Subgroups — #67563

Order 384.

80 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×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#4457	C₂	1	1	yes	maximal, characteristic
4	#1363	—	3	1	no	maximal, Sylow
5	#870	—	2	1	no	—
6	C₂×C₃×C₁₆	—	2	1	no	—
7	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	#818	C₄	1	1	yes	characteristic
9	#818	C₄	1	1	yes	characteristic
10	#817	—	2	1	no	—
11	C₄⋊(C₃⋊C₈)	—	2	1	no	—
12	#499	—	3	1	no	—
13	C₄⋊C₁₆	D₃	1	1	yes	characteristic
14	#334	—	3	1	no	—
15	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₃×C₄²	D₄	1	1	yes	characteristic
18	C₄⋊Dic₃	—	2	1	no	—
19	C₄⋊Dic₃	—	2	1	no	—
20	Dic₃⋊C₄	—	4	1	no	—
21	Dic₃⋊C₄	—	4	1	no	—
22	C₂×C₃⋊C₈	—	4	1	no	—
23	C₃×C₁₆	—	4	1	no	—
24	#125	—	6	1	no	—
25	C₂×C₁₆	—	2	1	no	—
26	#107	—	3	1	no	—
27	#107	—	3	1	no	—
28	#106	—	6	1	no	—
29	C₄⋊C₈	—	6	1	no	—
30	C₄×C₈	C₂×D₃	1	1	yes	characteristic
31	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₃×C₄	D₈	1	1	yes	characteristic
33	C₂×C₃×C₄	SD₁₆	1	1	yes	characteristic
34	C₂×Dic₃	—	4	1	no	—
35	C₂×Dic₃	—	4	1	no	—
36	C₃×C₈	—	2	1	no	—
37	C₃×C₈	SD₁₆	1	1	yes	characteristic
38	C₃×C₈	Q₁₆	1	1	yes	characteristic
39	C₃⋊C₈	—	8	1	no	—
40	C₂×C₈	—	12	1	no	—
41	C₂×C₈	C₄×D₃	1	1	yes	characteristic
42	C₂×C₈	#67	1	1	yes	characteristic
43	C₄⋊C₄	—	6	1	no	—
44	C₄⋊C₄	—	6	1	no	—
45	C₄⋊C₄	—	12	1	no	—
46	C₄⋊C₄	—	12	1	no	—
47	C₄²	D₁₂	1	1	yes	characteristic
48	C₁₆	—	4	1	no	—
49	C₂²×C₃	#102	1	1	yes	characteristic
50	C₃×C₄	—	2	1	no	—
51	C₃×C₄	—	2	1	no	—
52	C₃×C₄	#101	1	1	yes	characteristic
53	C₃×C₄	#103	1	1	yes	characteristic
54	Dic₃	—	8	1	no	—
55	Dic₃	—	8	1	no	—
56	C₂×C₄	—	12	1	no	—
57	C₂×C₄	—	12	1	no	—
58	C₂×C₄	#204	1	1	yes	characteristic
59	C₂×C₄	D₂₄	1	1	yes	characteristic
60	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
61	C₈	—	2	1	no	—
62	C₈	—	24	1	no	—
63	C₈	#214	1	1	yes	characteristic
64	C₈	#216	1	1	yes	characteristic
65	C₂×C₃	#332	1	1	yes	characteristic
66	C₂×C₃	#359	1	1	yes	characteristic
67	C₂×C₃	#360	1	1	yes	characteristic
68	C₂²	#821	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	1	no	—
71	C₄	—	24	1	no	—
72	C₄	—	24	1	no	—
73	C₄	#810	1	1	yes	characteristic
74	C₄	#824	1	1	yes	characteristic
75	C₃	#1363	1	1	yes	characteristic, complemented, Sylow
76	C₂	#4486	1	1	yes	characteristic
77	C₂	#4510	1	1	yes	characteristic
78	C₂	#4518	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#821
commutator	C₂×C₃×C₈	C₂×C₄
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂²×C₃	#102
cosocle	C₃×C₄×C₈	C₂²
hypercenter	C₄⋊C₁₆	D₃
hypocenter	C₃	#1363
LCS	G ▷ C₂×C₃×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
75. C₃	#1363	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×C₄⋊C₁₆	2
#4457	2
#1363	3