Subgroups — #63091

Order 256.

5447 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(7,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	#3585	C₂	1	30	yes	maximal, complemented
3	#3584	C₂	1	15	yes	maximal, complemented
4	C₂×#576	C₂	1	15	yes	maximal, complemented
5	#3573	C₂	1	1	yes	maximal, characteristic, complemented
6	#585	—	2	12	no	—
7	#585	C₂²	1	1	yes	characteristic
8	#585	C₂²	1	30	yes	complemented
9	C₂×H(5,𝔽₂)	—	2	20	no	—
10	C₂×H(5,𝔽₂)	C₂²	1	30	yes	complemented
11	#578	C₂²	1	20	yes	complemented
12	#577	C₂²	1	120	yes	complemented
13	#576	C₂²	1	60	yes	complemented
14	#576	C₂²	1	120	yes	complemented
15	#575	C₂²	1	90	yes	complemented
16	C₂×AGL(1,ℤ/8)	C₂²	1	90	yes	complemented
17	C₂×#135	C₂²	1	45	yes	complemented
18	C₂²×D₈	C₂²	1	15	yes	complemented
19	#568	C₂²	1	15	yes	complemented
20	C₂×#131	C₂²	1	15	yes	complemented
21	2₋¹⁺⁴	—	2	30	no	—
22	2₋¹⁺⁴	D₄	1	6	yes	—
23	2₋¹⁺⁴	C₂³	1	40	yes	complemented
24	H(5,𝔽₂)	—	2	120	no	—
25	H(5,𝔽₂)	—	2	90	no	—
26	H(5,𝔽₂)	D₄	1	10	yes	—
27	H(5,𝔽₂)	C₂³	1	120	yes	complemented
28	C₂×#41	—	2	120	no	—
29	C₂×#41	—	2	30	no	—
30	C₂×#41	C₂³	1	15	yes	—
31	C₂×#41	C₂³	1	90	yes	complemented
32	C₂²×D₄	—	2	90	no	—
33	C₂²×D₄	C₂³	1	30	yes	complemented
34	#137	C₂³	1	120	yes	complemented
35	AGL(1,ℤ/8)	C₂³	1	360	yes	complemented
36	#135	C₂³	1	60	yes	complemented
37	#135	C₂³	1	180	yes	complemented
38	C₂×Q₁₆	C₂³	1	15	yes	complemented
39	C₂×SD₁₆	C₂³	1	90	yes	complemented
40	C₂×D₈	C₂³	1	45	yes	complemented
41	C₂×D₈	C₂³	1	90	yes	complemented
42	#131	C₂³	1	20	yes	complemented
43	#131	C₂³	1	60	yes	complemented
44	C₂×M₁₆	C₂³	1	45	yes	complemented
45	C₂²×C₈	C₂³	1	15	yes	complemented
46	C₂⁴	—	2	30	no	—
47	#41	—	2	120	no	—
48	#41	—	2	120	no	—
49	#41	—	2	180	no	—
50	#41	C₂×D₄	1	60	yes	—
51	#41	C₂⁴	1	120	yes	—
52	#41	C₂⁴	1	20	yes	—
53	C₂×Q₈	—	2	40	no	—
54	C₂×Q₈	C₂×D₄	1	15	yes	—
55	C₂×Q₈	C₂⁴	1	30	yes	—
56	C₂×D₄	—	2	90	no	—
57	C₂×D₄	—	2	360	no	—
58	C₂×D₄	—	2	90	no	—
59	C₂×D₄	C₂×D₄	1	45	yes	—
60	C₂×D₄	C₂⁴	1	90	yes	—
61	C₂²×C₄	—	2	90	no	—
62	C₂²×C₄	C₂⁴	1	15	yes	—
63	Q₁₆	C₂⁴	1	6	yes	—
64	Q₁₆	C₂⁴	1	30	yes	—
65	SD₁₆	C₂⁴	1	120	yes	—
66	D₈	C₂⁴	1	10	yes	—
67	D₈	C₂⁴	1	90	yes	—
68	M₁₆	C₂⁴	1	60	yes	—
69	C₂×C₈	C₂⁴	1	15	yes	—
70	C₂×C₈	C₂⁴	1	45	yes	—
71	C₂³	—	2	90	no	—
72	C₂³	—	16	30	no	—
73	C₂³	C₂²×D₄	1	15	yes	—
74	Q₈	—	2	30	no	—
75	Q₈	C₂²×D₄	1	20	yes	—
76	Q₈	C₂⁵	1	12	yes	—
77	D₄	—	2	120	no	—
78	D₄	—	2	90	no	—
79	D₄	C₂²×D₄	1	60	yes	—
80	D₄	C₂⁵	1	20	yes	—
81	C₂×C₄	—	2	30	no	—
82	C₂×C₄	—	2	120	no	—
83	C₂×C₄	C₂²×D₄	1	45	yes	—
84	C₂×C₄	C₂⁵	1	15	yes	—
85	C₈	C₂⁵	1	1	yes	characteristic
86	C₈	C₂⁵	1	15	yes	—
87	C₂²	—	2	20	no	—
88	C₂²	—	4	15	no	—
89	C₂²	—	8	90	no	—
90	C₂²	C₂³×D₄	1	15	yes	—
91	C₄	—	2	12	no	—
92	C₄	C₂³×D₄	1	15	yes	—
93	C₄	C₂⁶	1	1	yes	characteristic
94	C₂	—	2	15	no	—
95	C₂	—	4	20	no	—
96	C₂	C₂⁴×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂⁴×D₄
commutator, Frattini, cosocle	C₄	C₂⁶
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ #585 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. H(7,𝔽₂)	C₂	semidirect
2. #3585	C₂	semidirect
3. #3584	C₂	semidirect
4. C₂×#576	C₂	semidirect
5. #3573	C₂	semidirect
8. #585	C₂²	semidirect
10. C₂×H(5,𝔽₂)	C₂²	semidirect
11. #578	C₂²	semidirect
12. #577	C₂²	semidirect
13. #576	C₂²	semidirect
14. #576	C₂²	semidirect
15. #575	C₂²	semidirect
16. C₂×AGL(1,ℤ/8)	C₂²	semidirect
17. C₂×#135	C₂²	semidirect
18. C₂²×D₈	C₂²	semidirect
19. #568	C₂²	semidirect
20. C₂×#131	C₂²	semidirect
23. 2₋¹⁺⁴	C₂³	semidirect
27. H(5,𝔽₂)	C₂³	semidirect
31. C₂×#41	C₂³	semidirect
33. C₂²×D₄	C₂³	semidirect
34. #137	C₂³	semidirect
35. AGL(1,ℤ/8)	C₂³	semidirect
36. #135	C₂³	semidirect
37. #135	C₂³	semidirect
38. C₂×Q₁₆	C₂³	semidirect
39. C₂×SD₁₆	C₂³	semidirect
40. C₂×D₈	C₂³	semidirect
41. C₂×D₈	C₂³	semidirect
42. #131	C₂³	semidirect
43. #131	C₂³	semidirect
44. C₂×M₁₆	C₂³	semidirect
45. C₂²×C₈	C₂³	semidirect

Maximal subgroups

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

subgroup	index
H(7,𝔽₂)	2
#3585	2
#3584	2
C₂×#576	2
#3573	2