Finite Group Databasebeta

← #5025772

Subgroups — #5025772

Order 512.

3769 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2607C211yesmaximal, characteristic, complemented
2C2×#2613C211yesmaximal, characteristic
3#15639C214yesmaximal, complemented
4#15631C214yesmaximal, complemented
5#15467C212yesmaximal, complemented
6#15428C214yesmaximal, complemented
7#15317C214yesmaximal, complemented
8#15284C212yesmaximal, complemented
9#15283C212yesmaximal, complemented
10#15259C212yesmaximal, complemented
11#14287C212yesmaximal, complemented
12#13902C212yesmaximal, complemented
13#13828C211yesmaximal, characteristic, complemented
14C24×D4C2211yescharacteristic
15C22×#124C2212yescomplemented
16C22×#123C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×#120C2211yescharacteristic
19#2613—28no—
20#2607—28no—
21C2×#399C2212yes—
22C2×#397C2212yescomplemented
23C2×#394C2214yes—
24C2×#394C2214yescomplemented
25C2×#393C2212yes—
26C2×#393C2212yescomplemented
27C2×#393C2214yescomplemented
28C2×#392C2214yes—
29C2×#391C2214yescomplemented
30C2×#390C2214yes—
31C2×#390C2214yes—
32C2×#390C2212yescomplemented
33C2×#388C2214yescomplemented
34C2×#387C2214yescomplemented
35C2×#381C2211yescharacteristic
36C2×#380C2212yes—
37C2×#379C2211yescharacteristic
38C2×#379C2211yescharacteristic
39C2×#379C2214yes—
40C2×#379C2214yes—
41C2×C4×C22⋊C4C2212yes—
42#1441C2214yescomplemented
43#1440C2214yes—
44#1440C2214yescomplemented
45#1440C2214yescomplemented
46#1440C2214yescomplemented
47#1440C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1438C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2212yescomplemented
56#1438C2214yescomplemented
57#1438C2212yescomplemented
58#1437C2212yes—
59#1437C2212yes—
60#1437C2212yescomplemented
61#1437C2212yescomplemented
62#1437C2214yescomplemented
63#1437C2212yescomplemented
64#1437C2212yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2212yescomplemented
68#1436C2212yescomplemented
69C26C2311yescharacteristic
70C26C2311yescharacteristic
71C23×Q8C2312yes—
72C23×D4—24no—
73C23×D4—28no—
74C23×D4—44no—
75C23×D4—44no—
76C23×D4C2314yes—
77C23×D4C2311yescharacteristic
78C23×D4C2312yes—
79C23×D4C2312yes—
80C24×C4C2311yescharacteristic
81C2×#124—44no—
82C2×#124—48no—
83C2×#123—24no—
84C2×#123—28no—
85C2×#120—24no—
86C2×#120—24no—
87C2×#120—44no—
88C2×#120—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2314yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2311yescharacteristic
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C42C2312yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117#399—28no—
118#397—44no—
119#394—216no—
120#394—48no—
121#393—28no—
122#393—48no—
123#393—44no—
124#392—216no—
125#391—216no—
126#390—48no—
127#390—48no—
128#390—44no—
129#388—48no—
130#387—216no—
131#381D418yes—
132#380—28no—
133#379—24no—
134#379—24no—
135#379—48no—
136#379—48no—
137C4×C22⋊C4—28no—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C25—22no—
156C25—22no—
157C25—24no—
158C25—24no—
159C25—24no—
160C25—24no—
161C25—42no—
162C25—42no—
163C25—44no—
164C25—44no—
165C25—44no—
166C25—44no—
167C25C2411yescharacteristic
168C25C2412yes—
169C25C2412yes—
170C25C2411yescharacteristic
171C22×Q8—24no—
172C22×Q8—28no—
173C22×Q8—42no—
174C22×Q8—44no—
175C22×Q8—42no—
176C22×D4—28no—
177C22×D4—28no—
178C22×D4—22no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—28no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—44no—
188C22×D4—44no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—42no—
192C22×D4—42no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—464no—
198C22×D4—416no—
199C22×D4—416no—
200C22×D4—416no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C2×D414yes—
204C23×C4#4118yes—
205C23×C4C2414yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214#124—432no—
215#123—416no—
216#120—416no—
217#120—416no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—28no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
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⋊C4C2×D414yes—
286C2×C22⋊C4C2×D414yes—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—44no—
290C2×C42—42no—
291C2×C42—44no—
292C2×C42—44no—
293C2×C42—44no—
294C2×C42—42no—
295#95—28no—
296#95—24no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—42no—
310#95—44no—
311#95C2×D4116yes—
312C24—24no—
313C24—22no—
314C24—22no—
315C24—24no—
316C24—28no—
317C24—28no—
318C24—22no—
319C24—22no—
320C24—24no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—21no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—416no—
341C24—416no—
342C24—44no—
343C24—44no—
344C24—416no—
345C24—416no—
346C24—44no—
347C24—44no—
348C24—88no—
349C24—88no—
350C24—84no—
351C24—84no—
352C24—84no—
353C24—84no—
354C24—84no—
355C24—84no—
356C24—88no—
357C24—84no—
358C24—84no—
359C24—88no—
360C24—84no—
361C24—84no—
362C24—88no—
363C24—84no—
364C24—84no—
365C24—88no—
366C24C22×D412yes—
367C24C2×#4114yes—
368C24H(5,𝔽2)14yes—
369C24H(5,𝔽2)14yes—
370C24C2511yescharacteristic
371C2×Q8—48no—
372C2×Q8—416no—
373C2×Q8—84no—
374C2×Q8—88no—
375C2×Q8—84no—
376C2×D4—232no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—416no—
380C2×D4—416no—
381C2×D4—416no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—4128no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—44no—
390C2×D4—416no—
391C2×D4—88no—
392C2×D4—88no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—88no—
396C2×D4—88no—
397C2×D4—88no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
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—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—416no—
447C22×C4#124116yescomplemented
448C22×C4C22×D414yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)18yes—
454C4⋊C4—416no—
455C4⋊C4—48no—
456C4⋊C4—88no—
457C4⋊C4—84no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—416no—
461C22⋊C4—416no—
462C22⋊C4—48no—
463C22⋊C4—48no—
464C22⋊C4—416no—
465C22⋊C4—416no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—88no—
469C22⋊C4—88no—
470C22⋊C4—88no—
471C22⋊C4—88no—
472C22⋊C4—88no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C22⋊C4—88no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C42—48no—
479C42—84no—
480C42—84no—
481C42—84no—
482C23—22no—
483C23—22no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—28no—
499C23—416no—
500C23—416no—
501C23—42no—
502C23—42no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—44no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—816no—
554C23—816no—
555C23—816no—
556C23—816no—
557C23—832no—
558C23—832no—
559C23—832no—
560C23—832no—
561C23—816no—
562C23—816no—
563C23C2×#12414yes—
564C23C23×D411yescharacteristic
565C23C22×#4112yes—
566C23C2×H(5,𝔽2)12yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)14yes—
569C23#58514yes—
570Q8—816no—
571D4—464no—
572D4—816no—
573D4—816no—
574D4—88no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—416no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
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—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4C2×#12418yes—
628C22—24no—
629C22—28no—
630C22—48no—
631C22—48no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—48no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—832no—
645C22—832no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22—84no—
652C22—84no—
653C22—84no—
654C22—81no—
655C22—84no—
656C22—84no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22C22×#12411yescharacteristic
661C22#348411yescharacteristic
662C22#348411yescharacteristic
663C22#348814yes—
664C22#349014yes—
665C22#350614yes—
666C22#350812yes—
667C22#351012yes—
668C22#353212yes—
669C22#353312yes—
670C22#353414yes—
671C22#354312yes—
672C22#354512yes—
673C22#355814yes—
674C4—28no—
675C4—44no—
676C4—48no—
677C4—44no—
678C4—81no—
679C4—84no—
680C4—84no—
681C4—84no—
682C4—82no—
683C2—44no—
684C2—84no—
685C2—84no—
686C2—161no—
687C2#3536711yescharacteristic
688C2#3555612yes—
689C2#4800514yes—
690C2#4801114yes—
691C2#5063512yes—
692C2#5085612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2613C2
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×#2607C2semidirect
3. #15639C2semidirect
4. #15631C2semidirect
5. #15467C2semidirect
6. #15428C2semidirect
7. #15317C2semidirect
8. #15284C2semidirect
9. #15283C2semidirect
10. #15259C2semidirect
11. #14287C2semidirect
12. #13902C2semidirect
13. #13828C2semidirect
15. C22×#124C22semidirect
22. C2×#397C22semidirect
24. C2×#394C22semidirect
26. C2×#393C22semidirect
27. C2×#393C22semidirect
29. C2×#391C22semidirect
32. C2×#390C22semidirect
33. C2×#388C22semidirect
34. C2×#387C22semidirect
42. #1441C22semidirect
44. #1440C22semidirect
45. #1440C22semidirect
46. #1440C22semidirect
47. #1440C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
55. #1438C22semidirect
56. #1438C22semidirect
57. #1438C22semidirect
60. #1437C22semidirect
61. #1437C22semidirect
62. #1437C22semidirect
63. #1437C22semidirect
64. #1437C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
447. C22×C4#124semidirect

Maximal subgroups

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

subgroupindex
C2×#26072
C2×#26132
#156392
#156312
#154672
#154282
#153172
#152842
#152832
#152592
#142872
#139022
#138282