C₄×C₈	1	1	yes	characteristic
35	C₃×C₄	C₄×C₈	1	1	yes	characteristic
36	C₃×C₄	M₃₂	1	4	yes	—
37	C₂×C₄	D₃×C₈	1	2	yes	—
38	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
39	C₈	—	6	4	no	—
40	C₈	D₃×C₈	1	2	yes	—
41	C₈	C₂×C₃⋊C₈	1	2	yes	—
42	C₂×C₃	C₈²	1	1	yes	characteristic
43	C₂×C₃	#346	1	2	yes	—
44	C₂²	C₈×Dic₃	1	1	yes	characteristic
45	C₄	#798	1	4	yes	—
46	C₄	C₄×C₃⋊C₈	1	1	yes	characteristic
47	C₄	C₈×Dic₃	1	1	yes	characteristic
48	C₃	#1311	1	1	yes	characteristic, complemented, Sylow
49	C₂	C₈×C₃⋊C₈	1	1	yes	characteristic
50	C₂	#4502	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄×C₈	C₂×D₃
commutator	C₂×C₃	C₈²
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₄×C₈
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#1311
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
16. C₃×C₁₆	C₈	semidirect
17. C₃⋊C₁₆	C₈	semidirect
32. C₁₆	C₃⋊C₈	semidirect
48. C₃	#1311	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
C₄×C₃⋊C₁₆	2
C₈×C₃⋊C₈	2
#1311	3