Subgroups — #67589

Order 384.

68 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₃₂	C₂	1	1	yes	maximal, characteristic
2	Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	#1397	—	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	C₄	1	2	yes	complemented
9	C₃⋊C₃₂	C₄	1	2	yes	complemented
10	C₂×C₃₂	—	3	1	no	—
11	C₂×C₃₂	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	C₂×C₄	1	1	yes	characteristic
17	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₃⋊C₁₆	C₈	1	2	yes	—
19	C₂×C₁₆	—	3	1	no	—
20	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
21	C₄×C₈	—	3	1	no	—
22	C₃₂	—	3	2	no	—
23	C₃₂	Dic₃	1	2	yes	complemented
24	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
25	C₂×Dic₃	C₁₆	1	2	yes	—
26	C₃×C₈	C₄²	1	1	yes	characteristic
27	C₃×C₈	C₂×C₈	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	—	3	2	no	—
33	C₁₆	C₄×D₃	1	1	yes	characteristic
34	C₁₆	C₂×Dic₃	1	1	yes	characteristic
35	C₂²×C₃	C₂×C₁₆	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	—	3	2	no	—
40	C₂×C₄	D₃×C₈	1	1	yes	characteristic
41	C₈	—	3	2	no	—
42	C₈	D₃×C₈	1	1	yes	characteristic
43	C₈	C₄×Dic₃	1	1	yes	characteristic
44	C₂×C₃	C₄×C₁₆	1	1	yes	characteristic
45	C₂×C₃	M₆₄	1	2	yes	—
46	C₂²	D₃×C₁₆	1	1	yes	characteristic
47	C₄	—	6	2	no	—
48	C₄	D₃×C₁₆	1	1	yes	characteristic
49	C₄	C₈×Dic₃	1	1	yes	characteristic
50	C₃	#1397	1	1	yes	characteristic, complemented, Sylow
51	C₂	#4447	1	2	yes	—
52	C₂	Dic₃×C₁₆	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂×D₃
commutator	C₂×C₃	C₄×C₁₆
Fitting	C₂×C₃×C₃₂	C₂
socle, prime radical	C₂²×C₃	C₂×C₁₆
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₂×C₃₂	D₃
hypocenter	C₃	#1397
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
8. C₃×C₃₂	C₄	semidirect
9. C₃⋊C₃₂	C₄	semidirect
23. C₃₂	Dic₃	semidirect
50. C₃	#1397	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₃₂	2
Dic₃×C₁₆	2
C₂×C₃⋊C₃₂	2
#1397	3