Finite Group Databasebeta

← #4456750

Subgroups — #4456750

Order 512.

2704 conjugacy classes of subgroups.

The subgroup lattice has 682 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×#2318C211yesmaximal, characteristic
5#16806C212yesmaximal, complemented
6#16805C212yesmaximal, complemented
7#16705C212yesmaximal, complemented
8#16701C212yesmaximal, complemented
9#16678C212yesmaximal, complemented
10#16567C212yesmaximal, complemented
11#16400C212yesmaximal, complemented
12#16394C212yesmaximal, complemented
13#14935C214yesmaximal, complemented
14#14912C214yesmaximal
15#14281C211yesmaximal, characteristic, complemented
16#13965C211yesmaximal, characteristic, complemented
17C2×#1445C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2564—28no—
23#2503—28no—
24#2486—28no—
25C2×#400C2212yes—
26C2×#400C2211yescharacteristic
27C2×#400C2212yes—
28C2×#399C2211yescharacteristic
29C2×#398C2212yes—
30C2×#397C2212yes—
31C2×#395C2211yescharacteristic
32C2×#393C2211yescharacteristic
33#2318—28no—
34C2×#391C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#386C2212yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2214yes—
44C2×#383C2211yescharacteristic
45C2×#382C2211yescharacteristic
46C2×#382C2212yes—
47C2×#381C2211yescharacteristic
48C2×#381C2214yes—
49C2×#381C2212yes—
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×#380C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95C2211yescharacteristic
56C22×#95C2211yescharacteristic
57C22×#95C2214yes—
58#1446C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2212yes—
64#1444C2212yes—
65#1444C2212yes—
66#1443C2214yes—
67#1443C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2212yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1441C2214yes—
77#1441C2214yes—
78#1441C2214yes—
79#1441C2214yes—
80#1440C2214yes—
81#1440C2214yes—
82#1439C2214yes—
83#1439C2214yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86C22⋊#95C2214yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C2×#123—28no—
95C2×#123—48no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4D418yes—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#400—24no—
131#400—28no—
132#400—28no—
133#399—24no—
134#398—28no—
135#397—28no—
136#395—24no—
137#393—24no—
138#391—28no—
139#389—28no—
140#388—28no—
141#387—28no—
142#387—28no—
143#386—28no—
144#386—28no—
145#385—28no—
146#385—28no—
147#385—48no—
148#383—24no—
149#382—28no—
150#382—24no—
151#381—28no—
152#381—216no—
153#381—24no—
154#380—28no—
155#380—24no—
156#380—28no—
157#380—24no—
158C4×C22⋊C4D418yes—
159C2×#95—24no—
160C2×#95—24no—
161C2×#95—48no—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C25C2411yescharacteristic
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—44no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4C2×D414yes—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—24no—
213C23×C4—28no—
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4#4114yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244#123—432no—
245C22⋊Q8—416no—
246C4×D4—232no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—416no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4C2×D414yes—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4C2×D414yes—
307C2×C22⋊C4C2×D414yes—
308C2×C22⋊C4C2×D418yes—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—44no—
312C2×C42C2×D414yes—
313C2×C42C2×D418yes—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95C2×D414yes—
346#95C2×D414yes—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24C22×D412yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C2×#4112yes—
361C24H(5,𝔽2)14yes—
362C242−1+414yes—
363C24C2511yescharacteristic
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8—84no—
367C2×Q8—84no—
368C2×D4—28no—
369C2×D4—28no—
370C2×D4—28no—
371C2×D4—48no—
372C2×D4—48no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—88no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—88no—
461C22×C4#121116yes—
462C22×C4C22×D414yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C42−1+414yes—
474C4⋊C4—28no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C22⋊C4—28no—
488C22⋊C4—216no—
489C22⋊C4—28no—
490C22⋊C4—48no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C42—28no—
498C42—48no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—28no—
507C23—28no—
508C23—28no—
509C23—28no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C2×#12114yes—
527C23#53514yes—
528C23#53614yes—
529C23#53718yes—
530C23#54218yes—
531C23C23×D411yescharacteristic
532C23C23×D411yescharacteristic
533C23C22×#4111yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23C2×2−1+412yes—
536C23C2×2−1+412yes—
537C23#58514yes—
538C23#58512yes—
539Q8—88no—
540D4—416no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—416no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—88no—
619C2×C4—816no—
620C2×C4—816no—
621C2×C4—816no—
622C2×C4C2×#12114yes—
623C2×C4C2×#12114yes—
624C22—24no—
625C22—24no—
626C22—216no—
627C22—28no—
628C22—28no—
629C22—28no—
630C22—28no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—44no—
635C22—44no—
636C22—84no—
637C22—82no—
638C22—82no—
639C22#262018yes—
640C22C22×#12111yescharacteristic
641C22C2×#53511yescharacteristic
642C22C2×#53611yescharacteristic
643C22C2×#53712yes—
644C22C2×#54212yes—
645C22#348512yes—
646C22#348512yes—
647C22#348912yes—
648C22#349112yes—
649C22#349911yescharacteristic
650C22#350011yescharacteristic
651C22#350712yes—
652C22#353314yes—
653C22#353514yes—
654C22#356314yes—
655C22#356414yes—
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—42no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—82no—
665C4—84no—
666C4—82no—
667C4—82no—
668C4—82no—
669C4—84no—
670C4—82no—
671C2—28no—
672C2—82no—
673C2C2×#262011yescharacteristic
674C2#3544711yescharacteristic
675C2#3562111yescharacteristic
676C2#3699914yes—
677C2#4136912yes—
678C2#4137312yes—
679C2#4140912yes—
680C2#4141112yes—
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. #16806C2semidirect
6. #16805C2semidirect
7. #16705C2semidirect
8. #16701C2semidirect
9. #16678C2semidirect
10. #16567C2semidirect
11. #16400C2semidirect
12. #16394C2semidirect
13. #14935C2semidirect
15. #14281C2semidirect
16. #13965C2semidirect
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×#23182
#168062
#168052
#167052
#167012
#166782
#165672
#164002
#163942
#149352
#149122
#142812
#139652
C2×#14452