Subgroups — #62849

Order 256.

3289 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3502	C₂	1	48	yes	maximal, complemented
2	#3458	C₂	1	12	yes	maximal
3	C₂×#519	C₂	1	3	yes	maximal
4	C₂³×Q₈	C₂²	1	1	yes	characteristic
5	Q₈²	C₂²	1	64	yes	complemented
6	#557	C₂²	1	192	yes	complemented
7	#544	C₂²	1	144	yes	—
8	#543	C₂²	1	16	yes	—
9	#533	C₂²	1	144	yes	—
10	#519	C₂²	1	48	yes	—
11	C₂×C₄×Q₈	C₂²	1	24	yes	—
12	C₂×#117	C₂²	1	18	yes	—
13	C₂²×Q₈	Q₈	1	16	yes	complemented
14	C₂²×Q₈	C₂³	1	12	yes	—
15	C₂³×C₄	C₂³	1	3	yes	—
16	C₄⋊Q₈	C₂³	1	288	yes	—
17	#125	C₂³	1	288	yes	—
18	C₂²⋊Q₈	C₂³	1	192	yes	—
19	C₄×Q₈	C₂³	1	192	yes	—
20	C₄×Q₈	C₂³	1	192	yes	—
21	#117	C₂³	1	144	yes	—
22	C₂×C₄⋊C₄	C₂³	1	36	yes	—
23	C₂×C₂²⋊C₄	C₂³	1	12	yes	—
24	C₂×C₄²	C₂³	1	36	yes	—
25	C₂⁴	C₂⁴	1	1	yes	characteristic
26	C₂×Q₈	C₂×Q₈	1	96	yes	—
27	C₂×Q₈	C₂⁴	1	64	yes	—
28	C₂×Q₈	C₂⁴	1	16	yes	—
29	C₂²×C₄	C₂×Q₈	1	24	yes	—
30	C₂²×C₄	C₂⁴	1	24	yes	—
31	C₂²×C₄	C₂⁴	1	18	yes	—
32	C₄⋊C₄	C₂⁴	1	144	yes	—
33	C₄⋊C₄	C₂⁴	1	192	yes	—
34	C₂²⋊C₄	C₂⁴	1	48	yes	—
35	C₄²	C₂⁴	1	144	yes	—
36	C₂³	—	4	2	no	—
37	C₂³	C₂²×Q₈	1	4	yes	—
38	C₂³	C₂⁵	1	3	yes	—
39	Q₈	—	4	64	no	—
40	Q₈	C₂²×Q₈	1	64	yes	—
41	C₂×C₄	—	4	24	no	—
42	C₂×C₄	C₂²×Q₈	1	72	yes	—
43	C₂×C₄	C₂⁵	1	48	yes	—
44	C₂×C₄	C₂⁵	1	12	yes	—
45	C₂²	—	2	6	no	—
46	C₂²	—	4	4	no	—
47	C₂²	C₂³×Q₈	1	6	yes	—
48	C₂²	C₂⁶	1	1	yes	characteristic
49	C₄	—	2	48	no	—
50	C₄	C₂³×Q₈	1	24	yes	—
51	C₂	—	2	6	no	—
52	C₂	C₂⁴×Q₈	1	1	yes	characteristic
53	C₂	2₋¹⁺⁶	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
prime radical	C₂⁴	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3502	C₂	semidirect
5. Q₈²	C₂²	semidirect
6. #557	C₂²	semidirect
13. C₂²×Q₈	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#3502	2
#3458	2
C₂×#519	2