Subgroups — #67604

Order 384.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₈	C₂	1	1	yes	maximal, characteristic
2	C₃×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#4506	C₂	1	1	yes	maximal, characteristic, complemented
4	#1419	—	3	1	no	maximal, Sylow
5	C₂×D₂₄	—	2	1	no	—
6	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
7	#819	—	2	1	no	—
8	D₄₈	—	2	1	no	—
9	D₄₈	C₄	1	2	yes	—
10	C₃×C₃₂	—	2	1	no	—
11	C₂×D₁₆	—	3	1	no	—
12	M₆₄	D₃	1	1	yes	characteristic, complemented
13	#368	—	3	1	no	—
14	C₂×D₁₂	—	4	1	no	—
15	C₂×C₃×C₈	D₄	1	1	yes	characteristic
16	#208	—	4	1	no	—
17	D₂₄	—	2	2	no	—
18	D₂₄	—	4	1	no	—
19	C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×D₈	—	6	1	no	—
22	D₁₆	—	3	2	no	—
23	D₁₆	—	6	1	no	—
24	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
25	#108	—	6	1	no	—
26	C₃₂	—	2	1	no	—
27	C₂²×D₃	—	8	1	no	—
28	C₂×C₃×C₄	D₈	1	1	yes	characteristic
29	D₁₂	—	4	2	no	—
30	D₁₂	—	8	1	no	—
31	C₃×C₈	C₂²⋊C₄	1	1	yes	characteristic
32	C₃×C₈	SD₁₆	1	1	yes	characteristic
33	C₃⋊C₈	—	8	1	no	—
34	C₂×D₄	—	12	1	no	—
35	D₈	—	6	2	no	—
36	D₈	—	12	1	no	—
37	M₁₆	—	12	1	no	—
38	C₂×C₈	D₁₂	1	1	yes	characteristic
39	C₁₆	C₄×D₃	1	1	yes	characteristic
40	C₁₆	#67	1	1	yes	characteristic
41	C₂²×C₃	SD₃₂	1	1	yes	characteristic
42	C₂×D₃	—	8	2	no	—
43	C₂×D₃	—	16	2	no	—
44	C₃×C₄	#102	1	1	yes	characteristic
45	C₃×C₄	D₁₆	1	1	yes	characteristic
46	C₂³	—	24	1	no	—
47	D₄	—	12	2	no	—
48	D₄	—	24	1	no	—
49	C₂×C₄	D₂₄	1	1	yes	characteristic
50	C₈	—	24	1	no	—
51	C₈	#204	1	1	yes	characteristic
52	C₈	(C₂×D₃)⋊C₄	1	1	yes	characteristic
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	#357	1	1	yes	characteristic
55	D₃	—	16	2	no	—
56	C₂²	—	24	2	no	—
57	C₂²	—	48	2	no	—
58	C₂²	#800	1	1	yes	characteristic
59	C₄	D₄₈	1	1	yes	characteristic
60	C₄	#821	1	1	yes	characteristic
61	C₃	#1419	1	1	yes	characteristic, complemented, Sylow
62	C₂	—	2	1	no	—
63	C₂	—	48	2	no	—
64	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	C₂×D₄₈	C₂
cosocle	C₂×C₃×C₁₆	C₂²
hypercenter	M₆₄	D₃
hypocenter	C₃	#1419
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. #4506	C₂	semidirect
12. M₆₄	D₃	semidirect
61. C₃	#1419	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₈	2
C₃×M₆₄	2
#4506	2
#1419	3