Subgroups — #63040

Order 256.

4254 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3549	C₂	1	27	yes	maximal, complemented
2	#3498	C₂	1	27	yes	maximal, complemented
3	#3483	C₂	1	9	yes	maximal, complemented
4	C₂×H(5,𝔽₂)	C₂²	1	18	yes	complemented
5	H(3,𝔽₄)	C₂²	1	36	yes	complemented
6	Q₈²	C₂²	1	3	yes	complemented
7	#555	C₂²	1	81	yes	complemented
8	#550	C₂²	1	54	yes	complemented
9	D₄²	C₂²	1	162	yes	complemented
10	#535	C₂²	1	162	yes	complemented
11	#532	C₂²	1	81	yes	complemented
12	C₂×#127	C₂²	1	27	yes	complemented
13	C₄×#41	C₂²	1	27	yes	complemented
14	H(5,𝔽₂)	—	4	72	no	—
15	C₂×#41	C₂³	1	27	yes	—
16	C₂×#41	C₂³	1	54	yes	—
17	C₂²×D₄	C₂³	1	162	yes	—
18	C₄⋊Q₈	C₂³	1	27	yes	—
19	#127	C₂³	1	81	yes	—
20	#127	C₂³	1	54	yes	—
21	#124	C₂³	1	162	yes	—
22	#121	C₂³	1	324	yes	—
23	#120	C₂³	1	216	yes	—
24	C₄×Q₈	C₂³	1	18	yes	—
25	C₄×D₄	C₂³	1	162	yes	—
26	#117	C₂³	1	81	yes	—
27	C₂×C₄²	C₂³	1	27	yes	—
28	C₂⁴	C₂⁴	1	36	yes	—
29	#41	—	2	108	no	—
30	#41	—	4	108	no	—
31	C₂×Q₈	C₂⁴	1	18	yes	—
32	C₂×Q₈	C₂⁴	1	3	yes	—
33	C₂×D₄	—	4	324	no	—
34	C₂×D₄	C₂⁴	1	81	yes	—
35	C₂×D₄	C₂⁴	1	162	yes	—
36	C₂²×C₄	C₂⁴	1	81	yes	—
37	C₄⋊C₄	C₂⁴	1	54	yes	—
38	C₂²⋊C₄	C₂⁴	1	162	yes	—
39	C₄²	C₂⁴	1	27	yes	—
40	C₄²	C₂⁴	1	27	yes	—
41	C₂³	—	4	108	no	—
42	C₂³	C₂⁵	1	27	yes	—
43	Q₈	—	4	18	no	—
44	Q₈	H(5,𝔽₂)	1	12	yes	complemented
45	D₄	—	2	162	no	—
46	D₄	—	4	162	no	—
47	C₂×C₄	—	2	162	no	—
48	C₂×C₄	C₂⁵	1	27	yes	—
49	C₂×C₄	C₂⁵	1	9	yes	—
50	C₂²	—	2	81	no	—
51	C₂²	—	16	36	no	—
52	C₂²	C₂⁶	1	1	yes	characteristic
53	C₄	—	2	27	no	—
54	C₄	C₂×H(5,𝔽₂)	1	18	yes	—
55	C₂	—	4	27	no	—
56	C₂	C₂²×H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
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. #3549	C₂	semidirect
2. #3498	C₂	semidirect
3. #3483	C₂	semidirect
4. C₂×H(5,𝔽₂)	C₂²	semidirect
5. H(3,𝔽₄)	C₂²	semidirect
6. Q₈²	C₂²	semidirect
7. #555	C₂²	semidirect
8. #550	C₂²	semidirect
9. D₄²	C₂²	semidirect
10. #535	C₂²	semidirect
11. #532	C₂²	semidirect
12. C₂×#127	C₂²	semidirect
13. C₄×#41	C₂²	semidirect
44. Q₈	H(5,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
#3549	2
#3498	2
#3483	2