Finite Group Databasebeta

← #619949

Subgroups — #619949

Order 512.

3948 conjugacy classes of subgroups.

The subgroup lattice has 681 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2849C211yesmaximal, characteristic
2C2×#2622C211yesmaximal, characteristic
3C2×#2321C211yesmaximal, characteristic
4C22×#381C211yesmaximal, characteristic, complemented
5#15717C214yesmaximal, complemented
6#15702C214yesmaximal, complemented
7#15560C214yesmaximal, complemented
8#15538C214yesmaximal, complemented
9#14935C212yesmaximal, complemented
10#14927C212yesmaximal, complemented
11#14681C211yesmaximal, characteristic, complemented
12#14341C211yesmaximal, characteristic, complemented
13#14338C211yesmaximal, characteristic, complemented
14#13992C212yesmaximal, complemented
15#13968C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×#123C2212yes—
18C22×#120C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2849—28no—
23#2622—28no—
24C2×#397C2214yes—
25C2×#394C2214yes—
26#2321—28no—
27C2×#390C2212yes—
28C2×#388C2212yes—
29C2×#388C2212yes—
30C2×#386C2214yes—
31C2×#386C2211yescharacteristic
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#385C2214yes—
35C2×#385C2211yescharacteristic
36C2×#385C2212yes—
37C2×#381—28no—
38C2×#381C2212yes—
39C2×#381C2212yes—
40C2×#381C2214yes—
41C2×#381C2214yes—
42C2×#381C2212yes—
43C2×#380C2212yes—
44C2×#379C2211yescharacteristic
45C2×#379C2211yescharacteristic
46C2×#379C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C22×#95C2212yes—
49C22×#95C2212yes—
50#1443C2214yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1440C2214yes—
60#1440C2214yes—
61#1440C2214yescomplemented
62#1440C2214yescomplemented
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yescomplemented
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yescomplemented
75C26C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2312yes—
78C24×C4—24no—
79C24×C4—28no—
80C24×C4—24no—
81C24×C4—24no—
82C24×C4—44no—
83C24×C4C2312yes—
84C24×C4C2314yes—
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#123—28no—
90C2×#123—28no—
91C2×#123—44no—
92C2×#120—24no—
93C2×#120—24no—
94C2×#120D418yes—
95C2×C4×D4—24no—
96C2×C4×D4—44no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4D418yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126#397—216no—
127#394—216no—
128#390—28no—
129#388—28no—
130#388—28no—
131#386—216no—
132#386—28no—
133#386—24no—
134#386—28no—
135#385—24no—
136#385—216no—
137#385—28no—
138#381—28no—
139#381—28no—
140#381—216no—
141#381—216no—
142#381—44no—
143#380—44no—
144#379—24no—
145#379—24no—
146#379—24no—
147C4×C22⋊C4—24no—
148C2×#95—28no—
149C2×#95—28no—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C25—24no—
173C25—24no—
174C25—44no—
175C25—44no—
176C25C2×D414yes—
177C25C2×D414yes—
178C25#4114yes—
179C25C2411yescharacteristic
180C25C2412yes—
181C25C2411yescharacteristic
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—44no—
189C22×D4—44no—
190C22×D4C2×D414yes—
191C22×D4C2×D418yes—
192C23×C4—232no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—432no—
219C23×C4—432no—
220C23×C4—416no—
221C23×C4—416no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4#4118yes—
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2414yes—
237#123—432no—
238#120—232no—
239C4×D4—416no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—416no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—416no—
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×D418yes—
298C2×C22⋊C4C2×D418yes—
299C2×C22⋊C4C2×D418yes—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—42no—
303C2×C42—44no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—28no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326C24—24no—
327C24—24no—
328C24—28no—
329C24—28no—
330C24—216no—
331C24—216no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24—21no—
341C24—22no—
342C24—21no—
343C24—22no—
344C24—24no—
345C24—24no—
346C24—48no—
347C24—48no—
348C24—416no—
349C24—416no—
350C24—416no—
351C24—416no—
352C24—416no—
353C24—416no—
354C24#121116yes—
355C24C22×D412yes—
356C24C22×D414yes—
357C24C22×D414yes—
358C24C22×D412yes—
359C24C22×D412yes—
360C24C2×#4114yes—
361C24C2×#4112yes—
362C24C2×#4114yes—
363C24H(5,𝔽2)14yes—
364C24C2511yescharacteristic
365C2×D4—28no—
366C2×D4—216no—
367C2×D4—216no—
368C2×D4—416no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—84no—
373C2×D4—84no—
374C2×D4—88no—
375C22×C4—232no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
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—48no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4—44no—
442C22×C4—416no—
443C22×C4—4128no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—84no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C2×#4118yes—
464C4⋊C4—416no—
465C4⋊C4—44no—
466C4⋊C4—84no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—216no—
470C22⋊C4—28no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—44no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—44no—
478C22⋊C4—44no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C42—44no—
487C42—84no—
488C23—28no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—28no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—24no—
504C23—24no—
505C23—28no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—232no—
511C23—22no—
512C23—24no—
513C23—28no—
514C23—432no—
515C23—432no—
516C23—464no—
517C23—416no—
518C23—416no—
519C23—464no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—41no—
529C23—41no—
530C23—44no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23C2×#12118yes—
539C23C2×#12114yes—
540C23#52514yes—
541C23#53414yes—
542C23#54618yes—
543C23#55118yes—
544C23C23×D411yescharacteristic
545C23C23×D411yescharacteristic
546C23C22×#4111yescharacteristic
547C23C22×#4112yes—
548C23C2×H(5,𝔽2)12yes—
549C23#58514yes—
550C23#58514yes—
551D4—416no—
552D4—816no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—464no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—432no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
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—88no—
616C2×C4—88no—
617C2×C4—88no—
618C2×C4—816no—
619C22—28no—
620C22—216no—
621C22—24no—
622C22—24no—
623C22—28no—
624C22—216no—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—464no—
629C22—44no—
630C22—44no—
631C22—48no—
632C22—48no—
633C22—44no—
634C22—44no—
635C22—48no—
636C22—48no—
637C22—42no—
638C22—48no—
639C22—44no—
640C22—82no—
641C22—81no—
642C22—81no—
643C22—84no—
644C22#253418yes—
645C22C22×#12111yescharacteristic
646C22C2×#52511yescharacteristic
647C22C2×#53411yescharacteristic
648C22#345514yes—
649C22C2×#54612yes—
650C22C2×#55112yes—
651C22#348512yes—
652C22#348512yes—
653C22#348514yes—
654C22#349114yes—
655C22#349314yes—
656C22#353314yes—
657C22#353314yes—
658C4—48no—
659C4—42no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—82no—
665C4—84no—
666C4—84no—
667C4—81no—
668C4—81no—
669C4—84no—
670C2—28no—
671C2—48no—
672C2—82no—
673C2C2×#253411yescharacteristic
674C2#3509312yes—
675C2#3544712yes—
676C2#3695511yescharacteristic
677C2#3696511yescharacteristic
678C2#3766714yes—
679C2#3785514yes—
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
4. C22×#381C2semidirect
5. #15717C2semidirect
6. #15702C2semidirect
7. #15560C2semidirect
8. #15538C2semidirect
9. #14935C2semidirect
10. #14927C2semidirect
11. #14681C2semidirect
12. #14341C2semidirect
13. #14338C2semidirect
14. #13992C2semidirect
15. #13968C2semidirect
50. #1443C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
61. #1440C22semidirect
62. #1440C22semidirect
65. #1439C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. #1434C22semidirect
74. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28492
C2×#26222
C2×#23212
C22×#3812
#157172
#157022
#155602
#155382
#149352
#149272
#146812
#143412
#143382
#139922
#139682