Finite Group Databasebeta

← #626009

Subgroups — #626009

Order 512.

2874 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2563C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic, complemented
4C2×#2566C211yesmaximal, characteristic
5C2×#2660C211yesmaximal, characteristic, complemented
6C2×#2326C211yesmaximal, characteristic
7#15825C218yesmaximal, complemented
8#15513C218yesmaximal, complemented
9#15244C211yesmaximal, characteristic
10#15244C211yesmaximal, characteristic
11#15240C211yesmaximal, characteristic, complemented
12#15240C211yesmaximal, characteristic, complemented
13#14708C211yesmaximal, characteristic
14#14687C211yesmaximal, characteristic, complemented
15#14354C211yesmaximal, characteristic, complemented
16#14273C211yesmaximal, characteristic
17#13759C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2660—28no—
24#2568—28no—
25#2566—28no—
26#2565—28no—
27#2563—28no—
28C2×#400C2214yes—
29C2×#398C2214yes—
30C2×#398C2214yes—
31C2×#396C2214yes—
32C2×#395C2214yes—
33C2×#393C2214yes—
34#2326—28no—
35C2×#391C2211yescharacteristic
36C2×#391C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2214yes—
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#388C2214yes—
47C2×#387C2214yes—
48C2×#385C2214yes—
49C2×#385C2214yes—
50C2×#384C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#382C2211yescharacteristic
57C2×#382C2211yescharacteristic
58C2×#382C2211yescharacteristic
59C2×#382C2211yescharacteristic
60C2×#382C2211yescharacteristic
61C2×#382C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#380C2214yes—
64C2×C4×C4⋊C4C2211yescharacteristic
65C2×C4×C4⋊C4C2211yescharacteristic
66C2×#376C2211yescharacteristic
67C2×#376C2211yescharacteristic
68C2×C43C2211yescharacteristic
69#1445C2218yes—
70#1445C2218yes—
71#1444C2218yes—
72#1439C2218yes—
73#1439C2218yes—
74#1439C2218yes—
75#1437C2218yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2218yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2312yes—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#125—24no—
91C2×#124—24no—
92C2×#124—24no—
93C2×#124Q818yescomplemented
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#400—216no—
134#398—216no—
135#398—216no—
136#396—216no—
137#395—216no—
138#393—216no—
139#391—216no—
140#391—24no—
141#391—24no—
142#391—24no—
143#391—24no—
144#391—24no—
145#390—24no—
146#390—24no—
147#390—24no—
148#390—24no—
149#390—24no—
150#388—216no—
151#387—216no—
152#385—216no—
153#385—216no—
154#384—24no—
155#384—24no—
156#384—24no—
157#384—24no—
158#384—24no—
159#382—24no—
160#382—24no—
161#382—24no—
162#382—24no—
163#382—24no—
164#382—24no—
165#382—24no—
166#382—24no—
167#380—216no—
168C4×C4⋊C4D418yes—
169C4×C4⋊C4D418yes—
170#376—24no—
171#376D418yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C43—24no—
193C25C2412yes—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—48no—
204C22×Q8C2×Q814yes—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2×Q814yes—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—48no—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C4⋊Q8—416no—
233#125—416no—
234#124—232no—
235C4×Q8—416no—
236C4×D4—416no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—48no—
275C2×C4⋊C4—48no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—48no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4C2×Q8116yes—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×Q814yes—
323C2×C42#4114yes—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—44no—
339#95—44no—
340#95C2×D418yes—
341#95C2×D418yes—
342#95C2×D418yes—
343#95C2×D418yes—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—28no—
351C24—28no—
352C24C22×Q814yes—
353C24C2511yescharacteristic
354C2×Q8—28no—
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—416no—
358C2×Q8—44no—
359C2×Q8—48no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—44no—
363C2×Q8—88no—
364C2×Q8—88no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—28no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
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—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—28no—
426C22×C4—416no—
427C22×C4—48no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×Q814yes—
446C22×C4C22×Q814yes—
447C22×C4C22×Q812yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4112yes—
450C22×C42−1+412yes—
451C22×C42−1+412yes—
452C4⋊C4—48no—
453C4⋊C4—48no—
454C4⋊C4—416no—
455C4⋊C4—44no—
456C4⋊C4—48no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—232no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C42—28no—
477C42—28no—
478C42—28no—
479C42—28no—
480C42—28no—
481C42—416no—
482C42—48no—
483C42C22⋊Q8116yes—
484C42#127116yes—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—216no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—48no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—48no—
501C23—48no—
502C23C23×D411yescharacteristic
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C23×Q811yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×2−1+411yescharacteristic
508C23C2×2−1+411yescharacteristic
509C23#58514yes—
510C23#58514yes—
511Q8—416no—
512Q8—816no—
513D4—416no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
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—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—816no—
581C2×C4C2×C22⋊Q818yes—
582C2×C4C2×C22⋊Q814yes—
583C2×C4C2×#12714yes—
584C2×C4C2×#12714yes—
585C2×C4C2×#12714yes—
586C2×C4#53614yes—
587C2×C4#54714yes—
588C2×C4#54714yes—
589C2×C4D4×Q814yes—
590C2×C4D4×Q814yes—
591C22—48no—
592C22—44no—
593C22—48no—
594C22—48no—
595C22—82no—
596C22—82no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22C22×C22⋊Q811yescharacteristic
601C22C22×#12711yescharacteristic
602C22C2×#53611yescharacteristic
603C22#344912yes—
604C22#344912yes—
605C22C2×#54711yescharacteristic
606C22C2×#54711yescharacteristic
607C22C2×D4×Q811yescharacteristic
608C22C2×D4×Q811yescharacteristic
609C22#349314yes—
610C22#350012yes—
611C22#350012yes—
612C22#350012yes—
613C22#350012yes—
614C22#350414yes—
615C22#356514yes—
616C22#356514yes—
617C4—24no—
618C4—24no—
619C4—28no—
620C4—44no—
621C4—44no—
622C4—44no—
623C4—44no—
624C4—44no—
625C4—44no—
626C4—82no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—82no—
632C4#254218yes—
633C2—84no—
634C2C2×#254211yescharacteristic
635C2#3532911yescharacteristic
636C2#3532911yescharacteristic
637C2#3562012yes—
638C2#3742312yes—
639C2#3787012yes—
640C2#3787112yes—
641C2#3961312yes—
642C2#3961312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2568C2semidirect
3. C2×#2565C2semidirect
5. C2×#2660C2semidirect
7. #15825C2semidirect
8. #15513C2semidirect
11. #15240C2semidirect
12. #15240C2semidirect
14. #14687C2semidirect
15. #14354C2semidirect
17. #13759C2semidirect
93. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#25632
C2×#25652
C2×#25662
C2×#26602
C2×#23262
#158252
#155132
#152442
#152402
#147082
#146872
#143542
#142732
#137592