Finite Group Databasebeta

← #10585861

Subgroups — #10585861

Order 512.

7849 conjugacy classes of subgroups; only the 3440 normal subgroups are shown.

normal subgroups only

This group has 7849 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

#subgroupG/Hclass sizemultiplicitynormal?properties
1#63093C211yesmaximal, characteristic, complemented
2#63091C211yesmaximal, characteristic, complemented
3#62637C2115yesmaximal, complemented
4#62636C2115yesmaximal, complemented
5#62635C2115yesmaximal, complemented
6C2×#3416C2115yesmaximal, complemented
7#62624C211yesmaximal, characteristic, complemented
82−1+6—21no—
9H(7,𝔽2)—21no—
10#3586—215no—
11#3585—215no—
12#3584C22115yescomplemented
13#3584C22115yescomplemented
14C2×#578C22115yescomplemented
15C2×#576C22115yescomplemented
16#3573C2211yescharacteristic
17#3417C22160yescomplemented
18#3416C22160yescomplemented
19#3416C221120yescomplemented
20#3416C22120yescomplemented
21#3415C22160yescomplemented
22#3414C22190yescomplemented
23C2×#510C22145yescomplemented
24C2×#509C22145yescomplemented
25C2×#508C22145yescomplemented
26C22×SD32C22115yescomplemented
27#3407C22115yescomplemented
28C2×#504C22115yescomplemented
29#585—215no—
30#585—215no—
31#585—46no—
32#585—410no—
33#585D411yescharacteristic
34C2×2−1+4—215no—
35C2×2−1+4—46no—
36C2×H(5,𝔽2)—215no—
37C2×H(5,𝔽2)—410no—
38#578—260no—
39#578C23120yes—
40#578C23160yescomplemented
41#577—260no—
42#577—260no—
43#576—260no—
44#576C23160yes—
45#576C23120yescomplemented
46#575—245no—
47#575—245no—
48C2×#137—245no—
49C2×AGL(1,ℤ/8)—245no—
50C2×#135C23145yes—
51C2×#135C23145yescomplemented
52C22×Q16C23115yescomplemented
53C22×D8C23115yes—
54#568D4115yes—
55C2×#131C23115yes—
56#510C231180yescomplemented
57#510C23160yescomplemented
58#509C231180yescomplemented
59#509C23160yescomplemented
60#508C23160yescomplemented
61#508C231180yescomplemented
62C2×Q32C23145yescomplemented
63C2×SD32C23145yescomplemented
64C2×SD32C23190yescomplemented
65C2×SD32C23115yescomplemented
66C2×D16C23145yescomplemented
67#504C23120yescomplemented
68#504C23160yescomplemented
69C2×M32C23145yescomplemented
70C22×C16C23115yescomplemented
712−1+4—220no—
722−1+4—260no—
732−1+4—460no—
742−1+4—415no—
752−1+4—415no—
762−1+4D816yes—
77H(5,𝔽2)—260no—
78H(5,𝔽2)—220no—
79H(5,𝔽2)—460no—
80H(5,𝔽2)—445no—
81H(5,𝔽2)—445no—
82H(5,𝔽2)D8110yes—
83C2×#41—245no—
84C2×#41—245no—
85C2×#41—460no—
86C2×#41—460no—
87C2×#41—415no—
88C2×#41—445no—
89C2×#41C2×D4115yes—
90C22×Q8—215no—
91C22×Q8—415no—
92C22×D4—215no—
93C22×D4—445no—
94#137—2180no—
95#137—260no—
96AGL(1,ℤ/8)—2180no—
97AGL(1,ℤ/8)—260no—
98#135—290no—
99#135—290no—
100#135C24160yes—
101#135C24160yes—
102C2×Q16—245no—
103C2×Q16C24145yes—
104C2×Q16C24115yes—
105C2×SD16—245no—
106C2×SD16—245no—
107C2×D8—245no—
108C2×D8C24145yes—
109C2×D8C24115yes—
110#131C2×D4160yes—
111#131C24120yes—
112C2×M16C2×D4145yes—
113C22×C8C24115yes—
114Q32C24160yes—
115SD32C24110yes—
116SD32C2416yes—
117SD32C24190yes—
118SD32C24130yes—
119D16C24160yes—
120M32C24160yes—
121C2×C16C24115yes—
122C2×C16C24145yes—
123C24—415no—
124#41—260no—
125#41—260no—
126#41—460no—
127#41—420no—
128#41—460no—
129#41—4180no—
130#41—490no—
131#41—490no—
132#41C2×D8160yes—
133#41C22×D4120yes—
134C2×Q8—215no—
135C2×Q8—245no—
136C2×Q8—460no—
137C2×Q8—420no—
138C2×Q8—445no—
139C2×Q8C2×D8115yes—
140C2×D4—245no—
141C2×D4—215no—
142C2×D4—445no—
143C2×D4—445no—
144C2×D4—4180no—
145C2×D4—460no—
146C2×D4—445no—
147C2×D4C2×D8145yes—
148C22×C4—415no—
149C22×C4—445no—
150C22×C4—445no—
151C22×C4C22×D4115yes—
152Q16—245no—
153Q16—215no—
154Q16C25110yes—
155Q16C2516yes—
156SD16—260no—
157SD16—260no—
158D8—245no—
159D8—215no—
160D8C2516yes—
161D8C25110yes—
162M16C22×D4160yes—
163C2×C8C22×D4145yes—
164C2×C8C25115yes—
165C16C2511yescharacteristic
166C16C25115yes—
167C23—415no—
168C23—445no—
169C23—3215no—
170C23C22×D8115yes—
171Q8—26no—
172Q8—210no—
173Q8—445no—
174Q8—415no—
175Q8C22×D8120yes—
176D4—210no—
177D4—26no—
178D4—460no—
179D4—460no—
180D4—445no—
181D4—415no—
182D4C22×D8160yes—
183C2×C4—415no—
184C2×C4—445no—
185C2×C4—460no—
186C2×C4—460no—
187C2×C4C22×D8145yes—
188C2×C4C23×D4115yes—
189C8C23×D4115yes—
190C8C2611yescharacteristic
191C22—415no—
192C22—46no—
193C22—410no—
194C22—1615no—
195C22—1645no—
196C22C23×D8115yes—
197C4—46no—
198C4—410no—
199C4C23×D8115yes—
200C4C24×D411yescharacteristic
201C2—215no—
202C2—86no—
203C2—810no—
204C2C24×D811yescharacteristic
subgroupG/H
center, socleC2C24×D8
commutator, Frattini, cosocleC8C26
LCSG ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2 ◁ #585 ◁ #3573 ◁ G
LECSG ▷ C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #63093C2semidirect
2. #63091C2semidirect
3. #62637C2semidirect
4. #62636C2semidirect
5. #62635C2semidirect
6. C2×#3416C2semidirect
7. #62624C2semidirect
12. #3584C22semidirect
13. #3584C22semidirect
14. C2×#578C22semidirect
15. C2×#576C22semidirect
17. #3417C22semidirect
18. #3416C22semidirect
19. #3416C22semidirect
20. #3416C22semidirect
21. #3415C22semidirect
22. #3414C22semidirect
23. C2×#510C22semidirect
24. C2×#509C22semidirect
25. C2×#508C22semidirect
26. C22×SD32C22semidirect
27. #3407C22semidirect
28. C2×#504C22semidirect
40. #578C23semidirect
45. #576C23semidirect
51. C2×#135C23semidirect
52. C22×Q16C23semidirect
56. #510C23semidirect
57. #510C23semidirect
58. #509C23semidirect
59. #509C23semidirect
60. #508C23semidirect
61. #508C23semidirect
62. C2×Q32C23semidirect
63. C2×SD32C23semidirect
64. C2×SD32C23semidirect
65. C2×SD32C23semidirect
66. C2×D16C23semidirect
67. #504C23semidirect
68. #504C23semidirect
69. C2×M32C23semidirect
70. C22×C16C23semidirect

Maximal subgroups

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

subgroupindex
#630932
#630912
#626372
#626362
#626352
C2×#34162
#626242