Subgroups — #7530

Order 256.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1709	C₂	1	1	yes	maximal, characteristic
2	#1408	C₂	1	2	yes	maximal
3	#494	C₄	1	1	yes	characteristic, complemented
4	#493	C₂²	1	1	yes	characteristic
5	#486	—	2	2	no	—
6	#353	—	2	2	no	—
7	C₈²	C₄	1	1	yes	characteristic, complemented
8	C₂×Q₁₆	—	4	1	no	—
9	C₂×D₈	—	4	1	no	—
10	C₄⋊Q₈	—	2	1	no	—
11	#127	D₄	1	2	yes	—
12	#102	—	4	4	no	—
13	#99	—	4	2	no	—
14	C₄×C₈	—	2	1	no	—
15	C₄×C₈	C₂×C₄	1	1	yes	characteristic
16	C₂×Q₈	—	4	1	no	—
17	C₂×D₄	—	2	2	no	—
18	C₂×D₄	—	4	2	no	—
19	Q₁₆	—	8	2	no	—
20	D₈	—	8	2	no	—
21	C₂×C₈	—	2	1	no	—
22	C₂×C₈	—	2	2	no	—
23	C₄⋊C₄	—	4	2	no	—
24	C₂²⋊C₄	—	8	2	no	—
25	C₄²	C₂²⋊C₄	1	1	yes	characteristic
26	C₂³	—	4	2	no	—
27	Q₈	—	8	2	no	—
28	D₄	—	8	2	no	—
29	D₄	—	8	4	no	—
30	C₂×C₄	—	2	1	no	—
31	C₂×C₄	—	8	1	no	—
32	C₂×C₄	—	16	2	no	—
33	C₂×C₄	#99	1	1	yes	characteristic
34	C₈	—	2	2	no	—
35	C₈	—	4	2	no	—
36	C₂²	—	8	2	no	—
37	C₂²	—	16	2	no	—
38	C₂²	#353	1	1	yes	characteristic
39	C₄	—	2	2	no	—
40	C₄	—	2	1	no	—
41	C₄	—	16	1	no	—
42	C₄	—	32	2	no	—
43	C₂	—	2	1	no	—
44	C₂	—	16	2	no	—
45	C₂	#1408	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#1408
commutator	C₄×C₈	C₂×C₄
Frattini, prime radical, cosocle	#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #493 ◁ G
LECS	G ▷ #493 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

N	H ≅ G/N	type
3. #494	C₄	semidirect
7. C₈²	C₄	semidirect

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

subgroup	index
#1709	2
#1408	2