Subgroups — #67606

Order 384.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4904	C₂	1	1	yes	maximal, characteristic
2	C₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#4505	C₂	1	1	yes	maximal, characteristic, complemented
4	#1418	—	3	1	no	maximal, Sylow
5	#904	—	2	1	no	—
6	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
7	#818	—	2	1	no	—
8	Dic₂₄	C₄	1	1	yes	characteristic, complemented
9	#800	—	2	1	no	—
10	D₄₈	C₄	1	1	yes	characteristic, complemented
11	C₃×C₃₂	—	2	1	no	—
12	#508	—	3	1	no	—
13	M₆₄	D₃	1	1	yes	characteristic, complemented
14	#367	—	3	1	no	—
15	#235	—	4	1	no	—
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₄⋊Dic₃	—	4	1	no	—
18	Dic₁₂	—	2	1	no	—
19	D₂₄	—	2	1	no	—
20	#204	—	4	1	no	—
21	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₃×C₁₆	D₄	1	1	yes	characteristic
23	#135	—	6	1	no	—
24	Q₃₂	—	3	1	no	—
25	SD₃₂	—	6	1	no	—
26	D₁₆	—	3	1	no	—
27	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
28	#107	—	6	1	no	—
29	C₃₂	—	2	1	no	—
30	C₂×C₃×C₄	D₈	1	1	yes	characteristic
31	#67	—	8	1	no	—
32	C₂×Dic₃	—	8	1	no	—
33	D₁₂	—	4	1	no	—
34	C₄×D₃	—	8	1	no	—
35	Dic₆	—	4	1	no	—
36	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
37	C₃×C₈	SD₁₆	1	1	yes	characteristic
38	#41	—	12	1	no	—
39	Q₁₆	—	6	1	no	—
40	SD₁₆	—	12	1	no	—
41	D₈	—	6	1	no	—
42	C₂×C₈	D₁₂	1	1	yes	characteristic
43	C₄⋊C₄	—	12	1	no	—
44	C₁₆	C₄×D₃	1	1	yes	characteristic
45	C₁₆	#67	1	1	yes	characteristic
46	C₂²×C₃	D₁₆	1	1	yes	characteristic
47	C₂×D₃	—	8	1	no	—
48	C₃×C₄	#102	1	1	yes	characteristic
49	C₃×C₄	SD₃₂	1	1	yes	characteristic
50	Dic₃	—	8	1	no	—
51	Dic₃	—	16	1	no	—
52	Q₈	—	12	1	no	—
53	D₄	—	12	1	no	—
54	D₄	—	24	1	no	—
55	C₂×C₄	—	24	1	no	—
56	C₂×C₄	—	24	1	no	—
57	C₂×C₄	D₂₄	1	1	yes	characteristic
58	C₈	#204	1	1	yes	characteristic
59	C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
60	C₂×C₃	—	2	1	no	—
61	C₂×C₃	#357	1	1	yes	characteristic
62	D₃	—	16	1	no	—
63	C₂²	—	24	1	no	—
64	C₂²	D₄₈	1	1	yes	characteristic
65	C₄	—	24	1	no	—
66	C₄	—	48	1	no	—
67	C₄	#800	1	1	yes	characteristic
68	C₄	#821	1	1	yes	characteristic
69	C₃	#1418	1	1	yes	characteristic, complemented, Sylow
70	C₂	—	2	1	no	—
71	C₂	—	48	1	no	—
72	C₂	#4509	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4509
commutator	C₃×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₃×M₆₄	C₂
socle	C₂×C₃	#357
prime radical	#4904	C₂
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	M₆₄	D₃
hypocenter	C₃	#1418
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. C₃×M₆₄	C₂	semidirect
3. #4505	C₂	semidirect
8. Dic₂₄	C₄	semidirect
10. D₄₈	C₄	semidirect
13. M₆₄	D₃	semidirect
69. C₃	#1418	semidirect

Maximal subgroups

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

subgroup	index
#4904	2
C₃×M₆₄	2
#4505	2
#1418	3