Subgroups — #67573

Order 384.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4683	C₂	1	1	yes	maximal, characteristic
2	C₃×#364	C₂	1	1	yes	maximal, characteristic
3	#4463	C₂	1	1	yes	maximal, characteristic
4	#1366	—	3	1	no	maximal, Sylow
5	#870	C₄	1	2	yes	—
6	C₃×M₃₂	—	2	1	no	—
7	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	#818	—	2	1	no	—
9	#817	—	2	1	no	—
10	#812	—	2	1	no	—
11	#499	—	3	1	no	—
12	#364	—	3	1	no	—
13	#364	D₃	1	1	yes	characteristic
14	C₂×C₃×C₈	D₄	1	1	yes	characteristic
15	C₂×C₃×C₈	D₄	1	1	yes	characteristic
16	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
17	C₄⋊Dic₃	—	2	2	no	—
18	Dic₃⋊C₄	—	4	2	no	—
19	C₃×C₁₆	—	4	1	no	—
20	C₃⋊C₁₆	—	4	1	no	—
21	#125	—	3	2	no	—
22	M₃₂	—	2	1	no	—
23	M₃₂	—	6	1	no	—
24	#107	—	6	1	no	—
25	#106	—	6	1	no	—
26	C₄×C₈	C₂×D₃	1	1	yes	characteristic
27	C₂×C₃×C₄	—	2	1	no	—
28	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×Dic₃	—	4	2	no	—
30	C₃×C₈	SD₁₆	1	1	yes	characteristic
31	C₃×C₈	SD₁₆	1	1	yes	characteristic
32	C₃×C₈	Q₁₆	1	1	yes	characteristic
33	C₃×C₈	Q₁₆	1	1	yes	characteristic
34	C₂×C₈	D₁₂	1	1	yes	characteristic
35	C₂×C₈	#67	1	1	yes	characteristic
36	C₄⋊C₄	—	6	2	no	—
37	C₄⋊C₄	—	12	2	no	—
38	C₄²	C₄×D₃	1	1	yes	characteristic
39	C₁₆	—	4	1	no	—
40	C₁₆	—	12	1	no	—
41	C₂²×C₃	#100	1	1	yes	characteristic
42	C₃×C₄	—	4	1	no	—
43	C₃×C₄	#103	1	1	yes	characteristic
44	C₃×C₄	#103	1	1	yes	characteristic
45	Dic₃	—	8	2	no	—
46	C₂×C₄	—	2	1	no	—
47	C₂×C₄	—	12	2	no	—
48	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
49	C₈	#204	1	1	yes	characteristic
50	C₈	Dic₁₂	1	1	yes	characteristic
51	C₈	#214	1	1	yes	characteristic
52	C₈	#216	1	1	yes	characteristic
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	#333	1	1	yes	characteristic
55	C₂²	#823	1	1	yes	characteristic
56	C₄	—	4	1	no	—
57	C₄	—	24	2	no	—
58	C₄	#810	1	1	yes	characteristic
59	C₄	#816	1	1	yes	characteristic
60	C₃	#1366	1	1	yes	characteristic, complemented, Sylow
61	C₂	—	2	1	no	—
62	C₂	#4481	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4481
commutator	C₃×C₄²	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#364	C₂
socle	C₂×C₃	#333
prime radical	C₂²×C₃	#100
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#364	D₃
hypocenter	C₃	#1366
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

Complements

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

N	H ≅ G/N	type
60. C₃	#1366	semidirect

Maximal subgroups

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

subgroup	index
#4683	2
C₃×#364	2
#4463	2
#1366	3