Subgroups — #918

Order 96.

60 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#239	C₂	1	1	yes	maximal, characteristic
2	D₃×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	#216	C₂	1	1	yes	maximal, characteristic, complemented
5	#215	C₂	1	1	yes	maximal, characteristic, complemented
6	#204	C₂	1	1	yes	maximal, characteristic, complemented
7	ΓL(1,𝔽₂₅)	C₂	1	1	yes	maximal, characteristic, complemented
8	#137	—	3	1	no	maximal, Sylow
9	C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₃×Q₈	C₂²	1	1	yes	characteristic
11	D₁₂	—	2	1	no	—
12	D₁₂	C₂²	1	1	yes	characteristic
13	C₄×D₃	—	2	1	no	—
14	C₄×D₃	—	2	1	no	—
15	C₄×D₃	C₂²	1	1	yes	characteristic
16	Dic₆	—	2	1	no	—
17	Dic₆	C₂²	1	1	yes	characteristic
18	C₃×C₈	C₂²	1	1	yes	characteristic
19	C₃⋊C₈	C₂²	1	1	yes	characteristic
20	#41	—	3	1	no	—
21	C₂×Q₈	—	3	1	no	—
22	Q₁₆	—	3	1	no	—
23	Q₁₆	D₃	1	1	yes	characteristic, complemented
24	SD₁₆	—	3	1	no	—
25	SD₁₆	—	3	1	no	—
26	M₁₆	—	3	1	no	—
27	C₂×D₃	—	2	1	no	—
28	C₂×D₃	D₄	1	1	yes	characteristic
29	C₃×C₄	—	2	1	no	—
30	C₃×C₄	—	2	1	no	—
31	C₃×C₄	C₂³	1	1	yes	characteristic
32	Dic₃	—	2	1	no	—
33	Dic₃	D₄	1	1	yes	characteristic
34	Q₈	—	3	1	no	—
35	Q₈	—	6	1	no	—
36	Q₈	C₂×D₃	1	1	yes	characteristic
37	Q₈	C₂×D₃	1	1	yes	characteristic
38	D₄	—	3	1	no	—
39	D₄	—	6	1	no	—
40	C₂×C₄	—	3	1	no	—
41	C₂×C₄	—	6	1	no	—
42	C₂×C₄	—	6	1	no	—
43	C₈	—	3	1	no	—
44	C₈	C₂×D₃	1	1	yes	characteristic
45	C₂×C₃	C₂×D₄	1	1	yes	characteristic
46	D₃	—	2	1	no	—
47	D₃	—	4	1	no	—
48	C₂²	—	3	1	no	—
49	C₂²	—	6	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	2	1	no	—
52	C₄	—	3	1	no	—
53	C₄	—	6	1	no	—
54	C₄	C₂²×D₃	1	1	yes	characteristic
55	C₃	#137	1	1	yes	characteristic, complemented, Sylow
56	C₂	—	6	1	no	—
57	C₂	—	12	1	no	—
58	C₂	D₃×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₃×D₄
commutator, cosocle	C₃×C₄	C₂³
Frattini	C₄	C₂²×D₃
Fitting	C₃×Q₁₆	C₂
socle	C₂×C₃	C₂×D₄
prime radical	#239	C₂
hypercenter	Q₁₆	D₃
hypocenter	C₃	#137
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
2. D₃×Q₈	C₂	semidirect
3. C₃×Q₁₆	C₂	semidirect
4. #216	C₂	semidirect
5. #215	C₂	semidirect
6. #204	C₂	semidirect
7. ΓL(1,𝔽₂₅)	C₂	semidirect
23. Q₁₆	D₃	semidirect
55. C₃	#137	semidirect

Maximal subgroups

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

subgroup	index
#239	2
D₃×Q₈	2
C₃×Q₁₆	2
#216	2
#215	2
#204	2
ΓL(1,𝔽₂₅)	2
#137	3