Finite Group Databasebeta

Subgroups — #10585860

Order 512.

7898 conjugacy classes of subgroups; only the 3440 normal subgroups are shown.

normal subgroups only

This group has 7898 conjugacy classes of subgroups — too many to draw. Only its 3440 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#63091	C₂	1	2	yes	maximal, complemented
2	#62636	C₂	1	30	yes	maximal, complemented
3	#62635	C₂	1	15	yes	maximal, complemented
4	C₂×#3415	C₂	1	15	yes	maximal, complemented
5	#62624	C₂	1	1	yes	maximal, characteristic, complemented
6	H(7,𝔽₂)	—	2	2	no	—
7	#3585	—	2	30	no	—
8	#3584	C₂²	1	30	yes	complemented
9	C₂×#576	C₂²	1	30	yes	complemented
10	#3573	C₂²	1	1	yes	characteristic
11	#3417	C₂²	1	20	yes	complemented
12	#3416	C₂²	1	120	yes	complemented
13	#3415	C₂²	1	60	yes	complemented
14	#3415	C₂²	1	120	yes	complemented
15	#3414	C₂²	1	90	yes	complemented
16	C₂×#509	C₂²	1	90	yes	complemented
17	C₂×#508	C₂²	1	45	yes	complemented
18	C₂²×D₁₆	C₂²	1	15	yes	complemented
19	#3407	C₂²	1	15	yes	complemented
20	C₂×#504	C₂²	1	15	yes	complemented
21	#585	—	2	30	no	—
22	#585	—	4	12	no	—
23	#585	D₄	1	1	yes	characteristic
24	C₂×H(5,𝔽₂)	—	2	30	no	—
25	C₂×H(5,𝔽₂)	—	4	20	no	—
26	#578	C₂³	1	40	yes	complemented
27	#577	—	2	120	no	—
28	#576	—	2	120	no	—
29	#576	C₂³	1	120	yes	complemented
30	#575	—	2	90	no	—
31	C₂×AGL(1,ℤ/8)	—	2	90	no	—
32	C₂×#135	C₂³	1	90	yes	complemented
33	C₂²×D₈	C₂³	1	30	yes	complemented
34	#568	D₄	1	15	yes	—
35	C₂×#131	C₂³	1	15	yes	—
36	#510	C₂³	1	120	yes	complemented
37	#509	C₂³	1	360	yes	complemented
38	#508	C₂³	1	60	yes	complemented
39	#508	C₂³	1	180	yes	complemented
40	C₂×Q₃₂	C₂³	1	15	yes	complemented
41	C₂×SD₃₂	C₂³	1	90	yes	complemented
42	C₂×D₁₆	C₂³	1	45	yes	complemented
43	C₂×D₁₆	C₂³	1	90	yes	complemented
44	#504	C₂³	1	20	yes	complemented
45	#504	C₂³	1	60	yes	complemented
46	C₂×M₃₂	C₂³	1	45	yes	complemented
47	C₂²×C₁₆	C₂³	1	15	yes	complemented
48	2₋¹⁺⁴	—	2	40	no	—
49	2₋¹⁺⁴	—	4	30	no	—
50	2₋¹⁺⁴	D₈	1	6	yes	—
51	H(5,𝔽₂)	—	2	120	no	—
52	H(5,𝔽₂)	—	4	120	no	—
53	H(5,𝔽₂)	—	4	90	no	—
54	H(5,𝔽₂)	D₈	1	10	yes	—
55	C₂×#41	—	2	90	no	—
56	C₂×#41	—	4	120	no	—
57	C₂×#41	—	4	30	no	—
58	C₂×#41	C₂×D₄	1	15	yes	—
59	C₂²×D₄	—	2	30	no	—
60	C₂²×D₄	—	4	90	no	—
61	#137	—	2	120	no	—
62	AGL(1,ℤ/8)	—	2	360	no	—
63	#135	—	2	180	no	—
64	#135	C₂⁴	1	120	yes	—
65	C₂×Q₁₆	C₂⁴	1	30	yes	—
66	C₂×SD₁₆	—	2	90	no	—
67	C₂×D₈	—	2	90	no	—
68	C₂×D₈	C₂⁴	1	90	yes	—
69	#131	C₂×D₄	1	60	yes	—
70	#131	C₂⁴	1	20	yes	—
71	C₂×M₁₆	C₂×D₄	1	45	yes	—
72	C₂²×C₈	C₂⁴	1	15	yes	—
73	Q₃₂	C₂⁴	1	6	yes	—
74	Q₃₂	C₂⁴	1	30	yes	—
75	SD₃₂	C₂⁴	1	120	yes	—
76	D₁₆	C₂⁴	1	10	yes	—
77	D₁₆	C₂⁴	1	90	yes	—
78	M₃₂	C₂⁴	1	60	yes	—
79	C₂×C₁₆	C₂⁴	1	15	yes	—
80	C₂×C₁₆	C₂⁴	1	45	yes	—
81	C₂⁴	—	4	30	no	—
82	#41	—	2	120	no	—
83	#41	—	4	120	no	—
84	#41	—	4	120	no	—
85	#41	—	4	180	no	—
86	#41	C₂×D₈	1	60	yes	—
87	#41	C₂²×D₄	1	20	yes	—
88	C₂×Q₈	—	2	30	no	—
89	C₂×Q₈	—	4	40	no	—
90	C₂×Q₈	C₂×D₈	1	15	yes	—
91	C₂×D₄	—	2	90	no	—
92	C₂×D₄	—	4	90	no	—
93	C₂×D₄	—	4	360	no	—
94	C₂×D₄	—	4	90	no	—
95	C₂×D₄	C₂×D₈	1	45	yes	—
96	C₂²×C₄	—	4	90	no	—
97	C₂²×C₄	C₂²×D₄	1	15	yes	—
98	Q₁₆	—	2	30	no	—
99	Q₁₆	C₂⁵	1	12	yes	—
100	SD₁₆	—	2	120	no	—
101	D₈	—	2	90	no	—
102	D₈	C₂⁵	1	20	yes	—
103	M₁₆	C₂²×D₄	1	60	yes	—
104	C₂×C₈	C₂²×D₄	1	45	yes	—
105	C₂×C₈	C₂⁵	1	15	yes	—
106	C₁₆	C₂⁵	1	1	yes	characteristic
107	C₁₆	C₂⁵	1	15	yes	—
108	C₂³	—	4	90	no	—
109	C₂³	—	32	30	no	—
110	C₂³	C₂²×D₈	1	15	yes	—
111	Q₈	—	2	12	no	—
112	Q₈	—	4	30	no	—
113	Q₈	C₂²×D₈	1	20	yes	—
114	D₄	—	2	20	no	—
115	D₄	—	4	120	no	—
116	D₄	—	4	90	no	—
117	D₄	C₂²×D₈	1	60	yes	—
118	C₂×C₄	—	4	30	no	—
119	C₂×C₄	—	4	120	no	—
120	C₂×C₄	C₂²×D₈	1	45	yes	—
121	C₂×C₄	C₂³×D₄	1	15	yes	—
122	C₈	C₂³×D₄	1	15	yes	—
123	C₈	C₂⁶	1	1	yes	characteristic
124	C₂²	—	4	15	no	—
125	C₂²	—	4	20	no	—
126	C₂²	—	16	90	no	—
127	C₂²	C₂³×D₈	1	15	yes	—
128	C₄	—	4	12	no	—
129	C₄	C₂³×D₈	1	15	yes	—
130	C₄	C₂⁴×D₄	1	1	yes	characteristic
131	C₂	—	2	15	no	—
132	C₂	—	8	20	no	—
133	C₂	C₂⁴×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂⁴×D₈
commutator, Frattini, cosocle	C₈	C₂⁶
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ #585 ◁ #3573 ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #63091	C₂	semidirect
2. #62636	C₂	semidirect
3. #62635	C₂	semidirect
4. C₂×#3415	C₂	semidirect
5. #62624	C₂	semidirect
8. #3584	C₂²	semidirect
9. C₂×#576	C₂²	semidirect
11. #3417	C₂²	semidirect
12. #3416	C₂²	semidirect
13. #3415	C₂²	semidirect
14. #3415	C₂²	semidirect
15. #3414	C₂²	semidirect
16. C₂×#509	C₂²	semidirect
17. C₂×#508	C₂²	semidirect
18. C₂²×D₁₆	C₂²	semidirect
19. #3407	C₂²	semidirect
20. C₂×#504	C₂²	semidirect
26. #578	C₂³	semidirect
29. #576	C₂³	semidirect
32. C₂×#135	C₂³	semidirect
33. C₂²×D₈	C₂³	semidirect
36. #510	C₂³	semidirect
37. #509	C₂³	semidirect
38. #508	C₂³	semidirect
39. #508	C₂³	semidirect
40. C₂×Q₃₂	C₂³	semidirect
41. C₂×SD₃₂	C₂³	semidirect
42. C₂×D₁₆	C₂³	semidirect
43. C₂×D₁₆	C₂³	semidirect
44. #504	C₂³	semidirect
45. #504	C₂³	semidirect
46. C₂×M₃₂	C₂³	semidirect
47. C₂²×C₁₆	C₂³	semidirect

Maximal subgroups

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

subgroup	index
#63091	2
#62636	2
#62635	2
C₂×#3415	2
#62624	2