Finite Group Databasebeta

← #5486623

Subgroups — #5486623

Order 512.

3290 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C211yesmaximal, characteristic, complemented
2#17232C214yesmaximal, complemented
3#16885C214yesmaximal, complemented
4#16368C214yesmaximal, complemented
5#15947C211yesmaximal, characteristic
6#15391C214yesmaximal, complemented
7#15266C214yesmaximal, complemented
8#15259C211yesmaximal, characteristic, complemented
9#15259C211yesmaximal, characteristic, complemented
10#14718C214yesmaximal, complemented
11#14662C211yesmaximal, characteristic, complemented
12#14657C211yesmaximal, characteristic, complemented
13#14134C211yesmaximal, characteristic, complemented
14C22×#124C2211yescharacteristic
15C22×#120C2211yescharacteristic
16C22×#120C2211yescharacteristic
17C22×#120C2212yes—
18C22×#120C2212yes—
19C23×C22⋊C4C2212yescomplemented
20C2×#394C2214yes—
21C2×#394C2214yes—
22C2×#393C2212yescomplemented
23C2×#393C2212yescomplemented
24C2×#386C2214yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#385C2214yescomplemented
28C2×#382C2212yescomplemented
29C2×#379C2212yescomplemented
30C2×#379C2212yescomplemented
31C2×#379C2212yescomplemented
32#1446C2212yescomplemented
33#1445C2214yescomplemented
34#1443C2214yescomplemented
35#1443C2214yescomplemented
36#1442C2214yescomplemented
37#1442C2214yescomplemented
38#1440C2214yescomplemented
39#1440C2214yescomplemented
40#1440C2212yescomplemented
41#1440C2212yescomplemented
42#1440C2212yescomplemented
43#1440C2214yescomplemented
44#1440C2214yescomplemented
45#1440C2214yescomplemented
46#1440C2212yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1439C2214yescomplemented
53#1438C2214yes—
54#1438C2211yescharacteristic
55#1438C2211yescharacteristic
56#1438C2214yes—
57#1438C2211yescharacteristic
58#1438C2211yescharacteristic
59#1438C2212yes—
60#1438C2212yes—
61#1438C2214yes—
62#1438C2214yes—
63#1437C2212yescomplemented
64#1437C2212yescomplemented
65C22⋊#95C2214yescomplemented
66C22⋊#95C2214yescomplemented
67C26C2311yescharacteristic
68C26C2311yescharacteristic
69C23×Q8C2311yescharacteristic
70C23×D4C2314yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2311yescharacteristic
74C24×C4C2314yes—
75C2×#124—24no—
76C2×#124—44no—
77C2×#120—24no—
78C2×#120—24no—
79C2×#120—48no—
80C2×#120—44no—
81C2×#120—44no—
82C2×#120—44no—
83C2×#120—48no—
84C2×#120—44no—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C22⋊C4—48no—
88C22×C22⋊C4—44no—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2311yescharacteristic
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C42C2311yescharacteristic
113#394—48no—
114#394—48no—
115#393—44no—
116#393—44no—
117#386—48no—
118#386—48no—
119#386—48no—
120#385—48no—
121#382—44no—
122#379—44no—
123#379—44no—
124#379—44no—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C25—22no—
145C25—24no—
146C25—22no—
147C25—24no—
148C25—28no—
149C25—44no—
150C25—42no—
151C25—44no—
152C25—42no—
153C25—44no—
154C25—42no—
155C25—42no—
156C25C2411yescharacteristic
157C25C2411yescharacteristic
158C25C2412yes—
159C25C2412yes—
160C22×Q8—22no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—42no—
164C22×D4—24no—
165C22×D4—28no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—28no—
169C22×D4—28no—
170C22×D4—28no—
171C22×D4—28no—
172C22×D4—24no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—42no—
180C22×D4—44no—
181C22×D4—44no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4#4114yes—
185C23×C4—28no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4C2414yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2414yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201#124—416no—
202#120—416no—
203#120—416no—
204#120—432no—
205#120—432no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—42no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—432no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C42—22no—
273C2×C42—24no—
274#95—28no—
275#95—44no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—42no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—42no—
288#95—42no—
289#95—44no—
290#95—44no—
291#95—44no—
292#95—44no—
293C24—22no—
294C24—24no—
295C24—24no—
296C24—22no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—28no—
301C24—28no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—24no—
306C24—24no—
307C24—24no—
308C24—24no—
309C24—22no—
310C24—22no—
311C24—22no—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—24no—
317C24—24no—
318C24—24no—
319C24—24no—
320C24—416no—
321C24—416no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24—48no—
326C24—48no—
327C24—48no—
328C24—88no—
329C24—88no—
330C24—88no—
331C24—82no—
332C24—82no—
333C24—82no—
334C24—82no—
335C24—88no—
336C24—84no—
337C24—88no—
338C24—88no—
339C24—84no—
340C24—84no—
341C24—84no—
342C24—84no—
343C24—88no—
344C24—88no—
345C24—84no—
346C24C2×#4112yes—
347C24C2×#4112yes—
348C24H(5,𝔽2)14yes—
349C24H(5,𝔽2)14yes—
350C24H(5,𝔽2)18yes—
351C24C2511yescharacteristic
352C2×Q8—44no—
353C2×Q8—82no—
354C2×Q8—82no—
355C2×Q8—88no—
356C2×D4—44no—
357C2×D4—44no—
358C2×D4—44no—
359C2×D4—48no—
360C2×D4—48no—
361C2×D4—416no—
362C2×D4—416no—
363C2×D4—416no—
364C2×D4—416no—
365C2×D4—48no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—88no—
369C2×D4—84no—
370C2×D4—88no—
371C2×D4—88no—
372C2×D4—88no—
373C2×D4—88no—
374C2×D4—84no—
375C2×D4—88no—
376C2×D4—88no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—416no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—88no—
433C22×C4—88no—
434C22×C4C2×#4112yes—
435C22×C4H(5,𝔽2)14yes—
436C4⋊C4—88no—
437C4⋊C4—88no—
438C22⋊C4—416no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—48no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—416no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—416no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—416no—
452C22⋊C4—88no—
453C22⋊C4—88no—
454C22⋊C4—88no—
455C22⋊C4—88no—
456C22⋊C4—84no—
457C22⋊C4—88no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C42—28no—
462C23—24no—
463C23—28no—
464C23—28no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—22no—
478C23—22no—
479C23—28no—
480C23—28no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—24no—
485C23—432no—
486C23—42no—
487C23—42no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—832no—
533C23—88no—
534C23—88no—
535C23—816no—
536C23—816no—
537C23—816no—
538C23—832no—
539C23—816no—
540C23—816no—
541C23—832no—
542C23—832no—
543C23H(3,𝔽4)18yes—
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×H(5,𝔽2)12yes—
548C23C2×H(5,𝔽2)14yes—
549C23#58514yes—
550Q8—88no—
551D4—88no—
552D4—816no—
553D4—832no—
554D4—816no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—28no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—832no—
608C22—28no—
609C22—216no—
610C22—44no—
611C22—44no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—48no—
616C22—48no—
617C22—44no—
618C22—44no—
619C22—42no—
620C22—42no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—48no—
625C22—48no—
626C22—44no—
627C22—44no—
628C22—44no—
629C22—44no—
630C22—832no—
631C22—832no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22—84no—
637C22—84no—
638C22—84no—
639C22—84no—
640C22#349011yescharacteristic
641C22#349011yescharacteristic
642C22#349011yescharacteristic
643C22#351314yes—
644C22C2×H(3,𝔽4)12yes—
645C22#353212yes—
646C22#353312yes—
647C22#353414yes—
648C22#353414yes—
649C22#353414yes—
650C22#353714yes—
651C22#353712yes—
652C22#354012yes—
653C22#354512yes—
654C4—42no—
655C4—44no—
656C4—82no—
657C4—82no—
658C4—84no—
659C4—84no—
660C4—84no—
661C4—84no—
662C4—82no—
663C2—48no—
664C2—82no—
665C2—82no—
666C2—84no—
667C2#4556812yes—
668C2#4638411yescharacteristic
669C2#4811312yes—
670C2#5181512yes—
671C2#5241014yes—
672C2#5438914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15947C2
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. #17232C2semidirect
3. #16885C2semidirect
4. #16368C2semidirect
6. #15391C2semidirect
7. #15266C2semidirect
8. #15259C2semidirect
9. #15259C2semidirect
10. #14718C2semidirect
11. #14662C2semidirect
12. #14657C2semidirect
13. #14134C2semidirect
19. C23×C22⋊C4C22semidirect
22. C2×#393C22semidirect
23. C2×#393C22semidirect
27. C2×#385C22semidirect
28. C2×#382C22semidirect
29. C2×#379C22semidirect
30. C2×#379C22semidirect
31. C2×#379C22semidirect
32. #1446C22semidirect
33. #1445C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1440C22semidirect
39. #1440C22semidirect
40. #1440C22semidirect
41. #1440C22semidirect
42. #1440C22semidirect
43. #1440C22semidirect
44. #1440C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1439C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
#172732
#172322
#168852
#163682
#159472
#153912
#152662
#152592
#147182
#146622
#146572
#141342