Finite Group Databasebeta

← #4929720

Subgroups — #4929720

Order 512.

3038 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2564C211yesmaximal, characteristic
2C2×#2503C211yesmaximal, characteristic, complemented
3C2×#2486C211yesmaximal, characteristic
4C2×#2275C211yesmaximal, characteristic
5#15097C212yesmaximal, complemented
6#15085C212yesmaximal, complemented
7#15050C212yesmaximal, complemented
8#15016C212yesmaximal, complemented
9#14908C212yesmaximal, complemented
10#14869C212yesmaximal, complemented
11#14841C212yesmaximal, complemented
12#14826C212yesmaximal, complemented
13#14281C211yesmaximal, characteristic, complemented
14#13968C214yesmaximal
15#13966C211yesmaximal, characteristic, complemented
16#13874C214yesmaximal, complemented
17C2×#1445C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×D4C2212yes—
21C22×C4×D4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2564—28no—
24#2503—28no—
25#2486—28no—
26C2×#400C2211yescharacteristic
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#399C2211yescharacteristic
30C2×#398C2212yes—
31C2×#397C2212yes—
32C2×#395C2211yescharacteristic
33C2×#393C2211yescharacteristic
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#387C2212yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#382C2212yes—
40C2×#381C2212yes—
41C2×#381C2214yes—
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45#2275—28no—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×#376C2211yescharacteristic
53C22×#95C2211yescharacteristic
54C22×#95C2214yes—
55C22×#95C2211yescharacteristic
56C22×#95C2211yescharacteristic
57#1446C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1443C2214yes—
64#1443C2214yes—
65#1442C2212yes—
66#1442C2214yes—
67#1442C2214yes—
68#1442C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2212yes—
73#1440C2214yes—
74#1440C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1434C2214yes—
80#1434C2214yes—
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C22⋊#95C2214yes—
85C22⋊#95C2214yes—
86C22⋊#95C2214yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94C24×C4C2311yescharacteristic
95C2×#123—28no—
96C2×#123—216no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C4×D4—28no—
101C2×C4×D4—24no—
102C2×C4×D4—28no—
103C2×C4×D4—24no—
104C2×C4×D4—28no—
105C2×C4×D4D418yes—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133#400—24no—
134#400—28no—
135#400—28no—
136#399—24no—
137#398—28no—
138#397—28no—
139#395—24no—
140#393—24no—
141#391—28no—
142#389—28no—
143#387—28no—
144#386—28no—
145#385—28no—
146#382—28no—
147#381—28no—
148#381—216no—
149#380—28no—
150#380—24no—
151#380—28no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—24no—
157C4×C22⋊C4D418yes—
158#376—24no—
159C2×#95—24no—
160C2×#95—24no—
161C2×#95—216no—
162C2×#95—24no—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C25C2411yescharacteristic
197C25C2411yescharacteristic
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—44no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4C2×D414yes—
206C22×D4#4118yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4#4118yes—
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250#123—432no—
251C22⋊Q8—416no—
252C4×D4—264no—
253C4×D4—232no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—416no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D414yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4C2×D414yes—
307C2×C22⋊C4C2×D414yes—
308C2×C42—24no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318C2×C42C2×D414yes—
319C2×C42C2×D418yes—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D414yes—
352#95C2×D414yes—
353C24—22no—
354C24—24no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24C22×D412yes—
363C24C22×D412yes—
364C24C22×D412yes—
365C24C2×#4114yes—
366C24C2×#4112yes—
367C24C2×#4114yes—
368C24C2×#4114yes—
369C24C2511yescharacteristic
370C2×Q8—48no—
371C2×Q8—44no—
372C2×Q8—84no—
373C2×Q8—84no—
374C2×D4—28no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—28no—
378C2×D4#124116yes—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
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—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—216no—
412C22×C4—216no—
413C22×C4—28no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4—84no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4—88no—
466C22×C4—84no—
467C22×C4—88no—
468C22×C4—88no—
469C22×C4#121116yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C42−1+414yes—
485C4⋊C4—216no—
486C4⋊C4—28no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—84no—
495C22⋊C4—216no—
496C22⋊C4—28no—
497C22⋊C4—216no—
498C22⋊C4—28no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C22⋊C4—84no—
504C42—216no—
505C42—28no—
506C42—48no—
507C42—84no—
508C42—84no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—24no—
514C23—28no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—22no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23C2×#12114yes—
536C23#52518yes—
537C23C2×#12414yes—
538C23C2×#12414yes—
539C23#52918yes—
540C23#53214yes—
541C23C23×D411yescharacteristic
542C23C23×D411yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4112yes—
545C23C22×#4112yes—
546C23C2×2−1+412yes—
547C23#58514yes—
548C23#58512yes—
549Q8—88no—
550D4—232no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—416no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
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—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—88no—
628C2×C4—816no—
629C2×C4—816no—
630C2×C4—816no—
631C2×C4—816no—
632C2×C4C2×#12114yes—
633C2×C4C2×#12114yes—
634C2×C4C2×#12414yes—
635C22—24no—
636C22—24no—
637C22—216no—
638C22—28no—
639C22—28no—
640C22—28no—
641C22—28no—
642C22—24no—
643C22—28no—
644C22—42no—
645C22—42no—
646C22—48no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22#285018yes—
651C22C22×#12111yescharacteristic
652C22C2×#52512yes—
653C22C22×#12411yescharacteristic
654C22C2×#52912yes—
655C22C2×#53211yescharacteristic
656C22#348512yes—
657C22#348512yes—
658C22#348912yes—
659C22#349112yes—
660C22#349114yes—
661C22#349514yes—
662C22#349911yescharacteristic
663C22#350011yescharacteristic
664C22#350712yes—
665C22#351514yes—
666C22#352114yes—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—42no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—84no—
678C4—82no—
679C4—82no—
680C4—84no—
681C4—82no—
682C2—28no—
683C2—44no—
684C2#3544711yescharacteristic
685C2#3546114yes—
686C2#3562111yescharacteristic
687C2#3601012yes—
688C2#3602112yes—
689C2#3604212yes—
690C2#3606112yes—
691C2C2×#285011yescharacteristic
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
2. C2×#2503C2semidirect
5. #15097C2semidirect
6. #15085C2semidirect
7. #15050C2semidirect
8. #15016C2semidirect
9. #14908C2semidirect
10. #14869C2semidirect
11. #14841C2semidirect
12. #14826C2semidirect
13. #14281C2semidirect
15. #13966C2semidirect
16. #13874C2semidirect
17. C2×#1445C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25642
C2×#25032
C2×#24862
C2×#22752
#150972
#150852
#150502
#150162
#149082
#148692
#148412
#148262
#142812
#139682
#139662
#138742
C2×#14452