Finite Group Databasebeta

← #10573989

Subgroups — #10573989

Order 512.

4584 conjugacy classes of subgroups.

The subgroup lattice has 249 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#53380C217yesmaximal, complemented
2#53307C217yesmaximal, complemented
3#53282C2121yesmaximal, complemented
4#53193C2121yesmaximal, complemented
5#53158C217yesmaximal, complemented
6#2860C2217yescomplemented
7#2856C2217yescomplemented
8#2853C22121yescomplemented
9#2850C22121yescomplemented
10#2846C2217yes—
11#2803C22121yescomplemented
12#2797C22121yescomplemented
13#2793C22121yescomplemented
14#2785C2217yescomplemented
15#2783C22121yescomplemented
16#2761C22121yescomplemented
17#2754C22121yescomplemented
18#2753C22121yescomplemented
19#2751C22121yescomplemented
20#2736C2217yes—
21#2735C22121yes—
22#2730C22121yescomplemented
23#2723C22121yescomplemented
24#2718C22121yescomplemented
25#2714C22121yescomplemented
26#2700C22121yescomplemented
27#2699C22121yescomplemented
28#2697C22121yes—
29#2688C22121yescomplemented
30#2685C22121yes—
31#2684C22121yes—
32#2681C22121yescomplemented
33#2678C22121yes—
34#2678C22121yes—
35#2676C22121yescomplemented
36#2675C2217yescomplemented
37#2673C22121yescomplemented
38#2670C22121yes—
39#2669C22121yes—
40#2668C22121yes—
41C26C2311yescharacteristic
42C2×#127C2317yes—
43C2×#126C2317yes—
44C2×#126C2317yescomplemented
45C2×#124C23121yes—
46C2×#123C23121yes—
47C2×C22⋊Q8C23121yescomplemented
48C2×#121C23121yes—
49C2×#121C23121yes—
50C2×#121C23121yescomplemented
51C2×#120C23121yes—
52C2×#120C23121yes—
53C2×#120C2317yes—
54C22×C42C2317yescomplemented
55#400C23121yescomplemented
56#399C23121yes—
57#398C23121yescomplemented
58#397C23121yes—
59#397C23121yes—
60#397C23121yes—
61#397C23121yes—
62#397C23121yescomplemented
63#396C2317yes—
64#396C23121yes—
65#396C23121yes—
66#396C2317yescomplemented
67#395C2317yescomplemented
68#394C23121yes—
69#394C23121yes—
70#394C23121yes—
71#394C23121yes—
72#394C23121yes—
73#394C23121yes—
74#394C23121yescomplemented
75#393C23121yes—
76#393C2317yes—
77#393C2317yescomplemented
78#392C23121yes—
79#392C23121yes—
80#392C23121yes—
81#392C23121yes—
82#392C2317yescomplemented
83#391C23121yescomplemented
84#390C23121yes—
85#390C23121yes—
86#390C23121yes—
87#390C23121yescomplemented
88#389C23121yescomplemented
89#388C23121yes—
90#388C23121yes—
91#388C23121yes—
92#388C23121yes—
93#388C23121yes—
94#388C23121yes—
95#388C23121yes—
96#388C23121yescomplemented
97#388C23121yescomplemented
98#388C23121yescomplemented
99#387C23121yescomplemented
100#387C23121yescomplemented
101#387C23121yescomplemented
102#387C23121yescomplemented
103#386C23121yes—
104#386C23121yes—
105#386C23121yes—
106#386C23121yes—
107#386C23121yescomplemented
108#386C23121yescomplemented
109#385C23121yescomplemented
110#385C23121yescomplemented
111#384C2317yescomplemented
112#382C23121yescomplemented
113#381C23121yescomplemented
114#379C23121yes—
115#379C23121yes—
116#379C2317yes—
117C43C2311yescharacteristic, complemented
118C25—87no—
119C25C2417yes—
120C22×Q8C2417yes—
121C22×D4C2417yes—
122C22×D4C24121yes—
123C22×D4C24121yes—
124C22×D4C24121yes—
125C23×C4C24121yes—
126#127—87no—
127#126—87no—
128#126—87no—
129#124—821no—
130#123—821no—
131C22⋊Q8—821no—
132#121—821no—
133#121—821no—
134#121—821no—
135#120—87no—
136#120—821no—
137#120—821no—
138C2×C4⋊C4C24121yes—
139C2×C4⋊C4C24121yes—
140C2×C4⋊C4C24121yes—
141C2×C4⋊C4C24121yes—
142C2×C4⋊C4C24121yes—
143C2×C22⋊C4C24121yes—
144C2×C22⋊C4C24121yes—
145C2×C22⋊C4C24121yes—
146C2×C22⋊C4C24121yes—
147C2×C22⋊C4C24121yes—
148C2×C22⋊C4C24121yes—
149C2×C22⋊C4C24121yes—
150C2×C22⋊C4C24121yes—
151C2×C22⋊C4C24121yes—
152C2×C22⋊C4C24121yes—
153C2×C22⋊C4C24121yes—
154C2×C22⋊C4C24121yes—
155C2×C42—87no—
156C2×C42C24121yes—
157C2×C42C2417yes—
158C2×C42C2417yes—
159#95C24121yes—
160#95C24121yes—
161#95C24121yes—
162#95C24121yes—
163#95C24121yes—
164#95C24121yes—
165#95C2417yes—
166#95C24121yes—
167C24—47no—
168C24—421no—
169C24—421no—
170C24—856no—
171C24C2517yes—
172C24C2517yes—
173C2×Q8—421no—
174C2×D4—421no—
175C2×D4—421no—
176C2×D4—421no—
177C2×D4—421no—
178C2×D4—421no—
179C2×D4—421no—
180C2×D4—421no—
181C2×D4—421no—
182C2×D4—421no—
183C2×D4—421no—
184C22×C4—421no—
185C22×C4—421no—
186C22×C4—421no—
187C22×C4C25121yes—
188C22×C4C25121yes—
189C22×C4C2517yes—
190C4⋊C4—421no—
191C4⋊C4—421no—
192C4⋊C4—421no—
193C4⋊C4—421no—
194C4⋊C4—421no—
195C22⋊C4—421no—
196C22⋊C4—421no—
197C22⋊C4—421no—
198C22⋊C4—421no—
199C22⋊C4—421no—
200C22⋊C4—421no—
201C22⋊C4—421no—
202C22⋊C4—421no—
203C22⋊C4—421no—
204C22⋊C4—421no—
205C22⋊C4—421no—
206C22⋊C4—421no—
207C42—421no—
208C42—47no—
209C42—47no—
210C23—27no—
211C23—221no—
212C23—221no—
213C23—221no—
214C23—27no—
215C23—221no—
216C23—814no—
217C23—842no—
218C23—842no—
219C23—864no—
220C23C2611yescharacteristic
221Q8—167no—
222D4—1621no—
223D4—1621no—
224D4—1621no—
225D4—167no—
226C2×C4—221no—
227C2×C4—221no—
228C2×C4—221no—
229C2×C4—221no—
230C2×C4—221no—
231C2×C4—221no—
232C2×C4—221no—
233C2×C4—842no—
234C22—421no—
235C22—47no—
236C22—421no—
237C22—421no—
238C22—421no—
239C22—47no—
240C22—856no—
241C22H(7,𝔽2)17yes—
242C4—421no—
243C4—421no—
244C4—47no—
245C2—87no—
246C2—87no—
247C2#6297317yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #53380C2semidirect
2. #53307C2semidirect
3. #53282C2semidirect
4. #53193C2semidirect
5. #53158C2semidirect
6. #2860C22semidirect
7. #2856C22semidirect
8. #2853C22semidirect
9. #2850C22semidirect
11. #2803C22semidirect
12. #2797C22semidirect
13. #2793C22semidirect
14. #2785C22semidirect
15. #2783C22semidirect
16. #2761C22semidirect
17. #2754C22semidirect
18. #2753C22semidirect
19. #2751C22semidirect
22. #2730C22semidirect
23. #2723C22semidirect
24. #2718C22semidirect
25. #2714C22semidirect
26. #2700C22semidirect
27. #2699C22semidirect
29. #2688C22semidirect
32. #2681C22semidirect
35. #2676C22semidirect
36. #2675C22semidirect
37. #2673C22semidirect
44. C2×#126C23semidirect
47. C2×C22⋊Q8C23semidirect
50. C2×#121C23semidirect
54. C22×C42C23semidirect
55. #400C23semidirect
57. #398C23semidirect
62. #397C23semidirect
66. #396C23semidirect
67. #395C23semidirect
74. #394C23semidirect
77. #393C23semidirect
82. #392C23semidirect
83. #391C23semidirect
87. #390C23semidirect
88. #389C23semidirect
96. #388C23semidirect
97. #388C23semidirect
98. #388C23semidirect
99. #387C23semidirect
100. #387C23semidirect
101. #387C23semidirect
102. #387C23semidirect
107. #386C23semidirect
108. #386C23semidirect
109. #385C23semidirect
110. #385C23semidirect
111. #384C23semidirect
112. #382C23semidirect
113. #381C23semidirect
117. C43C23semidirect

Maximal subgroups

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

subgroupindex
#533802
#533072
#532822
#531932
#531582