Subgroups — #67576

Order 384.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#366	C₂	1	1	yes	maximal, characteristic
2	#4504	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	#1424	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
6	C₃×#107	—	2	1	no	—
7	#818	—	2	1	no	—
8	C₃⋊C₃₂	C₄	1	2	yes	complemented
9	C₂×C₃₂	—	3	1	no	—
10	#366	—	3	1	no	—
11	#366	D₃	1	1	yes	characteristic
12	C₂×C₃×C₈	D₄	1	1	yes	characteristic
13	C₃×C₄⋊C₄	—	4	1	no	—
14	C₄⋊Dic₃	—	4	1	no	—
15	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
16	C₃×C₁₆	Q₈	1	1	yes	characteristic
17	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
18	#107	—	2	1	no	—
19	#107	—	6	1	no	—
20	C₃₂	—	3	2	no	—
21	C₂×C₃×C₄	—	8	1	no	—
22	C₂×C₃×C₄	D₈	1	1	yes	characteristic
23	C₂×Dic₃	—	8	1	no	—
24	C₃×C₈	C₄⋊C₄	1	1	yes	characteristic
25	C₃×C₈	Q₁₆	1	1	yes	characteristic
26	C₂×C₈	#67	1	1	yes	characteristic
27	C₄⋊C₄	—	4	1	no	—
28	C₄⋊C₄	—	12	1	no	—
29	C₁₆	Dic₆	1	1	yes	characteristic
30	C₁₆	C₄×D₃	1	1	yes	characteristic
31	C₂²×C₃	D₁₆	1	1	yes	characteristic
32	C₃×C₄	—	16	1	no	—
33	C₃×C₄	#107	1	1	yes	characteristic
34	C₃×C₄	Q₃₂	1	1	yes	characteristic
35	Dic₃	—	16	1	no	—
36	C₂×C₄	—	8	1	no	—
37	C₂×C₄	—	24	1	no	—
38	C₂×C₄	#213	1	1	yes	characteristic
39	C₈	Dic₃⋊C₄	1	1	yes	characteristic
40	C₈	#216	1	1	yes	characteristic
41	C₂×C₃	#366	1	1	yes	characteristic
42	C₂×C₃	SD₆₄	1	1	yes	characteristic
43	C₂×C₃	SD₆₄	1	1	yes	characteristic
44	C₂²	#826	1	1	yes	characteristic
45	C₄	—	16	1	no	—
46	C₄	—	48	1	no	—
47	C₄	#807	1	1	yes	characteristic
48	C₄	#829	1	1	yes	characteristic
49	C₃	#1424	1	1	yes	characteristic, complemented, Sylow
50	C₂	#4489	1	1	yes	characteristic
51	C₂	#4520	1	1	yes	characteristic
52	C₂	#4521	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#826
commutator	C₃×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₃×#366	C₂
socle, prime radical	C₂²×C₃	D₁₆
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	#366	D₃
hypocenter	C₃	#1424
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #366

Complements

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

N	H ≅ G/N	type
8. C₃⋊C₃₂	C₄	semidirect
49. C₃	#1424	semidirect

Maximal subgroups

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

subgroup	index
C₃×#366	2
#4504	2
C₂×C₃⋊C₃₂	2
#1424	3