Finite Group Databasebeta

← #5482447

Subgroups — #5482447

Order 512.

2475 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17232C214yesmaximal, complemented
2#17198C211yesmaximal, characteristic, complemented
3#16885C214yesmaximal, complemented
4#16831C214yesmaximal, complemented
5#16245C211yesmaximal, characteristic, complemented
6#15784C214yesmaximal, complemented
7#15579C214yesmaximal, complemented
8#15259C212yesmaximal
9#14727C214yesmaximal, complemented
10#14649C212yesmaximal, complemented
11#14134C211yesmaximal, characteristic
12C22×#124C2212yes—
13C22×#120C2211yescharacteristic
14C23×C22⋊C4C2212yes—
15C2×#397C2214yes—
16C2×#397C2214yes—
17C2×#393C2212yes—
18C2×#393C2212yes—
19C2×#386C2214yes—
20C2×#385C2214yes—
21C2×#379C2214yes—
22#1446C2214yescomplemented
23#1446C2214yescomplemented
24#1443C2214yescomplemented
25#1443C2214yescomplemented
26#1443C2214yescomplemented
27#1443C2214yescomplemented
28#1442C2214yescomplemented
29#1442C2214yescomplemented
30#1442C2214yescomplemented
31#1442C2214yescomplemented
32#1440C2211yescharacteristic
33#1440C2214yes—
34#1440C2211yescharacteristic
35#1440C2214yes—
36#1440C2214yes—
37#1440C2214yes—
38#1440C2214yes—
39#1440C2214yescomplemented
40#1440C2214yescomplemented
41#1439C2214yes—
42#1439C2214yes—
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1438C2212yes—
46#1438C2214yes—
47#1438C2214yes—
48#1438C2212yes—
49#1437C2212yes—
50#1437C2212yes—
51#1436C2214yescomplemented
52#1436C2214yescomplemented
53C22⋊#95C2214yes—
54C22⋊#95C2214yes—
55C22⋊#95C2214yes—
56C22⋊#95C2214yes—
57C26C2311yescharacteristic
58C23×Q8C2312yes—
59C23×D4C2312yes—
60C23×D4C2311yescharacteristic
61C24×C4C2314yes—
62C2×#124—48no—
63C2×#124—44no—
64C2×#120—24no—
65C2×#120—44no—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2314yes—
68C22×C22⋊C4—48no—
69C22×C22⋊C4—44no—
70C22×C22⋊C4C2314yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2312yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2311yescharacteristic
84C22×C42C2312yes—
85#397—48no—
86#397—48no—
87#393—28no—
88#393—44no—
89#386—48no—
90#385—48no—
91#379—48no—
92C2×#95C2314yes—
93C2×#95C2312yes—
94C2×#95C2312yes—
95C2×#95C2314yes—
96C2×#95C2314yes—
97C2×#95C2314yes—
98C2×#95C2314yes—
99C2×#95C2314yes—
100C2×#95C2314yes—
101C2×#95C2314yes—
102C2×#95C2314yes—
103C2×#95C2314yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C25—22no—
116C25—24no—
117C25—28no—
118C25—44no—
119C25—44no—
120C25C2411yescharacteristic
121C25C2411yescharacteristic
122C25C2412yes—
123C22×Q8—24no—
124C22×Q8—44no—
125C22×Q8—42no—
126C22×Q8—44no—
127C22×Q8—42no—
128C22×D4—24no—
129C22×D4—28no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—42no—
133C22×D4—42no—
134C22×D4—44no—
135C22×D4—44no—
136C23×C4—28no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—44no—
143C23×C4C2414yes—
144C23×C4C2414yes—
145C23×C4C2414yes—
146C23×C4C2414yes—
147C23×C4C2414yes—
148C23×C4C2414yes—
149C23×C4C2411yescharacteristic
150C23×C4C2412yes—
151#124—432no—
152#120—416no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4—44no—
158C2×C4⋊C4—44no—
159C2×C22⋊C4—28no—
160C2×C22⋊C4—28no—
161C2×C22⋊C4—28no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—28no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—44no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—432no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4#4114yes—
199C2×C42—44no—
200C2×C42—42no—
201#95—24no—
202#95—28no—
203#95—44no—
204#95—44no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—44no—
216#95—42no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224C24—22no—
225C24—24no—
226C24—22no—
227C24—24no—
228C24—24no—
229C24—24no—
230C24—24no—
231C24—28no—
232C24—24no—
233C24—22no—
234C24—24no—
235C24—24no—
236C24—21no—
237C24—24no—
238C24—22no—
239C24—24no—
240C24—24no—
241C24—416no—
242C24—416no—
243C24—416no—
244C24—48no—
245C24—48no—
246C24—88no—
247C24—88no—
248C24—88no—
249C24—84no—
250C24—88no—
251C24C2×#4112yes—
252C24H(5,𝔽2)14yes—
253C24H(5,𝔽2)18yes—
254C24C2511yescharacteristic
255C2×Q8—48no—
256C2×Q8—84no—
257C2×Q8—88no—
258C2×Q8—84no—
259C2×Q8—88no—
260C2×D4—48no—
261C2×D4—416no—
262C2×D4—44no—
263C2×D4—48no—
264C2×D4—84no—
265C2×D4—88no—
266C2×D4—84no—
267C2×D4—88no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—21no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—416no—
318C22×C4—88no—
319C22×C4—88no—
320C22×C4—88no—
321C22×C4—88no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4C2×#4114yes—
325C4⋊C4—88no—
326C4⋊C4—88no—
327C22⋊C4—44no—
328C22⋊C4—48no—
329C22⋊C4—416no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—416no—
333C22⋊C4—416no—
334C22⋊C4—416no—
335C22⋊C4—88no—
336C22⋊C4—88no—
337C22⋊C4—88no—
338C22⋊C4—88no—
339C22⋊C4—88no—
340C22⋊C4—88no—
341C42—48no—
342C23—24no—
343C23—28no—
344C23—28no—
345C23—24no—
346C23—28no—
347C23—28no—
348C23—28no—
349C23—28no—
350C23—28no—
351C23—28no—
352C23—22no—
353C23—24no—
354C23—28no—
355C23—432no—
356C23—42no—
357C23—42no—
358C23—44no—
359C23—42no—
360C23—44no—
361C23—42no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—42no—
367C23—42no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—41no—
375C23—42no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23—44no—
385C23—832no—
386C23—88no—
387C23—832no—
388C23—832no—
389C23H(3,𝔽4)18yes—
390C23C22×#4111yescharacteristic
391C23C2×H(5,𝔽2)12yes—
392C23C2×H(5,𝔽2)14yes—
393C23#58514yes—
394C23#58514yes—
395Q8—816no—
396D4—816no—
397D4—88no—
398C2×C4—24no—
399C2×C4—28no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—41no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—832no—
449C22—28no—
450C22—216no—
451C22—44no—
452C22—48no—
453C22—44no—
454C22—48no—
455C22—48no—
456C22—48no—
457C22—48no—
458C22—44no—
459C22—42no—
460C22—48no—
461C22—832no—
462C22—84no—
463C22—84no—
464C22—82no—
465C22—84no—
466C22—81no—
467C22—84no—
468C22—84no—
469C22#349011yescharacteristic
470C22#351012yes—
471C22#351314yes—
472C22C2×H(3,𝔽4)12yes—
473C22#353312yes—
474C22#353314yes—
475C22#353414yes—
476C22#353414yes—
477C22#356014yes—
478C22#356014yes—
479C22#356814yes—
480C4—44no—
481C4—84no—
482C4—84no—
483C4—84no—
484C4—84no—
485C4—81no—
486C4—84no—
487C4—84no—
488C2—48no—
489C2—82no—
490C2—161no—
491C2#4556812yes—
492C2#4645711yescharacteristic
493C2#4825214yes—
494C2#5253014yes—
495C2#5580514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17232C2semidirect
2. #17198C2semidirect
3. #16885C2semidirect
4. #16831C2semidirect
5. #16245C2semidirect
6. #15784C2semidirect
7. #15579C2semidirect
9. #14727C2semidirect
10. #14649C2semidirect
22. #1446C22semidirect
23. #1446C22semidirect
24. #1443C22semidirect
25. #1443C22semidirect
26. #1443C22semidirect
27. #1443C22semidirect
28. #1442C22semidirect
29. #1442C22semidirect
30. #1442C22semidirect
31. #1442C22semidirect
39. #1440C22semidirect
40. #1440C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
51. #1436C22semidirect
52. #1436C22semidirect

Maximal subgroups

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

subgroupindex
#172322
#171982
#168852
#168312
#162452
#157842
#155792
#152592
#147272
#146492
#141342