Finite Group Databasebeta

← #765944

Subgroups — #765944

Order 512.

3508 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2412C212yesmaximal
2C22×#397C211yesmaximal, characteristic, complemented
3C2×#2321C211yesmaximal, characteristic
4#15502C214yesmaximal, complemented
5#15494C214yesmaximal, complemented
6#15486C214yesmaximal, complemented
7#15485C214yesmaximal, complemented
8#15073C211yesmaximal, characteristic, complemented
9#15069C211yesmaximal, characteristic, complemented
10#14955C211yesmaximal, characteristic, complemented
11#14945C211yesmaximal, characteristic, complemented
12#13992C212yesmaximal, complemented
13#13968C212yesmaximal, complemented
14#13869C212yesmaximal, complemented
15#13794C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#120C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2212yes—
22C23×C22⋊C4C2211yescharacteristic
23#2412—216no—
24C2×#397—28no—
25C2×#397C2214yes—
26C2×#397C2214yes—
27C2×#397C2214yes—
28C2×#394C2214yes—
29#2321—28no—
30C2×#390C2212yes—
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#386C2211yescharacteristic
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2214yes—
37C2×#385C2211yescharacteristic
38C2×#385C2212yes—
39C2×#381C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#380C2214yes—
43C2×#379C2211yescharacteristic
44C2×C4×C22⋊C4C2211yescharacteristic
45C22×#95C2211yescharacteristic
46C22×#95C2212yes—
47#1444C2214yescomplemented
48#1443C2214yescomplemented
49#1443C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1440C2214yes—
57#1440C2214yes—
58#1440C2214yescomplemented
59#1440C2214yescomplemented
60#1439C2214yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63#1434C2214yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2212yes—
66C22⋊#95C2214yes—
67C22⋊#95C2214yes—
68C22⋊#95C2211yescharacteristic
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yescomplemented
74C26C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2312yes—
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×#123—28no—
84C2×#123—28no—
85C2×#123—44no—
86C2×#120—24no—
87C2×#120—24no—
88C2×#120D418yes—
89C2×C4×D4—24no—
90C2×C4×D4—44no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—28no—
99C22×C22⋊C4—28no—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—28no—
102C22×C22⋊C4—28no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4D418yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125#397—216no—
126#397—216no—
127#397—216no—
128#394—216no—
129#390—28no—
130#388—28no—
131#388—28no—
132#386—28no—
133#386—216no—
134#386—24no—
135#386—28no—
136#385—24no—
137#385—28no—
138#381—28no—
139#381—28no—
140#381—44no—
141#380—216no—
142#379—24no—
143C4×C22⋊C4—24no—
144C2×#95—28no—
145C2×#95—24no—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25—24no—
169C25—24no—
170C25—44no—
171C25—44no—
172C25C2×D414yes—
173C25C2×D414yes—
174C25#4114yes—
175C25C2411yescharacteristic
176C25C2412yes—
177C25C2411yescharacteristic
178C22×D4—22no—
179C22×D4—22no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4C2×D414yes—
187C22×D4C2×D418yes—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—28no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228#123—432no—
229#120—232no—
230C4×D4—416no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—416no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—432no—
274C2×C22⋊C4—416no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—432no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C2×D418yes—
289C2×C22⋊C4C2×D414yes—
290C2×C22⋊C4C2×D418yes—
291C2×C22⋊C4C2×D418yes—
292C2×C22⋊C4#4118yes—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—42no—
296C2×C42—44no—
297#95—28no—
298#95—28no—
299#95—28no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—28no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319C24—24no—
320C24—24no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—28no—
330C24—28no—
331C24—216no—
332C24—216no—
333C24—21no—
334C24—22no—
335C24—21no—
336C24—22no—
337C24—24no—
338C24—24no—
339C24—48no—
340C24—48no—
341C24—416no—
342C24—416no—
343C24—88no—
344C24—88no—
345C24—88no—
346C24—88no—
347C24#121116yes—
348C24C22×D412yes—
349C24C22×D414yes—
350C24C22×D414yes—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C2×#4114yes—
354C24C2×#4112yes—
355C24C2×#4114yes—
356C24H(5,𝔽2)14yes—
357C24C2511yescharacteristic
358C2×D4—28no—
359C2×D4—216no—
360C2×D4—216no—
361C2×D4—416no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—48no—
365C2×D4—84no—
366C2×D4—84no—
367C2×D4—88no—
368C22×C4—28no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—416no—
427C22×C4—44no—
428C22×C4—416no—
429C22×C4—416no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—84no—
441C22×C4—88no—
442C22×C4—88no—
443C22×C4—88no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4118yes—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—44no—
459C4⋊C4—84no—
460C4⋊C4—88no—
461C22⋊C4—216no—
462C22⋊C4—28no—
463C22⋊C4—216no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—416no—
467C22⋊C4—416no—
468C22⋊C4—44no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—48no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—88no—
478C42—44no—
479C42—84no—
480C23—24no—
481C23—24no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—28no—
497C23—28no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—232no—
503C23—22no—
504C23—24no—
505C23—28no—
506C23—432no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—832no—
526C23—832no—
527C23—88no—
528C23—816no—
529C23—88no—
530C23C2×#12118yes—
531C23C2×#12114yes—
532C23#53214yes—
533C23#53414yes—
534C23#54618yes—
535C23#55318yes—
536C23C23×D411yescharacteristic
537C23C23×D411yescharacteristic
538C23C22×#4111yescharacteristic
539C23C22×#4112yes—
540C23C2×H(5,𝔽2)12yes—
541C23#58514yes—
542C23#58514yes—
543D4—416no—
544D4—816no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—416no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
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—88no—
607C2×C4—832no—
608C2×C4—88no—
609C2×C4—816no—
610C2×C4—816no—
611C22—216no—
612C22—216no—
613C22—24no—
614C22—28no—
615C22—28no—
616C22—24no—
617C22—28no—
618C22—28no—
619C22—28no—
620C22—48no—
621C22—48no—
622C22—48no—
623C22—48no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—44no—
628C22—42no—
629C22—48no—
630C22—44no—
631C22—832no—
632C22—84no—
633C22—81no—
634C22—82no—
635C22—81no—
636C22#254318yes—
637C22C22×#12111yescharacteristic
638C22C2×#53211yescharacteristic
639C22C2×#53411yescharacteristic
640C22C2×#54612yes—
641C22C2×#55312yes—
642C22#348514yes—
643C22#348512yes—
644C22#348512yes—
645C22#349114yes—
646C22#352114yes—
647C22#352114yes—
648C22#353314yes—
649C22#354514yes—
650C4—42no—
651C4—48no—
652C4—44no—
653C4—44no—
654C4—42no—
655C4—42no—
656C4—84no—
657C4—84no—
658C4—81no—
659C4—84no—
660C4—82no—
661C4—81no—
662C2—28no—
663C2—48no—
664C2—82no—
665C2C2×#254311yescharacteristic
666C2#3544712yes—
667C2#3546712yes—
668C2#3695511yescharacteristic
669C2#3711911yescharacteristic
670C2#4352514yes—
671C2#4359714yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#397C2semidirect
4. #15502C2semidirect
5. #15494C2semidirect
6. #15486C2semidirect
7. #15485C2semidirect
8. #15073C2semidirect
9. #15069C2semidirect
10. #14955C2semidirect
11. #14945C2semidirect
12. #13992C2semidirect
13. #13968C2semidirect
14. #13869C2semidirect
15. #13794C2semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
58. #1440C22semidirect
59. #1440C22semidirect
60. #1439C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. #1434C22semidirect
73. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24122
C22×#3972
C2×#23212
#155022
#154942
#154862
#154852
#150732
#150692
#149552
#149452
#139922
#139682
#138692
#137942