Subgroups — #62694

Order 256.

3320 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#3531	C₂	1	18	yes	maximal, complemented
2	#3529	C₂	1	30	yes	maximal, complemented
3	#3444	C₂	1	15	yes	maximal, complemented
4	#564	C₂²	1	6	yes	complemented
5	#563	C₂²	1	90	yes	complemented
6	#562	C₂²	1	90	yes	complemented
7	H(3,𝔽₄)	C₂²	1	10	yes	complemented
8	#560	C₂²	1	60	yes	complemented
9	#533	C₂²	1	45	yes	complemented
10	#532	C₂²	1	45	yes	—
11	#529	C₂²	1	180	yes	complemented
12	#525	C₂²	1	90	yes	complemented
13	C₄×#41	C₂²	1	30	yes	complemented
14	C₂²×C₄²	C₂²	1	5	yes	complemented
15	C₂×#41	C₂³	1	30	yes	—
16	C₂³×C₄	C₂³	1	15	yes	—
17	C₄⋊Q₈	C₂³	1	45	yes	—
18	#127	C₂³	1	15	yes	—
19	#126	C₂³	1	120	yes	—
20	#125	C₂³	1	90	yes	—
21	#124	C₂³	1	90	yes	—
22	#123	C₂³	1	180	yes	—
23	C₂²⋊Q₈	C₂³	1	180	yes	—
24	#121	C₂³	1	180	yes	—
25	#120	C₂³	1	60	yes	—
26	C₄×Q₈	C₂³	1	60	yes	—
27	C₄×D₄	C₂³	1	180	yes	—
28	#117	C₂³	1	90	yes	—
29	C₂×C₄²	C₂³	1	45	yes	—
30	C₂×C₄²	C₂³	1	15	yes	—
31	C₂⁴	C₂⁴	1	5	yes	—
32	#41	—	4	60	no	—
33	C₂×Q₈	C₂⁴	1	30	yes	—
34	C₂×D₄	C₂⁴	1	90	yes	—
35	C₂²×C₄	—	4	30	no	—
36	C₂²×C₄	C₂⁴	1	45	yes	—
37	C₂²×C₄	C₂⁴	1	45	yes	—
38	C₄⋊C₄	C₂⁴	1	180	yes	—
39	C₂²⋊C₄	C₂⁴	1	180	yes	—
40	C₄²	C₂⁴	1	45	yes	—
41	C₄²	C₂⁴	1	15	yes	—
42	C₄²	C₂⁴	1	15	yes	—
43	C₄²	C₂⁴	1	1	yes	characteristic
44	C₂³	—	4	15	no	—
45	C₂³	C₂⁵	1	15	yes	—
46	Q₈	—	4	30	no	—
47	D₄	—	4	90	no	—
48	C₂×C₄	—	2	90	no	—
49	C₂×C₄	—	4	45	no	—
50	C₂×C₄	C₂⁵	1	45	yes	—
51	C₂×C₄	C₂⁵	1	3	yes	—
52	C₂²	—	2	45	no	—
53	C₂²	—	4	20	no	—
54	C₂²	C₂⁶	1	1	yes	characteristic
55	C₄	—	2	45	no	—
56	C₄	#585	1	6	yes	—
57	C₂	—	4	15	no	—
58	C₂	C₂×#585	1	3	yes	—

	subgroup	G/H
center	C₄²	C₂⁴
commutator, Frattini, socle, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #3531	C₂	semidirect
2. #3529	C₂	semidirect
3. #3444	C₂	semidirect
4. #564	C₂²	semidirect
5. #563	C₂²	semidirect
6. #562	C₂²	semidirect
7. H(3,𝔽₄)	C₂²	semidirect
8. #560	C₂²	semidirect
9. #533	C₂²	semidirect
11. #529	C₂²	semidirect
12. #525	C₂²	semidirect
13. C₄×#41	C₂²	semidirect
14. C₂²×C₄²	C₂²	semidirect

Maximal subgroups

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

subgroup	index
#3531	2
#3529	2
#3444	2