Subgroups — #69044

Order 384.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4682	C₂	1	2	yes	maximal
2	C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#4504	C₂	1	4	yes	maximal
4	#2252	—	3	1	no	maximal, Sylow
5	#869	—	2	2	no	—
6	C₂×C₃×C₁₆	C₂²	1	2	yes	—
7	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
8	#818	—	2	2	no	—
9	#818	C₂²	1	4	yes	—
10	#500	—	3	2	no	—
11	#366	—	3	4	no	—
12	C₄×C₁₆	D₃	1	1	yes	characteristic
13	C₂×Dic₆	—	4	2	no	—
14	C₂×C₃×C₈	D₄	1	1	yes	characteristic
15	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
16	C₃×C₄²	D₄	1	1	yes	characteristic
17	C₄⋊Dic₃	—	2	4	no	—
18	C₄⋊Dic₃	—	4	2	no	—
19	C₃×C₁₆	Q₈	1	4	yes	complemented
20	C₄⋊Q₈	—	6	2	no	—
21	C₂×C₁₆	C₂×D₃	1	2	yes	—
22	#107	—	3	4	no	—
23	#107	—	6	2	no	—
24	C₄×C₈	C₂×D₃	1	1	yes	characteristic
25	C₂×C₃×C₄	D₈	1	1	yes	characteristic
26	C₂×C₃×C₄	D₈	1	1	yes	characteristic
27	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
28	C₂×Dic₃	—	4	4	no	—
29	Dic₆	—	8	4	no	—
30	C₃×C₈	Q₁₆	1	2	yes	—
31	C₃×C₈	C₂×Q₈	1	2	yes	—
32	C₂×Q₈	—	12	2	no	—
33	C₂×C₈	D₁₂	1	1	yes	characteristic
34	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
35	C₄⋊C₄	—	6	4	no	—
36	C₄⋊C₄	—	12	2	no	—
37	C₄²	D₁₂	1	1	yes	characteristic
38	C₁₆	Dic₆	1	4	yes	complemented
39	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
40	C₃×C₄	D₁₆	1	2	yes	—
41	C₃×C₄	Q₃₂	1	2	yes	—
42	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
43	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
44	Dic₃	—	8	4	no	—
45	Q₈	—	24	4	no	—
46	C₂×C₄	—	12	4	no	—
47	C₂×C₄	D₂₄	1	1	yes	characteristic
48	C₂×C₄	D₂₄	1	1	yes	characteristic
49	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
50	C₈	Dic₁₂	1	2	yes	—
51	C₈	C₂×Dic₆	1	2	yes	—
52	C₂×C₃	#500	1	1	yes	characteristic
53	C₂×C₃	C₂×D₁₆	1	1	yes	characteristic
54	C₂×C₃	C₂×Q₃₂	1	1	yes	characteristic
55	C₂²	C₂×D₂₄	1	1	yes	characteristic
56	C₄	—	24	4	no	—
57	C₄	D₄₈	1	2	yes	—
58	C₄	Dic₂₄	1	2	yes	—
59	C₄	#869	1	1	yes	characteristic
60	C₄	C₂×Dic₁₂	1	1	yes	characteristic
61	C₃	#2252	1	1	yes	characteristic, complemented, Sylow
62	C₂	#4682	1	1	yes	characteristic
63	C₂	C₂×D₄₈	1	1	yes	characteristic
64	C₂	C₂×Dic₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₂₄
commutator, cosocle	C₂×C₃×C₈	C₂³
Frattini	C₂×C₈	C₂²×D₃
Fitting	C₃×C₄×C₁₆	C₂
socle, prime radical	C₂²×C₃	C₂×D₈
hypercenter	C₄×C₁₆	D₃
hypocenter	C₃	#2252
LCS	G ▷ C₂×C₃×C₈ ▷ 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
19. C₃×C₁₆	Q₈	semidirect
38. C₁₆	Dic₆	semidirect
61. C₃	#2252	semidirect

Maximal subgroups

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

subgroup	index
#4682	2
C₃×C₄×C₁₆	2
#4504	2
#2252	3