Finite Group Databasebeta

← #4125875

Subgroups — #4125875

Order 512.

2782 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C211yesmaximal, characteristic, complemented
2#16147C214yesmaximal, complemented
3#16012C214yesmaximal, complemented
4#15948C211yesmaximal, characteristic
5#15770C214yesmaximal, complemented
6#15262C211yesmaximal, characteristic, complemented
7#15259C211yesmaximal, characteristic
8#14169C211yesmaximal, characteristic, complemented
9#14167C211yesmaximal, characteristic, complemented
10#14134C211yesmaximal, characteristic
11#14066C214yesmaximal, complemented
12#13988C214yesmaximal, complemented
13#13873C214yesmaximal, complemented
14C22×#124C2211yescharacteristic
15C22×#123C2211yescharacteristic
16C22×#120C2211yescharacteristic
17C23×C22⋊C4C2212yes—
18C2×#395C2212yescomplemented
19C2×#393C2212yes—
20C2×#388C2214yes—
21C2×#388C2214yes—
22C2×#386C2214yes—
23C2×#386C2212yes—
24C2×#385C2212yescomplemented
25C2×#380C2214yes—
26C2×#379C2212yes—
27C2×#379C2212yes—
28C2×C4×C22⋊C4C2214yes—
29C2×C4×C22⋊C4C2212yescomplemented
30C2×#376C2212yescomplemented
31#1446C2212yescomplemented
32#1445C2214yescomplemented
33#1445C2214yescomplemented
34#1443C2214yescomplemented
35#1443C2214yescomplemented
36#1442C2214yescomplemented
37#1440C2212yes—
38#1440C2212yes—
39#1440C2212yes—
40#1440C2214yes—
41#1440C2214yes—
42#1440C2212yescomplemented
43#1440C2212yescomplemented
44#1439C2214yes—
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yescomplemented
48#1438C2211yescharacteristic
49#1438C2211yescharacteristic
50#1438C2211yescharacteristic
51#1438C2212yes—
52#1438C2211yescharacteristic
53#1438C2214yes—
54#1437C2214yes—
55#1437C2214yes—
56#1437C2212yes—
57#1437C2214yescomplemented
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2212yescomplemented
62#1434C2214yescomplemented
63#1434C2214yescomplemented
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C26C2311yescharacteristic
68C23×Q8C2311yescharacteristic
69C23×D4C2312yes—
70C23×D4C2311yescharacteristic
71C24×C4C2314yes—
72C24×C4C2311yescharacteristic
73C2×#124—24no—
74C2×#124—44no—
75C2×#123—24no—
76C2×#123—44no—
77C2×#120—24no—
78C2×#120—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C22⋊C4—216no—
83C22×C22⋊C4—44no—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2312yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2314yes—
105C22×C42C2314yes—
106#395—44no—
107#393—44no—
108#388—216no—
109#388—48no—
110#386—216no—
111#386—44no—
112#385—44no—
113#380—48no—
114#379—44no—
115#379—44no—
116C4×C22⋊C4—44no—
117C4×C22⋊C4—48no—
118#376—44no—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C25—22no—
142C25—24no—
143C25—28no—
144C25—44no—
145C25—42no—
146C25—42no—
147C25C2411yescharacteristic
148C25C2411yescharacteristic
149C25C2412yes—
150C22×Q8—22no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×D4—24no—
155C22×D4—28no—
156C22×D4—24no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—42no—
160C22×D4—42no—
161C22×D4—44no—
162C22×D4#4114yes—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—28no—
166C23×C4—28no—
167C23×C4—28no—
168C23×C4—44no—
169C23×C4—44no—
170C23×C4—44no—
171C23×C4—44no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4C2412yes—
175C23×C4C2412yes—
176C23×C4C2414yes—
177C23×C4C2414yes—
178C23×C4C2412yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2411yescharacteristic
182C23×C4C2412yes—
183C23×C4C2412yes—
184#124—416no—
185#123—416no—
186#120—416no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—432no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—42no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4#41116yes—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—28no—
247C2×C42—28no—
248C2×C42—44no—
249C2×C42—44no—
250C2×C42—44no—
251C2×C42—44no—
252#95—28no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—28no—
257#95—28no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—44no—
264#95—42no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—42no—
271#95—44no—
272#95—42no—
273#95—44no—
274C24—22no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—22no—
279C24—24no—
280C24—24no—
281C24—28no—
282C24—24no—
283C24—24no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—416no—
293C24—416no—
294C24—48no—
295C24—48no—
296C24—48no—
297C24—48no—
298C24—88no—
299C24—88no—
300C24—88no—
301C24—82no—
302C24—82no—
303C24—84no—
304C24—84no—
305C24C2×#4112yes—
306C24C2×#4112yes—
307C24C2×#4118yes—
308C24H(5,𝔽2)14yes—
309C24C2511yescharacteristic
310C2×Q8—44no—
311C2×Q8—82no—
312C2×Q8—82no—
313C2×Q8—88no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—44no—
317C2×D4—48no—
318C2×D4—416no—
319C2×D4—48no—
320C2×D4—88no—
321C2×D4—84no—
322C2×D4—88no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—416no—
377C22×C4—416no—
378C22×C4—416no—
379C22×C4—88no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—88no—
385C22×C4—88no—
386C22×C4C2×#4118yes—
387C22×C4C2×#4118yes—
388C22×C4C2×#4112yes—
389C22×C4H(5,𝔽2)14yes—
390C22×C4H(5,𝔽2)14yes—
391C4⋊C4—48no—
392C4⋊C4—88no—
393C4⋊C4—88no—
394C22⋊C4—232no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—416no—
400C22⋊C4—416no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—44no—
406C22⋊C4—44no—
407C22⋊C4—48no—
408C22⋊C4—88no—
409C22⋊C4—88no—
410C22⋊C4—84no—
411C22⋊C4—84no—
412C22⋊C4—84no—
413C42—28no—
414C42—416no—
415C42—88no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—24no—
420C23—28no—
421C23—28no—
422C23—28no—
423C23—28no—
424C23—24no—
425C23—28no—
426C23—28no—
427C23—28no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—432no—
435C23—42no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—42no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—42no—
458C23—42no—
459C23—44no—
460C23—44no—
461C23—42no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—832no—
466C23—88no—
467C23—816no—
468C23—816no—
469C23—832no—
470C23#55118yes—
471C23C22×#4114yes—
472C23C22×#4111yescharacteristic
473C23C2×H(5,𝔽2)12yes—
474C23C2×H(5,𝔽2)12yes—
475C23C2×H(5,𝔽2)12yes—
476C23#58514yes—
477Q8—88no—
478D4—88no—
479D4—816no—
480C2×C4—28no—
481C2×C4—28no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—24no—
488C2×C4—22no—
489C2×C4—28no—
490C2×C4—28no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—42no—
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—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—88no—
538C2×C4—832no—
539C22—28no—
540C22—216no—
541C22—44no—
542C22—44no—
543C22—48no—
544C22—48no—
545C22—48no—
546C22—48no—
547C22—48no—
548C22—42no—
549C22—42no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—832no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22C2×#55112yes—
562C22#347514yes—
563C22#348112yes—
564C22#348112yes—
565C22#349011yescharacteristic
566C22#349011yescharacteristic
567C22#349011yescharacteristic
568C22#349414yes—
569C22#351314yes—
570C22#351414yes—
571C22#353312yes—
572C22#353714yes—
573C22#354012yes—
574C22#355112yes—
575C4—44no—
576C4—42no—
577C4—48no—
578C4—48no—
579C4—44no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4—84no—
585C2—48no—
586C2—82no—
587C2—82no—
588C2#4638411yescharacteristic
589C2#4725714yes—
590C2#5919312yes—
591C2#5946814yes—
592C2#5962112yes—
593C2#5998812yes—
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. #16147C2semidirect
3. #16012C2semidirect
5. #15770C2semidirect
6. #15262C2semidirect
8. #14169C2semidirect
9. #14167C2semidirect
11. #14066C2semidirect
12. #13988C2semidirect
13. #13873C2semidirect
18. C2×#395C22semidirect
24. C2×#385C22semidirect
29. C2×C4×C22⋊C4C22semidirect
30. C2×#376C22semidirect
31. #1446C22semidirect
32. #1445C22semidirect
33. #1445C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
42. #1440C22semidirect
43. #1440C22semidirect
47. #1439C22semidirect
57. #1437C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect

Maximal subgroups

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

subgroupindex
#172732
#161472
#160122
#159482
#157702
#152622
#152592
#141692
#141672
#141342
#140662
#139882
#138732