Subgroups — #67470

Order 384.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#335	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#4458	C₂	1	1	yes	maximal, characteristic
4	#1372	—	3	1	no	maximal, Sylow
5	C₃×C₄⋊C₈	—	2	1	no	—
6	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊C₁₆	C₄	1	1	yes	characteristic
8	C₂×C₃⋊C₁₆	C₄	1	1	yes	characteristic
9	C₄⋊(C₃⋊C₈)	—	2	1	no	—
10	C₄×C₁₆	—	3	1	no	—
11	#335	—	3	1	no	—
12	#335	D₃	1	1	yes	characteristic
13	C₂×C₃×C₈	—	4	1	no	—
14	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₃×C₈	Q₈	1	1	yes	characteristic
16	C₃×C₄²	D₄	1	1	yes	characteristic
17	C₂×C₃⋊C₈	—	4	1	no	—
18	C₃⋊C₁₆	—	2	1	no	—
19	C₃⋊C₁₆	C₈	1	2	yes	complemented
20	C₂×C₁₆	—	3	1	no	—
21	C₂×C₁₆	—	3	1	no	—
22	C₄⋊C₈	—	2	1	no	—
23	C₄⋊C₈	—	6	1	no	—
24	C₄×C₈	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃×C₄	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₃×C₄	D₈	1	1	yes	characteristic
27	C₂×C₃×C₄	Q₁₆	1	1	yes	characteristic
28	C₃×C₈	—	2	1	no	—
29	C₃×C₈	—	8	1	no	—
30	C₃×C₈	C₂×C₈	1	1	yes	characteristic
31	C₃×C₈	M₁₆	1	1	yes	characteristic
32	C₃⋊C₈	—	8	1	no	—
33	C₂×C₈	—	4	1	no	—
34	C₂×C₈	—	12	1	no	—
35	C₂×C₈	Dic₆	1	1	yes	characteristic
36	C₂×C₈	C₄×D₃	1	1	yes	characteristic
37	C₄²	#67	1	1	yes	characteristic
38	C₁₆	—	3	2	no	—
39	C₁₆	—	6	1	no	—
40	C₂²×C₃	#107	1	1	yes	characteristic
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	C₄⋊C₈	1	1	yes	characteristic
43	C₃×C₄	#108	1	1	yes	characteristic
44	C₃×C₄	SD₃₂	1	1	yes	characteristic
45	C₃×C₄	SD₃₂	1	1	yes	characteristic
46	C₂×C₄	Dic₃⋊C₄	1	1	yes	characteristic
47	C₂×C₄	#213	1	1	yes	characteristic
48	C₂×C₄	#216	1	1	yes	characteristic
49	C₈	—	2	1	no	—
50	C₈	—	8	1	no	—
51	C₈	—	24	1	no	—
52	C₈	D₃×C₈	1	1	yes	characteristic
53	C₈	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
54	C₂×C₃	#335	1	1	yes	characteristic
55	C₂×C₃	#367	1	1	yes	characteristic
56	C₂×C₃	#368	1	1	yes	characteristic
57	C₂²	#807	1	1	yes	characteristic
58	C₄	—	2	1	no	—
59	C₄	Dic₃⋊C₈	1	1	yes	characteristic
60	C₄	#822	1	1	yes	characteristic
61	C₄	#827	1	1	yes	characteristic
62	C₄	#828	1	1	yes	characteristic
63	C₃	#1372	1	1	yes	characteristic, complemented, Sylow
64	C₂	#4479	1	1	yes	characteristic
65	C₂	#4490	1	1	yes	characteristic
66	C₂	#4493	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#213
commutator	C₃×C₈	C₂×C₈
Frattini	C₄×C₈	C₂×D₃
Fitting	C₃×#335	C₂
socle, prime radical	C₂²×C₃	#107
cosocle	C₃×C₄×C₈	C₂²
hypercenter	#335	D₃
hypocenter	C₃	#1372
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
19. C₃⋊C₁₆	C₈	semidirect
63. C₃	#1372	semidirect

Maximal subgroups

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

subgroup	index
C₃×#335	2
C₄×C₃⋊C₁₆	2
#4458	2
#1372	3