Finite Group Databasebeta

← #4861309

Subgroups — #4861309

Order 512.

2942 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic, complemented
2C2×#2635C211yesmaximal, characteristic
3#17144C212yesmaximal, complemented
4#17084C212yesmaximal, complemented
5#16895C212yesmaximal, complemented
6#16837C214yesmaximal, complemented
7#16665C212yesmaximal, complemented
8#16364C212yesmaximal, complemented
9#16270C214yesmaximal, complemented
10#16253C212yesmaximal, complemented
11#15730C212yesmaximal, complemented
12#15635C212yesmaximal, complemented
13#14902C212yesmaximal, complemented
14#14873C212yesmaximal, complemented
15#13959C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20#2635—28no—
21#2409—28no—
22C2×#400C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#397C2214yes—
25C2×#395C2212yes—
26C2×#394C2211yescharacteristic
27C2×#394C2212yes—
28C2×#394C2214yes—
29C2×#393C2212yes—
30C2×#393C2212yes—
31C2×#393C2212yes—
32C2×#393C2214yes—
33C2×#392C2211yescharacteristic
34C2×#392C2212yes—
35C2×#391C2212yes—
36C2×#388C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#385C2212yes—
41C2×#383C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#376C2211yescharacteristic
45#1446C2212yes—
46#1446C2212yes—
47#1446C2214yes—
48#1445C2212yes—
49#1444C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2212yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2214yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2212yes—
70#1437C2214yes—
71#1437C2212yes—
72#1437C2212yes—
73#1436C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C24×C4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×#121—24no—
93C2×#121—42no—
94C2×#121—42no—
95C2×C4×D4—42no—
96C2×C4×D4—42no—
97C2×C4×D4D418yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#400—24no—
123#399—24no—
124#397—48no—
125#395—28no—
126#394—24no—
127#394—28no—
128#394—48no—
129#393—28no—
130#393—44no—
131#393—44no—
132#393—48no—
133#392—28no—
134#392—24no—
135#391—28no—
136#388—28no—
137#386—28no—
138#386—28no—
139#386—48no—
140#385—28no—
141#383—42no—
142#381—24no—
143#380—24no—
144#376—42no—
145C2×#95C2311yescharacteristic
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—28no—
177C22×Q8—28no—
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—44no—
181C22×Q8—44no—
182C22×Q8—42no—
183C22×Q8—42no—
184C22×Q8—416no—
185C22×Q8—432no—
186C22×Q8—432no—
187C22×Q8—432no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—41no—
195C22×D4—41no—
196C22×D4—42no—
197C22×D4—41no—
198C22×D4—41no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4C2×D414yes—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—44no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4C2×D414yes—
219C23×C4C2×D414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C22⋊Q8—432no—
233#121—416no—
234C4×D4—416no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D414yes—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—42no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—44no—
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—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×D414yes—
293C2×C22⋊C4C2×D414yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—42no—
297C2×C42—44no—
298C2×C42C2×D414yes—
299#95—24no—
300#95—24no—
301#95—22no—
302#95—22no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325C24—21no—
326C24—21no—
327C24—24no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—21no—
332C24—21no—
333C24—22no—
334C24—22no—
335C24—24no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24C22×D412yes—
348C24C22×D412yes—
349C242−1+414yes—
350C24C2511yescharacteristic
351C2×Q8—216no—
352C2×Q8—416no—
353C2×Q8—416no—
354C2×Q8—416no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—416no—
359C2×Q8—4128no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×D4—28no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—416no—
370C2×D4—48no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—88no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—88no—
380C2×D4—88no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
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—22no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—416no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—84no—
449C22×C4—84no—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4H(5,𝔽2)18yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C4⋊C4—44no—
459C4⋊C4—416no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—84no—
463C4⋊C4—88no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—416no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C42—28no—
481C42—48no—
482C23—22no—
483C23—22no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—44no—
499C23—42no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23#53614yes—
535C23C23×D411yescharacteristic
536C23C2×H(5,𝔽2)11yescharacteristic
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×H(5,𝔽2)14yes—
539C23C2×2−1+412yes—
540C23#58512yes—
541C23#58514yes—
542Q8—432no—
543Q8—432no—
544D4—416no—
545D4—88no—
546D4—816no—
547D4—816no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—416no—
564C2×C4—432no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
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—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
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—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—44no—
613C2×C4—88no—
614C2×C4#53414yes—
615C22—28no—
616C22—24no—
617C22—48no—
618C22—42no—
619C22—42no—
620C22—44no—
621C22—48no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—44no—
627C22—84no—
628C22—84no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—82no—
633C22—82no—
634C22—82no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—84no—
639C22—84no—
640C22C2×#53411yescharacteristic
641C22C2×#53611yescharacteristic
642C22#344911yescharacteristic
643C22#348412yes—
644C22#350012yes—
645C22#353212yes—
646C22#353212yes—
647C22#354012yes—
648C22#354112yes—
649C22#354114yes—
650C22#354112yes—
651C22#355114yes—
652C22#355112yes—
653C22#356114yes—
654C22#356214yes—
655C4—24no—
656C4—42no—
657C4—42no—
658C4—42no—
659C4—48no—
660C4—42no—
661C4—84no—
662C4—82no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4—84no—
667C2—44no—
668C2—82no—
669C2—162no—
670C2#3671411yescharacteristic
671C2#3741911yescharacteristic
672C2#3743811yescharacteristic
673C2#4049112yes—
674C2#4215212yes—
675C2#5631214yes—
676C2#5889114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2635C2
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×#2409C2semidirect
3. #17144C2semidirect
4. #17084C2semidirect
5. #16895C2semidirect
6. #16837C2semidirect
7. #16665C2semidirect
8. #16364C2semidirect
9. #16270C2semidirect
10. #16253C2semidirect
11. #15730C2semidirect
12. #15635C2semidirect
13. #14902C2semidirect
14. #14873C2semidirect
15. #13959C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#26352
#171442
#170842
#168952
#168372
#166652
#163642
#162702
#162532
#157302
#156352
#149022
#148732
#139592