Finite Group Databasebeta

← #4420464

Subgroups — #4420464

Order 512.

2396 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2642C211yesmaximal, characteristic, complemented
3C2×#2633C211yesmaximal, characteristic
4C2×#2353C211yesmaximal, characteristic
5#16805C212yesmaximal, complemented
6#16803C212yesmaximal, complemented
7#16400C212yesmaximal, complemented
8#16394C212yesmaximal, complemented
9#15738C214yesmaximal, complemented
10#15495C214yesmaximal
11#15161C212yesmaximal, complemented
12#15050C212yesmaximal, complemented
13#14841C212yesmaximal, complemented
14#14826C212yesmaximal, complemented
15#14663C211yesmaximal, characteristic, complemented
16#14289C211yesmaximal, characteristic, complemented
17#13952C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2642—28no—
22#2633—28no—
23#2563—28no—
24C2×#400C2212yes—
25C2×#400C2214yes—
26C2×#400C2212yes—
27C2×#399C2211yescharacteristic
28C2×#399C2211yescharacteristic
29C2×#398C2212yes—
30C2×#397C2211yescharacteristic
31C2×#397C2211yescharacteristic
32C2×#394C2212yes—
33C2×#393C2212yes—
34C2×#393C2212yes—
35#2353—28no—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#390C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#383C2211yescharacteristic
45C2×#383C2211yescharacteristic
46C2×#382C2212yes—
47C2×#381C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×#376C2211yescharacteristic
55C22×#95C2211yescharacteristic
56#1446C2212yes—
57#1446C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2214yes—
62#1445C2212yes—
63#1444C2212yes—
64#1443C2214yes—
65#1443C2214yes—
66#1442C2214yes—
67#1442C2212yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2212yes—
73#1440C2214yes—
74#1440C2214yes—
75#1440C2214yes—
76#1440C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82C4⋊#95C2212yes—
83#1434C2214yes—
84#1434C2212yes—
85C22⋊#95C2214yes—
86C22⋊#95C2214yes—
87C23×Q8C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×#125—24no—
96C2×#125—24no—
97C2×#125—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4D418yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#400—28no—
129#400—28no—
130#400—216no—
131#399—24no—
132#399—24no—
133#398—28no—
134#397—24no—
135#397—24no—
136#394—28no—
137#393—28no—
138#393—28no—
139#391—28no—
140#391—28no—
141#390—28no—
142#388—28no—
143#387—28no—
144#386—28no—
145#385—28no—
146#385—28no—
147#383—24no—
148#383—24no—
149#382—28no—
150#381—24no—
151#380—28no—
152#380—28no—
153#380—24no—
154C4×C4⋊C4—24no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—24no—
157#376D418yes—
158C2×#95—24no—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C25C2411yescharacteristic
193C25C2411yescharacteristic
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—41no—
204C22×D4—41no—
205C22×D4C2×D414yes—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—42no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—416no—
235#125—416no—
236C4×D4—232no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4C2×D414yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42C2×D418yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42#4114yes—
308#95—22no—
309#95—24no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95C2×D414yes—
339#95C2×D414yes—
340#95C2×D418yes—
341C24—21no—
342C24—21no—
343C24—24no—
344C24—21no—
345C24—24no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—24no—
352C24—24no—
353C24—21no—
354C24—21no—
355C24C22×D412yes—
356C24C22×D412yes—
357C242−1+412yes—
358C242−1+412yes—
359C24C2511yescharacteristic
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—88no—
364C2×Q8—88no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
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—22no—
439C22×C4—24no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—84no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C4⋊C4—28no—
465C4⋊C4—48no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—84no—
473C4⋊C4—84no—
474C4⋊C4—84no—
475C4⋊C4—88no—
476C22⋊C4—28no—
477C22⋊C4—28no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C22⋊C4—84no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C42—48no—
488C42—48no—
489C42—84no—
490C42#121116yes—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—28no—
500C23—28no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—24no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23#53614yes—
525C23C23×D411yescharacteristic
526C23C23×D411yescharacteristic
527C23C22×#4111yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23C2×2−1+411yescharacteristic
530C23#58511yescharacteristic
531C23#58511yescharacteristic
532C23#58514yes—
533C23#58514yes—
534Q8—816no—
535D4—416no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
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—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
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—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—88no—
613C2×C4—88no—
614C2×C4—816no—
615C2×C4C2×#12114yes—
616C2×C4C2×#12114yes—
617C2×C4C2×#12114yes—
618C22—28no—
619C22—24no—
620C22—44no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22C22×#12111yescharacteristic
637C22C2×#53611yescharacteristic
638C22#344912yes—
639C22#344911yescharacteristic
640C22#348512yes—
641C22#348911yescharacteristic
642C22#348911yescharacteristic
643C22#349511yescharacteristic
644C22#349511yescharacteristic
645C22#350012yes—
646C22#350012yes—
647C22#351012yes—
648C22#351012yes—
649C22#355314yes—
650C22#356414yes—
651C22#356614yes—
652C22#356614yes—
653C4—24no—
654C4—24no—
655C4—24no—
656C4—42no—
657C4—42no—
658C4—44no—
659C4—44no—
660C4—82no—
661C4—82no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4—82no—
667C4—84no—
668C2—44no—
669C2—82no—
670C2#3530611yescharacteristic
671C2#3532711yescharacteristic
672C2#3561711yescharacteristic
673C2#4137412yes—
674C2#4138512yes—
675C2#4230412yes—
676C2#4239112yes—
677C2#5092012yes—
678C2#5122812yes—
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×#2642C2semidirect
5. #16805C2semidirect
6. #16803C2semidirect
7. #16400C2semidirect
8. #16394C2semidirect
9. #15738C2semidirect
11. #15161C2semidirect
12. #15050C2semidirect
13. #14841C2semidirect
14. #14826C2semidirect
15. #14663C2semidirect
16. #14289C2semidirect
17. #13952C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#26422
C2×#26332
C2×#23532
#168052
#168032
#164002
#163942
#157382
#154952
#151612
#150502
#148412
#148262
#146632
#142892
#139522