Finite Group Databasebeta

← #4861917

Subgroups — #4861917

Order 512.

2966 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×#2409C211yesmaximal, characteristic, complemented
2C2×#2631C211yesmaximal, characteristic
3#17198C212yesmaximal, complemented
4#17105C212yesmaximal, complemented
5#16895C212yesmaximal, complemented
6#16696C212yesmaximal, complemented
7#16559C212yesmaximal, complemented
8#16546C214yesmaximal, complemented
9#16268C214yesmaximal, complemented
10#16263C212yesmaximal, complemented
11#15568C212yesmaximal, complemented
12#15420C212yesmaximal, complemented
13#14873C212yesmaximal, complemented
14#14765C212yesmaximal, complemented
15#13940C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#124C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2631—28no—
21#2409—28no—
22C2×#397C2212yes—
23C2×#395C2212yes—
24C2×#394C2214yes—
25C2×#393C2211yescharacteristic
26C2×#393C2211yescharacteristic
27C2×#393C2212yes—
28C2×#393C2212yes—
29C2×#393C2214yes—
30C2×#393C2212yes—
31C2×#392C2211yescharacteristic
32C2×#392C2212yes—
33C2×#392C2211yescharacteristic
34C2×#391C2212yes—
35C2×#388C2214yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#385C2212yes—
41C2×#383C2211yescharacteristic
42C2×#381C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45#1446C2214yes—
46#1446C2212yes—
47#1446C2212yes—
48#1446C2214yes—
49#1445C2214yes—
50#1444C2212yes—
51#1443C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1441C2214yes—
57#1440C2212yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2212yes—
70#1438C2214yes—
71#1437C2212yes—
72#1436C2212yes—
73#1436C2214yes—
74#1434C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8—24no—
78C23×Q8—42no—
79C23×Q8—42no—
80C23×Q8—48no—
81C23×Q8C2312yes—
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×#124—24no—
92C2×#124—42no—
93C2×#124—42no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—48no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4D418yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122#397—28no—
123#395—28no—
124#394—48no—
125#393—24no—
126#393—24no—
127#393—28no—
128#393—44no—
129#393—44no—
130#393—48no—
131#392—28no—
132#392—24no—
133#392—24no—
134#391—28no—
135#388—48no—
136#386—28no—
137#386—28no—
138#386—28no—
139#386—48no—
140#385—28no—
141#383—24no—
142#381—42no—
143#380—24no—
144C4×C22⋊C4—42no—
145C2×#95C2311yescharacteristic
146C2×#95C2311yescharacteristic
147C2×#95C2312yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25C2412yes—
172C25C2411yescharacteristic
173C25C2411yescharacteristic
174C22×Q8—24no—
175C22×Q8—24no—
176C22×Q8—22no—
177C22×Q8—28no—
178C22×Q8—28no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—44no—
182C22×Q8—44no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—41no—
186C22×Q8—41no—
187C22×Q8—44no—
188C22×Q8—416no—
189C22×Q8—432no—
190C22×Q8—432no—
191C22×Q8—432no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—41no—
199C22×D4—41no—
200C22×D4—42no—
201C22×D4—41no—
202C22×D4—41no—
203C22×D4—42no—
204C22×D4—44no—
205C22×D4—42no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4C2×D414yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—28no—
217C23×C4—44no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236#124—416no—
237C22⋊Q8—432no—
238C4×D4—416no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
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⋊C4C2×D414yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
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—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
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—42no—
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—24no—
304#95—28no—
305#95—28no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—42no—
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—24no—
335C24—22no—
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—44no—
360C2×Q8—4128no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—84no—
364C2×Q8—84no—
365C2×Q8—84no—
366C2×D4—28no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—48no—
372C2×D4—48no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—84no—
380C2×D4—88no—
381C2×D4—84no—
382C2×D4—84no—
383C2×D4—88no—
384C2×D4—88no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
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—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—416no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—416no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4H(5,𝔽2)18yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C4⋊C4—44no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—88no—
466C22⋊C4—44no—
467C22⋊C4—44no—
468C22⋊C4—416no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—416no—
473C22⋊C4—48no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C22⋊C4—84no—
479C22⋊C4—84no—
480C22⋊C4—88no—
481C22⋊C4—84no—
482C22⋊C4—88no—
483C42—28no—
484C42—84no—
485C23—22no—
486C23—22no—
487C23—24no—
488C23—24no—
489C23—28no—
490C23—24no—
491C23—24no—
492C23—28no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—42no—
512C23—44no—
513C23—44no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23#53614yes—
538C23C23×D411yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×H(5,𝔽2)14yes—
542C23C2×2−1+412yes—
543C23#58512yes—
544C23#58514yes—
545Q8—432no—
546Q8—432no—
547Q8—88no—
548D4—416no—
549D4—88no—
550D4—816no—
551D4—816no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—416no—
568C2×C4—432no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
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—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—88no—
618C2×C4#53414yes—
619C22—28no—
620C22—24no—
621C22—48no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—48no—
626C22—42no—
627C22—44no—
628C22—44no—
629C22—42no—
630C22—42no—
631C22—84no—
632C22—84no—
633C22—84no—
634C22—84no—
635C22—82no—
636C22—82no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—84no—
644C22C2×#53411yescharacteristic
645C22C2×#53611yescharacteristic
646C22#344911yescharacteristic
647C22#348412yes—
648C22#350012yes—
649C22#353212yes—
650C22#353212yes—
651C22#354014yes—
652C22#354012yes—
653C22#354112yes—
654C22#354114yes—
655C22#354112yes—
656C22#355112yes—
657C22#356114yes—
658C22#356214yes—
659C4—24no—
660C4—42no—
661C4—42no—
662C4—42no—
663C4—48no—
664C4—42no—
665C4—84no—
666C4—82no—
667C4—82no—
668C4—84no—
669C4—82no—
670C4—84no—
671C2—44no—
672C2—82no—
673C2—162no—
674C2#3670211yescharacteristic
675C2#3741911yescharacteristic
676C2#3743811yescharacteristic
677C2#4049112yes—
678C2#4050712yes—
679C2#5798614yes—
680C2#5889114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2631C2
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. #17198C2semidirect
4. #17105C2semidirect
5. #16895C2semidirect
6. #16696C2semidirect
7. #16559C2semidirect
8. #16546C2semidirect
9. #16268C2semidirect
10. #16263C2semidirect
11. #15568C2semidirect
12. #15420C2semidirect
13. #14873C2semidirect
14. #14765C2semidirect
15. #13940C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#26312
#171982
#171052
#168952
#166962
#165592
#165462
#162682
#162632
#155682
#154202
#148732
#147652
#139402