Subgroups — #69045

Order 384.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4683	C₂	1	2	yes	maximal
2	C₃×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#4505	C₂	1	2	yes	maximal
4	#4504	C₂	1	2	yes	maximal
5	#2253	—	3	1	no	maximal, Sylow
6	#870	—	2	2	no	—
7	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	#818	C₂²	1	4	yes	—
11	#817	—	2	2	no	—
12	#499	—	3	2	no	—
13	#367	—	3	2	no	—
14	#366	—	3	2	no	—
15	C₄×C₁₆	D₃	1	1	yes	characteristic
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
18	C₃×C₄²	D₄	1	1	yes	characteristic
19	C₄⋊Dic₃	—	2	4	no	—
20	Dic₃⋊C₄	—	4	4	no	—
21	C₃×C₁₆	Q₈	1	2	yes	—
22	C₃×C₁₆	Q₈	1	2	yes	—
23	#125	—	6	2	no	—
24	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
25	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
26	#107	—	3	4	no	—
27	#106	—	6	2	no	—
28	C₄×C₈	C₂×D₃	1	1	yes	characteristic
29	C₂×C₃×C₄	D₈	1	2	yes	—
30	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₃	—	4	4	no	—
32	C₃×C₈	Q₁₆	1	2	yes	—
33	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
34	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₈	D₁₂	1	1	yes	characteristic
36	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
37	C₄⋊C₄	—	6	4	no	—
38	C₄⋊C₄	—	12	4	no	—
39	C₄²	D₁₂	1	1	yes	characteristic
40	C₁₆	Dic₆	1	2	yes	—
41	C₁₆	Dic₆	1	2	yes	—
42	C₂²×C₃	C₂×D₈	1	1	yes	characteristic
43	C₃×C₄	—	2	2	no	—
44	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
45	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
46	Dic₃	—	8	4	no	—
47	C₂×C₄	—	12	4	no	—
48	C₂×C₄	D₂₄	1	2	yes	—
49	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
50	C₈	Dic₁₂	1	2	yes	—
51	C₈	C₂×Dic₆	1	1	yes	characteristic
52	C₈	C₂×Dic₆	1	1	yes	characteristic
53	C₂×C₃	#500	1	1	yes	characteristic
54	C₂×C₃	#508	1	2	yes	—
55	C₂²	C₂×D₂₄	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	24	4	no	—
58	C₄	#869	1	1	yes	characteristic
59	C₄	C₂×Dic₁₂	1	1	yes	characteristic
60	C₃	#2253	1	1	yes	characteristic, complemented, Sylow
61	C₂	#4682	1	1	yes	characteristic
62	C₂	#4904	1	2	yes	—

	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	#2253
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
60. C₃	#2253	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
#4505	2
#4504	2
#2253	3