Finite Group Databasebeta

← #729367

Subgroups — #729367

Order 512.

3254 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2569C211yesmaximal, characteristic, complemented
2C22×#393C211yesmaximal, characteristic
3#16378C214yesmaximal, complemented
4#15637C214yesmaximal, complemented
5#15459C214yesmaximal, complemented
6#15026C212yesmaximal, complemented
7#15004C214yesmaximal, complemented
8#14820C212yesmaximal
9#14338C211yesmaximal, characteristic
10#13971C212yesmaximal
11#13966C212yesmaximal, complemented
12#13922C211yesmaximal, characteristic, complemented
13#13922C211yesmaximal, characteristic, complemented
14#13802C211yesmaximal, characteristic, complemented
15#13775C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2212yes—
19#2569—28no—
20C2×#400C2212yescomplemented
21C2×#398C2212yescomplemented
22C2×#397C2214yes—
23C2×#393—28no—
24C2×#393C2214yes—
25C2×#393C2212yes—
26C2×#393C2212yes—
27C2×#391C2214yescomplemented
28C2×#391C2214yescomplemented
29C2×#388C2214yes—
30C2×#387C2212yescomplemented
31C2×#387C2212yescomplemented
32C2×#385C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×#379C2212yes—
37C2×#379C2211yescharacteristic
38C2×#379C2211yescharacteristic
39C2×C4×C22⋊C4C2211yescharacteristic
40C2×C4×C22⋊C4C2211yescharacteristic
41C2×C4×C22⋊C4C2212yes—
42C22×#95C2212yes—
43C22×#95C2211yescharacteristic
44#1446C2212yescomplemented
45#1446C2212yescomplemented
46#1446C2214yescomplemented
47#1443C2214yescomplemented
48#1443C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1440C2214yes—
54#1440C2214yes—
55#1440C2214yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1437C2214yes—
62#1437C2214yes—
63#1437C2214yes—
64#1436C2214yescomplemented
65#1434C2214yescomplemented
66#1434C2214yescomplemented
67#1434C2212yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2211yescharacteristic
75C22⋊#95C2212yes—
76C4×#95C2212yescomplemented
77C26C2311yescharacteristic
78C23×Q8—24no—
79C23×Q8—24no—
80C23×Q8—24no—
81C23×Q8—48no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—24no—
96C22×C22⋊C4—24no—
97C22×C22⋊C4—24no—
98C22×C22⋊C4—48no—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2311yescharacteristic
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C42C2314yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116#400—28no—
117#398—28no—
118#397—48no—
119#393—216no—
120#393—28no—
121#393—28no—
122#391—48no—
123#391—48no—
124#388—48no—
125#387—28no—
126#387—28no—
127#385—28no—
128#380—24no—
129#380—28no—
130#380—24no—
131#379—28no—
132#379—24no—
133#379—24no—
134C4×C22⋊C4—24no—
135C4×C22⋊C4—24no—
136C4×C22⋊C4—28no—
137C2×#95—28no—
138C2×#95D418yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C25—22no—
169C25—22no—
170C25—22no—
171C25—22no—
172C25—22no—
173C25—22no—
174C25—44no—
175C25—44no—
176C25#4114yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—28no—
186C22×Q8—28no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—432no—
190C22×Q8—432no—
191C22×Q8—432no—
192C22×Q8—416no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2×D418yes—
208C23×C4C2×D414yes—
209C23×C4#4114yes—
210C23×C4#4114yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—432no—
272C2×C22⋊C4—416no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4#4114yes—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—28no—
283C2×C42—44no—
284C2×C42—44no—
285C2×C42—44no—
286C2×C42—44no—
287#95—28no—
288#95—28no—
289#95—24no—
290#95—22no—
291#95—22no—
292#95—24no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—28no—
297#95—24no—
298#95—24no—
299#95—28no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95C2×D418yes—
314#95C2×D414yes—
315#95C2×D414yes—
316C24—24no—
317C24—24no—
318C24—28no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—24no—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—28no—
330C24—28no—
331C24—416no—
332C24—48no—
333C24—48no—
334C24—416no—
335C24—44no—
336C24—44no—
337C24—44no—
338C24—44no—
339C24—84no—
340C24—84no—
341C24—84no—
342C24—84no—
343C24—88no—
344C24—88no—
345C24C22×D412yes—
346C24C2×#4114yes—
347C24C2×#4112yes—
348C24C2×#4112yes—
349C24C2×#4112yes—
350C24C2×#4112yes—
351C242−1+412yes—
352C242−1+412yes—
353C24C2511yescharacteristic
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—416no—
357C2×Q8—416no—
358C2×Q8—416no—
359C2×Q8—416no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—4128no—
363C2×Q8—48no—
364C2×Q8—48no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—216no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—48no—
428C22×C4—48no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—88no—
436C22×C4—88no—
437C22×C4—84no—
438C22×C4—84no—
439C22×C4#124116yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4114yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4H(5,𝔽2)18yes—
450C4⋊C4—48no—
451C4⋊C4—48no—
452C4⋊C4—84no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C4⋊C4—84no—
456C22⋊C4—28no—
457C22⋊C4—416no—
458C22⋊C4—416no—
459C22⋊C4—416no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—44no—
463C22⋊C4—48no—
464C22⋊C4—44no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—88no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C42—84no—
471C42—84no—
472C42—88no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—24no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—432no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—42no—
501C23—416no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—816no—
509C23—816no—
510C23—88no—
511C23—88no—
512C23—816no—
513C23—816no—
514C23—816no—
515C23C2×#12414yes—
516C23#52914yes—
517C23#54114yes—
518C23#54714yes—
519C23#54814yes—
520C23#55214yes—
521C23#55214yes—
522C23C23×D411yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4111yescharacteristic
525C23C22×#4111yescharacteristic
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)14yes—
528C23C2×2−1+411yescharacteristic
529C23C2×2−1+411yescharacteristic
530C23#58514yes—
531Q8—432no—
532Q8—432no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—416no—
551C2×C4—432no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
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—816no—
601C2×C4—88no—
602C2×C4C2×#12414yes—
603C2×C4C2×#12414yes—
604C22—24no—
605C22—28no—
606C22—24no—
607C22—28no—
608C22—28no—
609C22—28no—
610C22—28no—
611C22—28no—
612C22—48no—
613C22—48no—
614C22—44no—
615C22—44no—
616C22—44no—
617C22—832no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—82no—
622C22—84no—
623C22—84no—
624C22#248918yes—
625C22C22×#12411yescharacteristic
626C22C2×#52911yescharacteristic
627C22#344912yes—
628C22C2×#54111yescharacteristic
629C22C2×#54711yescharacteristic
630C22C2×#54811yescharacteristic
631C22C2×#55211yescharacteristic
632C22C2×#55211yescharacteristic
633C22#348412yes—
634C22#349012yes—
635C22#349012yes—
636C22#350514yes—
637C22#351012yes—
638C22#351012yes—
639C22#351414yes—
640C22#354114yes—
641C22#356114yes—
642C4—24no—
643C4—24no—
644C4—48no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—82no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4—82no—
653C4—84no—
654C4—82no—
655C2—28no—
656C2—84no—
657C2C2×#248911yescharacteristic
658C2#3533811yescharacteristic
659C2#3536711yescharacteristic
660C2#3570612yes—
661C2#3804412yes—
662C2#3971612yes—
663C2#4054112yes—
664C2#4307812yes—
665C2#4396212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2569C2semidirect
3. #16378C2semidirect
4. #15637C2semidirect
5. #15459C2semidirect
6. #15026C2semidirect
7. #15004C2semidirect
11. #13966C2semidirect
12. #13922C2semidirect
13. #13922C2semidirect
14. #13802C2semidirect
20. C2×#400C22semidirect
21. C2×#398C22semidirect
27. C2×#391C22semidirect
28. C2×#391C22semidirect
30. C2×#387C22semidirect
31. C2×#387C22semidirect
44. #1446C22semidirect
45. #1446C22semidirect
46. #1446C22semidirect
47. #1443C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
64. #1436C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
67. #1434C22semidirect
76. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25692
C22×#3932
#163782
#156372
#154592
#150262
#150042
#148202
#143382
#139712
#139662
#139222
#138022
#137752