Finite Group Databasebeta

← #6633279

Subgroups — #6633279

Order 512.

1854 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37387C211yesmaximal, characteristic, complemented
2#35554C211yesmaximal, characteristic
3#32914C212yesmaximal, complemented
4#31637C218yesmaximal, complemented
5#30741C218yesmaximal, complemented
6#29723C212yesmaximal, complemented
7#29332C212yesmaximal, complemented
8#25717C212yesmaximal
9#23942C212yesmaximal, complemented
10#23942C212yesmaximal, complemented
11#21913C211yesmaximal, characteristic, complemented
12C2×#541C2212yes—
13C22×#127C2211yescharacteristic
14C2×#518C2211yescharacteristic
15C2×#492C2214yes—
16C2×#461C2214yescomplemented
17C2×#450C2214yes—
18C2×#450C2214yes—
19#2857C2212yes—
20#2726—24no—
21#2722—24no—
22#2621C2212yes—
23#2565C2214yescomplemented
24#2546C2214yescomplemented
25#2445—24no—
26#2445C2218yes—
27#2405—24no—
28#2309C2211yescharacteristic
29#2309C2211yescharacteristic
30#2309C2211yescharacteristic
31#2296C2211yescharacteristic
32#2296C2211yescharacteristic
33#2099C2214yescomplemented
34#2073C2214yescomplemented
35#2055C2218yescomplemented
36#2043C2214yescomplemented
37#2025C2214yescomplemented
38#2025C2214yescomplemented
39#2014C2218yes—
40#2013C2214yescomplemented
41#2013C2214yescomplemented
42#2013C2214yescomplemented
43#1957C2214yes—
44#1933C2218yescomplemented
45#1916C2218yescomplemented
46#1880C2218yescomplemented
47#1880C2218yescomplemented
48#1877C2218yescomplemented
49#1873C2214yes—
50#1873C2214yes—
51#1818C2212yes—
52#1804C2214yescomplemented
53#1798C2212yes—
54C4×C22⋊C8C2212yes—
55C23×D4—22no—
56C23×D4C2311yescharacteristic
57C23×D4C2311yescharacteristic
58C22×SD16C2318yes—
59C22×SD16C2318yes—
60#541—48no—
61C2×C4⋊Q8C2312yes—
62C2×C4⋊Q8C2312yes—
63C2×C4⋊Q8C2312yes—
64C2×#127—22no—
65C2×#127—44no—
66C2×#127C2314yes—
67C2×#126—24no—
68C2×#125—24no—
69C2×#125C2312yes—
70C2×#123—28no—
71C2×C22⋊Q8C2314yes—
72#518—44no—
73C2×C4×Q8C2314yes—
74C2×C4×D4—22no—
75C2×C4×D4C2314yes—
76C2×#117C2311yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C22⋊C4—22no—
81C22×C42C2311yescharacteristic
82#492—48no—
83#461—48no—
84#450—48no—
85#450—48no—
86C2×C4⋊C8C2314yes—
87C2×#103C2318yes—
88C2×#103C2318yes—
89C2×#102C2318yes—
90C2×#102C2318yes—
91C2×C22⋊C8C2314yes—
92C2×C4×C8C2314yes—
93#398—28no—
94#398—24no—
95#398—28no—
96#397—24no—
97#394—28no—
98#394—28no—
99#391C2314yes—
100#391C2314yes—
101#390—22no—
102#390—22no—
103#390—22no—
104#390C2314yes—
105#390C2314yes—
106#390C2314yes—
107#389—28no—
108#389—24no—
109#389D414yes—
110#389C2314yes—
111#387—28no—
112#386—28no—
113#385—22no—
114#385D414yes—
115#384C2312yes—
116#382C2312yes—
117#380—24no—
118#379—22no—
119#379—22no—
120#379—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4C2314yes—
123#340C2318yes—
124#340C2318yes—
125#336C2314yes—
126#336C2314yes—
127#336C2314yes—
128#336C2314yes—
129C25—21no—
130C25—21no—
131C22×Q8C2414yes—
132C22×D4—24no—
133C22×D4—21no—
134C22×D4—21no—
135C22×D4—28no—
136C22×D4—24no—
137C22×D4—24no—
138C22×D4—42no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—48no—
142C22×D4—88no—
143C22×D4C2412yes—
144C22×D4C2412yes—
145C23×C4—24no—
146C23×C4C2×D412yes—
147C23×C4C2411yescharacteristic
148C2×SD16—216no—
149C2×SD16—48no—
150C2×SD16—48no—
151C2×SD16—48no—
152C2×SD16—48no—
153C2×SD16—48no—
154C22×C8C2418yes—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158#127—28no—
159#127—44no—
160#127—48no—
161#126—416no—
162#125—44no—
163#125—416no—
164#123—816no—
165C22⋊Q8—28no—
166C22⋊Q8—28no—
167C4×Q8—44no—
168C4×Q8—44no—
169C4×D4—28no—
170C4×D4—44no—
171C4×D4—48no—
172#117—24no—
173#117—42no—
174#117—42no—
175#117—42no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D418yes—
186C2×C4⋊C4C2×D412yes—
187C2×C4⋊C4C2412yes—
188C2×C4⋊C4C2412yes—
189C2×C4⋊C4C2414yes—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—48no—
198C2×C22⋊C4C2×D412yes—
199C2×C22⋊C4C2×D412yes—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—24no—
203C2×C42—42no—
204C2×C42C2×D412yes—
205C2×C42C2412yes—
206C2×C42C2412yes—
207C2×C42C2412yes—
208C4⋊C8—44no—
209#103—48no—
210#103—48no—
211#102—48no—
212#102—48no—
213C22⋊C8—28no—
214C4×C8—44no—
215#95—24no—
216#95—28no—
217#95—24no—
218#95—28no—
219#95C2×D418yes—
220C24—24no—
221C24—22no—
222C24—22no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24—84no—
229C24—84no—
230C24C22×D411yescharacteristic
231C2×Q8—28no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—28no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—24no—
242C2×D4—416no—
243C2×D4—42no—
244C2×D4—416no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—416no—
250C2×D4—88no—
251C2×D4—88no—
252C2×D4—832no—
253C2×D4H(5,𝔽2)14yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—28no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—48no—
263C22×C4—88no—
264C22×C4C22×D414yes—
265C22×C4C22×D414yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D414yes—
268C22×C4H(5,𝔽2)14yes—
269C22×C4C2511yescharacteristic
270SD16—816no—
271SD16—816no—
272C2×C8—28no—
273C2×C8—28no—
274C2×C8—28no—
275C4⋊C4—24no—
276C4⋊C4—216no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—48no—
283C4⋊C4—48no—
284C4⋊C4—48no—
285C4⋊C4—48no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—84no—
289C4⋊C4—84no—
290C22⋊C4—24no—
291C22⋊C4—42no—
292C22⋊C4—48no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—48no—
296C22⋊C4—88no—
297C22⋊C4—88no—
298C22⋊C4—88no—
299C42—24no—
300C42—24no—
301C42—24no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—48no—
306C23—21no—
307C23—21no—
308C23—24no—
309C23—24no—
310C23—48no—
311C23—42no—
312C23—42no—
313C23—48no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—48no—
318C23—84no—
319C23—88no—
320C23—88no—
321C23—164no—
322C23—164no—
323C23—164no—
324C23—164no—
325C23#53512yes—
326C23C23×D411yescharacteristic
327Q8—48no—
328Q8—48no—
329D4—28no—
330D4—48no—
331D4—44no—
332D4—48no—
333D4—832no—
334D4—84no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—28no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—48no—
354C2×C4—168no—
355C2×C4#53512yes—
356C2×C4#57718yes—
357C2×C4C2×H(5,𝔽2)12yes—
358C2×C4C2×H(5,𝔽2)12yes—
359C2×C4#58512yes—
360C8—48no—
361C22—22no—
362C22—24no—
363C22—42no—
364C22—42no—
365C22—44no—
366C22—48no—
367C22—84no—
368C22—84no—
369C22—82no—
370C22—84no—
371C22—84no—
372C22—84no—
373C22—168no—
374C22—168no—
375C22C2×#53511yescharacteristic
376C22#348411yescharacteristic
377C22#348711yescharacteristic
378C22C2×#57714yes—
379C4—22no—
380C4—28no—
381C4—44no—
382C4—44no—
383C4—84no—
384C4—84no—
385C4—82no—
386C4—82no—
387C4#353414yes—
388C2—42no—
389C2—84no—
390C2—162no—
391C2#3671011yescharacteristic
392C2#6082012yes—
393C2#6163112yes—
394C2#6193312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37387C2semidirect
3. #32914C2semidirect
4. #31637C2semidirect
5. #30741C2semidirect
6. #29723C2semidirect
7. #29332C2semidirect
9. #23942C2semidirect
10. #23942C2semidirect
11. #21913C2semidirect
16. C2×#461C22semidirect
23. #2565C22semidirect
24. #2546C22semidirect
33. #2099C22semidirect
34. #2073C22semidirect
35. #2055C22semidirect
36. #2043C22semidirect
37. #2025C22semidirect
38. #2025C22semidirect
40. #2013C22semidirect
41. #2013C22semidirect
42. #2013C22semidirect
44. #1933C22semidirect
45. #1916C22semidirect
46. #1880C22semidirect
47. #1880C22semidirect
48. #1877C22semidirect
52. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#373872
#355542
#329142
#316372
#307412
#297232
#293322
#257172
#239422
#219132