Subgroups — #68050

Order 384.

214 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×ΓL(1,𝔽₂₅)	C₂	1	3	yes	maximal
2	C₃×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
3	#4455	C₂	1	3	yes	maximal, complemented
4	#1455	—	3	1	no	maximal, Sylow
5	C₂×ΓL(1,𝔽₂₅)	C₄	1	6	yes	—
6	C₄²×D₃	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₈	C₂²	1	3	yes	—
8	#815	C₄	1	6	yes	—
9	C₄×C₃⋊C₈	C₂²	1	3	yes	—
10	C₄×M₁₆	—	3	3	no	—
11	#322	—	3	3	no	—
12	C₈²	D₃	1	1	yes	characteristic, complemented
13	C₂×C₄×D₃	C₂×C₄	1	3	yes	—
14	C₂×C₃×C₈	C₂×C₄	1	6	yes	—
15	C₃×C₄²	C₂³	1	1	yes	characteristic
16	C₄×Dic₃	C₂×C₄	1	3	yes	—
17	C₂×C₃⋊C₈	C₂×C₄	1	6	yes	—
18	ΓL(1,𝔽₂₅)	—	2	12	no	—
19	C₂×M₁₆	—	3	6	no	—
20	C₂×C₄²	—	3	1	no	—
21	#97	—	3	6	no	—
22	C₄×C₈	—	3	3	no	—
23	C₄×C₈	C₂×D₃	1	3	yes	—
24	C₂²×D₃	C₄²	1	1	yes	characteristic
25	C₂×C₃×C₄	C₂²×C₄	1	3	yes	—
26	C₂×Dic₃	C₄²	1	3	yes	—
27	C₄×D₃	—	2	6	no	—
28	C₃×C₈	M₁₆	1	12	yes	complemented
29	C₃⋊C₈	—	2	6	no	—
30	C₂²×C₄	—	3	3	no	—
31	M₁₆	—	6	12	no	—
32	C₂×C₈	—	3	6	no	—
33	C₂×C₈	C₄×D₃	1	6	yes	—
34	C₄²	—	3	3	no	—
35	C₄²	C₂²×D₃	1	1	yes	characteristic
36	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
37	C₂×D₃	—	2	3	no	—
38	C₃×C₄	C₂×M₁₆	1	6	yes	—
39	Dic₃	—	2	3	no	—
40	C₂³	—	3	1	no	—
41	C₂×C₄	—	3	3	no	—
42	C₂×C₄	—	6	6	no	—
43	C₂×C₄	C₂×C₄×D₃	1	3	yes	—
44	C₈	—	6	6	no	—
45	C₈	ΓL(1,𝔽₂₅)	1	12	yes	complemented
46	C₂×C₃	C₄×M₁₆	1	3	yes	—
47	D₃	—	4	1	no	—
48	C₂²	—	6	3	no	—
49	C₂²	C₄²×D₃	1	1	yes	characteristic
50	C₄	—	6	3	no	—
51	C₄	C₂×ΓL(1,𝔽₂₅)	1	6	yes	—
52	C₃	#1455	1	1	yes	characteristic, complemented, Sylow
53	C₂	—	12	1	no	—
54	C₂	C₄×ΓL(1,𝔽₂₅)	1	3	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃
commutator, socle	C₂²×C₃	C₂×C₄²
Fitting	C₃×C₈²	C₂
prime radical	C₂²×D₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	C₈²	D₃
hypocenter	C₃	#1455
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
2. C₃×C₈²	C₂	semidirect
3. #4455	C₂	semidirect
12. C₈²	D₃	semidirect
28. C₃×C₈	M₁₆	semidirect
45. C₈	ΓL(1,𝔽₂₅)	semidirect
52. C₃	#1455	semidirect

Maximal subgroups

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

subgroup	index
C₄×ΓL(1,𝔽₂₅)	2
C₃×C₈²	2
#4455	2
#1455	3