Subgroups — #67597

Order 384.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₂₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃₂	C₂	1	1	yes	maximal, characteristic
3	#4504	C₂	1	1	yes	maximal, characteristic
4	#1416	—	3	1	no	maximal, Sylow
5	C₂×Dic₁₂	—	2	1	no	—
6	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
7	#818	—	2	1	no	—
8	Dic₂₄	—	2	1	no	—
9	Dic₂₄	C₄	1	2	yes	complemented
10	C₃×C₃₂	—	2	1	no	—
11	C₂×Q₃₂	—	3	1	no	—
12	C₂×C₃₂	D₃	1	1	yes	characteristic
13	#366	—	3	1	no	—
14	C₂×Dic₆	—	4	1	no	—
15	C₂×C₃×C₈	D₄	1	1	yes	characteristic
16	C₄⋊Dic₃	—	4	1	no	—
17	Dic₁₂	—	2	2	no	—
18	Dic₁₂	—	4	1	no	—
19	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×Q₁₆	—	6	1	no	—
22	Q₃₂	—	3	2	no	—
23	Q₃₂	—	6	1	no	—
24	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
25	#107	—	6	1	no	—
26	C₃₂	—	2	1	no	—
27	C₂×C₃×C₄	D₈	1	1	yes	characteristic
28	C₂×Dic₃	—	8	1	no	—
29	C₂×Dic₃	—	8	1	no	—
30	Dic₆	—	4	2	no	—
31	Dic₆	—	8	1	no	—
32	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
33	C₃×C₈	SD₁₆	1	1	yes	characteristic
34	C₂×Q₈	—	12	1	no	—
35	Q₁₆	—	6	2	no	—
36	Q₁₆	—	12	1	no	—
37	C₂×C₈	D₁₂	1	1	yes	characteristic
38	C₄⋊C₄	—	12	1	no	—
39	C₁₆	C₄×D₃	1	1	yes	characteristic
40	C₁₆	#67	1	1	yes	characteristic
41	C₂²×C₃	D₁₆	1	1	yes	characteristic
42	C₃×C₄	#102	1	1	yes	characteristic
43	C₃×C₄	SD₃₂	1	1	yes	characteristic
44	Dic₃	—	8	2	no	—
45	Dic₃	—	16	1	no	—
46	Q₈	—	12	2	no	—
47	Q₈	—	24	1	no	—
48	C₂×C₄	—	24	1	no	—
49	C₂×C₄	—	24	1	no	—
50	C₂×C₄	D₂₄	1	1	yes	characteristic
51	C₈	#204	1	1	yes	characteristic
52	C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
53	C₂×C₃	#357	1	1	yes	characteristic
54	C₂×C₃	SD₆₄	1	1	yes	characteristic
55	C₂×C₃	Q₆₄	1	1	yes	characteristic
56	C₂²	D₄₈	1	1	yes	characteristic
57	C₄	—	24	2	no	—
58	C₄	—	48	1	no	—
59	C₄	#800	1	1	yes	characteristic
60	C₄	#821	1	1	yes	characteristic
61	C₃	#1416	1	1	yes	characteristic, complemented, Sylow
62	C₂	#4449	1	1	yes	characteristic
63	C₂	Dic₄₈	1	1	yes	characteristic
64	C₂	#4509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄₈
commutator	C₃×C₁₆	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₃
Fitting	C₂×C₃×C₃₂	C₂
socle, prime radical	C₂²×C₃	D₁₆
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	C₂×C₃₂	D₃
hypocenter	C₃	#1416
LCS	G ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
9. Dic₂₄	C₄	semidirect
61. C₃	#1416	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₂₄	2
C₂×C₃×C₃₂	2
#4504	2
#1416	3