Subgroups — #4919

Order 192.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#919	C₂	1	2	yes	maximal, complemented
2	C₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#828	C₂	1	2	yes	maximal, complemented
4	D₄₈	C₂	1	1	yes	maximal, characteristic, complemented
5	D₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#508	—	3	1	no	maximal, Sylow
7	#239	—	2	2	no	—
8	C₃×Q₁₆	C₂²	1	2	yes	—
9	#215	—	2	2	no	—
10	D₂₄	C₂²	1	2	yes	—
11	D₃×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₃⋊C₁₆	C₂²	1	1	yes	characteristic
14	#135	—	3	2	no	—
15	Q₃₂	D₃	1	1	yes	characteristic, complemented
16	SD₃₂	—	3	2	no	—
17	D₁₆	—	3	1	no	—
18	C₂×C₁₆	—	3	1	no	—
19	C₃×Q₈	—	2	2	no	—
20	D₁₂	—	2	2	no	—
21	D₁₂	—	4	2	no	—
22	C₄×D₃	—	4	2	no	—
23	C₄×D₃	D₄	1	1	yes	characteristic
24	C₃×C₈	C₂³	1	1	yes	characteristic
25	C₃⋊C₈	D₄	1	1	yes	characteristic
26	#41	—	6	2	no	—
27	Q₁₆	C₂×D₃	1	2	yes	—
28	SD₁₆	—	6	2	no	—
29	D₈	—	3	2	no	—
30	C₂×C₈	—	3	1	no	—
31	C₁₆	—	3	1	no	—
32	C₁₆	C₂×D₃	1	1	yes	characteristic
33	C₂×D₃	—	4	2	no	—
34	C₂×D₃	D₈	1	1	yes	characteristic
35	C₃×C₄	—	4	2	no	—
36	C₃×C₄	C₂×D₄	1	1	yes	characteristic
37	Dic₃	D₈	1	1	yes	characteristic
38	Q₈	—	2	2	no	—
39	D₄	—	6	2	no	—
40	D₄	—	12	2	no	—
41	C₂×C₄	—	3	1	no	—
42	C₂×C₄	—	12	2	no	—
43	C₈	—	3	1	no	—
44	C₈	C₂²×D₃	1	1	yes	characteristic
45	C₂×C₃	C₂×D₈	1	1	yes	characteristic
46	D₃	—	2	1	no	—
47	D₃	—	8	2	no	—
48	C₂²	—	3	1	no	—
49	C₂²	—	12	2	no	—
50	C₄	—	3	1	no	—
51	C₄	—	4	2	no	—
52	C₄	D₃×D₄	1	1	yes	characteristic
53	C₃	#508	1	1	yes	characteristic, complemented, Sylow
54	C₂	—	6	1	no	—
55	C₂	—	24	2	no	—
56	C₂	D₃×D₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×D₈
commutator, cosocle	C₃×C₈	C₂³
Frattini	C₈	C₂²×D₃
Fitting	C₃×Q₃₂	C₂
socle	C₂×C₃	C₂×D₈
hypercenter	Q₃₂	D₃
hypocenter	C₃	#508
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

Complements

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

N	H ≅ G/N	type
1. #919	C₂	semidirect
2. C₃×Q₃₂	C₂	semidirect
3. #828	C₂	semidirect
4. D₄₈	C₂	semidirect
5. D₃×C₁₆	C₂	semidirect
15. Q₃₂	D₃	semidirect
53. C₃	#508	semidirect

Maximal subgroups

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

subgroup	index
#919	2
C₃×Q₃₂	2
#828	2
D₄₈	2
D₃×C₁₆	2
#508	3