Finite Group Databasebeta

← #5486631

Subgroups — #5486631

Order 512.

2306 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C211yesmaximal, characteristic, complemented
2#17231C214yesmaximal, complemented
3#16935C214yesmaximal, complemented
4#16546C214yesmaximal, complemented
5#16099C214yesmaximal, complemented
6#15948C211yesmaximal, characteristic
7#15282C214yesmaximal, complemented
8#15262C211yesmaximal, characteristic, complemented
9#15259C211yesmaximal, characteristic
10#14755C214yesmaximal, complemented
11#14662C211yesmaximal, characteristic, complemented
12#14657C211yesmaximal, characteristic
13#14167C211yesmaximal, characteristic, complemented
14C22×#124C2211yescharacteristic
15C22×#123C2211yescharacteristic
16C22×#120C2211yescharacteristic
17C2×#397C2214yes—
18C2×#395C2212yescomplemented
19C2×#393C2212yes—
20C2×#388C2214yes—
21C2×#388C2214yes—
22C2×#386C2212yes—
23C2×#385C2212yescomplemented
24C2×#382C2212yescomplemented
25C2×#379C2212yes—
26C2×#379C2212yes—
27C2×#379C2212yes—
28C2×C4×C22⋊C4C2212yescomplemented
29#1446C2212yescomplemented
30#1445C2214yescomplemented
31#1443C2214yescomplemented
32#1443C2214yescomplemented
33#1443C2214yescomplemented
34#1443C2214yescomplemented
35#1441C2214yescomplemented
36#1441C2214yescomplemented
37#1440C2212yes—
38#1440C2212yes—
39#1440C2212yes—
40#1440C2214yes—
41#1440C2214yes—
42#1440C2214yes—
43#1440C2212yescomplemented
44#1440C2212yescomplemented
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yescomplemented
53#1439C2214yescomplemented
54#1438C2211yescharacteristic
55#1438C2214yes—
56#1438C2211yescharacteristic
57#1438C2211yescharacteristic
58#1438C2212yes—
59#1438C2214yes—
60#1438C2211yescharacteristic
61#1437C2214yes—
62#1437C2212yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2212yescomplemented
67C26C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C2×#124—24no—
73C2×#124—44no—
74C2×#123—24no—
75C2×#123—44no—
76C2×#120—24no—
77C2×#120—44no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C22⋊C4C2312yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2311yescharacteristic
89C22×C22⋊C4C2311yescharacteristic
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C42C2311yescharacteristic
103C22×C42C2314yes—
104#397—48no—
105#395—44no—
106#393—44no—
107#388—48no—
108#388—48no—
109#386—44no—
110#385—44no—
111#382—44no—
112#379—44no—
113#379—44no—
114#379—44no—
115C4×C22⋊C4—44no—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2312yes—
120C2×#95C2314yes—
121C2×#95C2312yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C25—22no—
139C25—24no—
140C25—44no—
141C25—44no—
142C25—42no—
143C25—42no—
144C25C2411yescharacteristic
145C25C2411yescharacteristic
146C25C2412yes—
147C22×Q8—22no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×D4—24no—
152C22×D4—24no—
153C22×D4—42no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—42no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4#4114yes—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C2412yes—
165C23×C4C2414yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2411yescharacteristic
172C23×C4C2412yes—
173C23×C4C2412yes—
174#124—416no—
175#123—416no—
176#120—416no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—42no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—42no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C42—22no—
240C2×C42—24no—
241C2×C42—44no—
242C2×C42—44no—
243C2×C42—44no—
244#95—28no—
245#95—28no—
246#95—28no—
247#95—44no—
248#95—44no—
249#95—44no—
250#95—44no—
251#95—44no—
252#95—44no—
253#95—44no—
254#95—44no—
255#95—42no—
256#95—42no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—44no—
262#95—42no—
263#95—44no—
264#95—44no—
265#95—44no—
266C24—28no—
267C24—22no—
268C24—24no—
269C24—24no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—48no—
286C24—48no—
287C24—82no—
288C24—82no—
289C24—88no—
290C24—88no—
291C24—84no—
292C24—88no—
293C24—84no—
294C24—84no—
295C24—84no—
296C24—88no—
297C24—88no—
298C24C2×#4112yes—
299C24C2×#4112yes—
300C24H(5,𝔽2)14yes—
301C24C2511yescharacteristic
302C2×Q8—44no—
303C2×Q8—82no—
304C2×Q8—82no—
305C2×Q8—88no—
306C2×D4—44no—
307C2×D4—44no—
308C2×D4—44no—
309C2×D4—48no—
310C2×D4—48no—
311C2×D4—88no—
312C2×D4—84no—
313C2×D4—88no—
314C2×D4—88no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—88no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4C2×#4112yes—
374C22×C4H(5,𝔽2)14yes—
375C22×C4H(5,𝔽2)14yes—
376C4⋊C4—48no—
377C4⋊C4—88no—
378C4⋊C4—88no—
379C4⋊C4—88no—
380C22⋊C4—416no—
381C22⋊C4—48no—
382C22⋊C4—48no—
383C22⋊C4—48no—
384C22⋊C4—48no—
385C22⋊C4—416no—
386C22⋊C4—44no—
387C22⋊C4—44no—
388C22⋊C4—48no—
389C22⋊C4—48no—
390C22⋊C4—48no—
391C22⋊C4—88no—
392C22⋊C4—88no—
393C22⋊C4—84no—
394C22⋊C4—88no—
395C22⋊C4—84no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C22⋊C4—88no—
400C42—28no—
401C42—88no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—24no—
408C23—22no—
409C23—28no—
410C23—28no—
411C23—28no—
412C23—24no—
413C23—28no—
414C23—24no—
415C23—42no—
416C23—42no—
417C23—44no—
418C23—44no—
419C23—42no—
420C23—44no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—44no—
426C23—42no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—42no—
438C23—44no—
439C23—44no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—88no—
452C23—816no—
453C23—832no—
454C23—832no—
455C23—816no—
456C23—816no—
457C23C22×#4111yescharacteristic
458C23C2×H(5,𝔽2)12yes—
459C23C2×H(5,𝔽2)12yes—
460C23C2×H(5,𝔽2)12yes—
461C23#58514yes—
462C23#58514yes—
463Q8—88no—
464D4—88no—
465D4—816no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—88no—
524C22—48no—
525C22—48no—
526C22—42no—
527C22—42no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22—44no—
533C22—44no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—832no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22#349011yescharacteristic
547C22#349011yescharacteristic
548C22#349011yescharacteristic
549C22#352114yes—
550C22#352912yes—
551C22#353312yes—
552C22#353314yes—
553C22#353714yes—
554C22#354012yes—
555C22#354012yes—
556C22#354012yes—
557C22#354512yes—
558C22#356014yes—
559C22#356214yes—
560C4—44no—
561C4—42no—
562C4—44no—
563C4—84no—
564C4—84no—
565C4—84no—
566C4—84no—
567C4—82no—
568C4—82no—
569C4—82no—
570C2—82no—
571C2—84no—
572C2—82no—
573C2#4573112yes—
574C2#4638411yescharacteristic
575C2#4894912yes—
576C2#5180312yes—
577C2#5248214yes—
578C2#5506114yes—
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. #17273C2semidirect
2. #17231C2semidirect
3. #16935C2semidirect
4. #16546C2semidirect
5. #16099C2semidirect
7. #15282C2semidirect
8. #15262C2semidirect
10. #14755C2semidirect
11. #14662C2semidirect
13. #14167C2semidirect
18. C2×#395C22semidirect
23. C2×#385C22semidirect
24. C2×#382C22semidirect
28. C2×C4×C22⋊C4C22semidirect
29. #1446C22semidirect
30. #1445C22semidirect
31. #1443C22semidirect
32. #1443C22semidirect
33. #1443C22semidirect
34. #1443C22semidirect
35. #1441C22semidirect
36. #1441C22semidirect
43. #1440C22semidirect
44. #1440C22semidirect
52. #1439C22semidirect
53. #1439C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect

Maximal subgroups

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

subgroupindex
#172732
#172312
#169352
#165462
#160992
#159482
#152822
#152622
#152592
#147552
#146622
#146572
#141672