Finite Group Databasebeta

← #766202

Subgroups — #766202

Order 512.

3804 conjugacy classes of subgroups.

The subgroup lattice has 677 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#393C211yesmaximal, characteristic, complemented
2C2×#2321C211yesmaximal, characteristic
3#15770C214yesmaximal, complemented
4#15586C214yesmaximal, complemented
5#15506C214yesmaximal, complemented
6#15476C214yesmaximal, complemented
7#15026C211yesmaximal, characteristic, complemented
8#14946C211yesmaximal, characteristic, complemented
9#14874C211yesmaximal, characteristic, complemented
10#14873C211yesmaximal, characteristic, complemented
11#14134C212yesmaximal
12#14040C212yesmaximal, complemented
13#14016C212yesmaximal, complemented
14#13877C212yesmaximal, complemented
15#13794C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×#120C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C2×#395C2214yescomplemented
23C2×#393—28no—
24C2×#393C2214yes—
25C2×#393C2214yes—
26C2×#393C2214yescomplemented
27#2321—28no—
28C2×#390C2212yes—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#386C2211yescharacteristic
32C2×#386C2212yes—
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2214yes—
36C2×#385C2211yescharacteristic
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#380C2214yes—
40C2×#379C2211yescharacteristic
41C2×C4×C22⋊C4C2211yescharacteristic
42C22×#95C2211yescharacteristic
43C22×#95C2212yes—
44#1446C2214yescomplemented
45#1446C2214yescomplemented
46#1445C2214yescomplemented
47#1445C2214yescomplemented
48#1443C2214yescomplemented
49#1443C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1440C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yescomplemented
56#1439C2214yescomplemented
57#1438C2214yes—
58#1436C2214yescomplemented
59#1436C2214yescomplemented
60#1434C2214yescomplemented
61C22⋊#95C2212yes—
62C22⋊#95C2211yescharacteristic
63C22⋊#95C2212yes—
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2214yes—
66C22⋊#95C2212yes—
67C22⋊#95C2214yes—
68C22⋊#95C2212yes—
69C22⋊#95C2214yes—
70C22⋊#95C2214yes—
71C22⋊#95C2212yes—
72C22⋊#95C2214yescomplemented
73C26C2311yescharacteristic
74C23×Q8—24no—
75C23×Q8—44no—
76C23×Q8—48no—
77C23×Q8C2314yes—
78C23×D4C2311yescharacteristic
79C23×D4C2312yes—
80C24×C4C2312yes—
81C24×C4C2312yes—
82C24×C4C2314yes—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#120—24no—
87C2×#120—24no—
88C2×#120Q818yescomplemented
89C2×C4×D4—24no—
90C2×C4×D4—44no—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—28no—
97C22×C22⋊C4—28no—
98C22×C22⋊C4—48no—
99C22×C22⋊C4—44no—
100C22×C22⋊C4—44no—
101C22×C22⋊C4D418yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2311yescharacteristic
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#395—48no—
122#393—216no—
123#393—216no—
124#393—48no—
125#390—28no—
126#388—28no—
127#388—28no—
128#386—24no—
129#386—28no—
130#386—28no—
131#386—48no—
132#386—44no—
133#385—24no—
134#385—28no—
135#385—44no—
136#380—48no—
137#379—24no—
138C4×C22⋊C4—24no—
139C2×#95—28no—
140C2×#95—24no—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C25—24no—
164C25—24no—
165C25—44no—
166C25—44no—
167C25C2×D414yes—
168C25C2×Q814yes—
169C25#4114yes—
170C25C2411yescharacteristic
171C25C2412yes—
172C25C2411yescharacteristic
173C22×Q8—28no—
174C22×Q8—28no—
175C22×Q8—28no—
176C22×Q8—44no—
177C22×Q8—44no—
178C22×Q8—44no—
179C22×Q8—44no—
180C22×Q8—432no—
181C22×Q8—416no—
182C22×Q8—464no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—24no—
189C22×D4—44no—
190C22×D4—44no—
191C22×D4C2×Q814yes—
192C22×D4C2×Q818yescomplemented
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—28no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4—42no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C2×D414yes—
218C23×C4C2×D414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231#120—232no—
232C4×D4—416no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—432no—
269C2×C22⋊C4—416no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—432no—
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⋊C4C2×D418yes—
287C2×C22⋊C4C2×D418yes—
288C2×C22⋊C4C2×Q814yes—
289C2×C22⋊C4C2×Q818yescomplemented
290C2×C22⋊C4#4118yes—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—42no—
294C2×C42—44no—
295#95—28no—
296#95—28no—
297#95—28no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—44no—
306#95—44no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317C24—24no—
318C24—24no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—28no—
328C24—28no—
329C24—216no—
330C24—216no—
331C24—21no—
332C24—22no—
333C24—21no—
334C24—24no—
335C24—22no—
336C24—24no—
337C24—48no—
338C24—48no—
339C24—416no—
340C24—416no—
341C24—88no—
342C24—88no—
343C24—88no—
344C24—88no—
345C24C22⋊Q8116yes—
346C24C22×D414yes—
347C24C22×D412yes—
348C24C22×Q812yes—
349C24C22×Q814yes—
350C24C22×Q812yes—
351C24C2×#4114yes—
352C24C2×#4112yes—
353C24C2×#4114yes—
354C24H(5,𝔽2)14yes—
355C24C2511yescharacteristic
356C2×Q8—416no—
357C2×Q8—416no—
358C2×Q8—416no—
359C2×Q8—416no—
360C2×Q8—416no—
361C2×Q8—416no—
362C2×Q8—4128no—
363C2×Q8—416no—
364C2×D4—28no—
365C2×D4—216no—
366C2×D4—216no—
367C2×D4—416no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—84no—
372C2×D4—84no—
373C2×D4—88no—
374C22×C4—216no—
375C22×C4—28no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—416no—
433C22×C4—416no—
434C22×C4—416no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—84no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—88no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×Q814yes—
457C22×C4C22×Q812yes—
458C22×C4C2×#4118yes—
459C22×C4H(5,𝔽2)18yes—
460C4⋊C4—44no—
461C4⋊C4—84no—
462C4⋊C4—88no—
463C4⋊C4—88no—
464C22⋊C4—28no—
465C22⋊C4—216no—
466C22⋊C4—216no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—416no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—84no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—84no—
480C22⋊C4—84no—
481C42—44no—
482C42—84no—
483C23—24no—
484C23—24no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—28no—
505C23—232no—
506C23—22no—
507C23—24no—
508C23—28no—
509C23—432no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—832no—
529C23—832no—
530C23—88no—
531C23—816no—
532C23—88no—
533C23C2×C22⋊Q818yes—
534C23C2×C22⋊Q814yes—
535C23#53214yes—
536C23#53414yes—
537C23#55118yes—
538C23#55418yes—
539C23C23×D411yescharacteristic
540C23C23×Q811yescharacteristic
541C23C22×#4111yescharacteristic
542C23C22×#4112yes—
543C23C2×H(5,𝔽2)12yes—
544C23C2×H(5,𝔽2)14yes—
545C23#58514yes—
546Q8—464no—
547D4—416no—
548D4—816no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—432no—
567C2×C4—416no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
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—88no—
612C2×C4—832no—
613C2×C4—88no—
614C2×C4—816no—
615C22—28no—
616C22—216no—
617C22—216no—
618C22—24no—
619C22—28no—
620C22—24no—
621C22—28no—
622C22—28no—
623C22—28no—
624C22—48no—
625C22—48no—
626C22—48no—
627C22—48no—
628C22—44no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—42no—
633C22—44no—
634C22—48no—
635C22—832no—
636C22—84no—
637C22—81no—
638C22—82no—
639C22—81no—
640C22#256418yes—
641C22C22×C22⋊Q811yescharacteristic
642C22C2×#53211yescharacteristic
643C22C2×#53411yescharacteristic
644C22C2×#55112yes—
645C22C2×#55412yes—
646C22#348412yes—
647C22#348512yes—
648C22#350114yes—
649C22#350514yes—
650C22#351414yes—
651C22#351414yes—
652C22#353214yes—
653C22#354514yes—
654C4—42no—
655C4—44no—
656C4—48no—
657C4—44no—
658C4—48no—
659C4—42no—
660C4—42no—
661C4—84no—
662C4—81no—
663C4—82no—
664C4—84no—
665C4—81no—
666C2—28no—
667C2—48no—
668C2—82no—
669C2C2×#256411yescharacteristic
670C2#3534912yes—
671C2#3546612yes—
672C2#3670211yescharacteristic
673C2#3711911yescharacteristic
674C2#4355014yes—
675C2#4357414yes—
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
1. C22×#393C2semidirect
3. #15770C2semidirect
4. #15586C2semidirect
5. #15506C2semidirect
6. #15476C2semidirect
7. #15026C2semidirect
8. #14946C2semidirect
9. #14874C2semidirect
10. #14873C2semidirect
12. #14040C2semidirect
13. #14016C2semidirect
14. #13877C2semidirect
15. #13794C2semidirect
22. C2×#395C22semidirect
26. C2×#393C22semidirect
44. #1446C22semidirect
45. #1446C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1443C22semidirect
49. #1443C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
58. #1436C22semidirect
59. #1436C22semidirect
60. #1434C22semidirect
72. C22⋊#95C22semidirect
88. C2×#120Q8semidirect
192. C22×D4C2×Q8semidirect
289. C2×C22⋊C4C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#3932
C2×#23212
#157702
#155862
#155062
#154762
#150262
#149462
#148742
#148732
#141342
#140402
#140162
#138772
#137942