Finite Group Databasebeta

← #6484999

Subgroups — #6484999

Order 512.

1549 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35247C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#33099C212yesmaximal, complemented
4#33094C212yesmaximal
5#29913C212yesmaximal, complemented
6#29912C212yesmaximal
7#26329C214yesmaximal, complemented
8#26291C214yesmaximal, complemented
9#25063C212yesmaximal, complemented
10#25057C212yesmaximal, complemented
11#24995C212yesmaximal, complemented
12#24993C212yesmaximal, complemented
13#21982C211yesmaximal, characteristic, complemented
14#21867C211yesmaximal, characteristic
15#21820C211yesmaximal, characteristic, complemented
16#21388C211yesmaximal, characteristic
17#21365C211yesmaximal, characteristic, complemented
18C2×#529C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2855C2212yes—
23#2854C2212yes—
24#2637—22no—
25#2592C2212yes—
26#2584C2212yes—
27#2582C2211yescharacteristic
28#2582C2211yescharacteristic
29#2538C2211yescharacteristic
30#2538C2211yescharacteristic
31#2486—21no—
32#2413—22no—
33#2409—21no—
34#2355—21no—
35#2355—21no—
36#2355—22no—
37#2355C2212yes—
38#2355C2212yes—
39#2355C2214yes—
40#2323—21no—
41#2323—21no—
42C2×#391—21no—
43C2×#387—22no—
44#2285C2211yescharacteristic
45C2×#384C2211yescharacteristic
46C2×#382C2212yes—
47C2×C4×C4⋊C4—21no—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2101C2214yes—
52#2097C2214yes—
53#2092C2214yes—
54#2085C2214yes—
55#2085C2214yes—
56#2074C2214yes—
57#2073C2214yes—
58#1992C2212yes—
59#1989C2212yes—
60#1985C2214yes—
61#1984C2212yes—
62#1984C2212yes—
63#1983C2212yes—
64#1959C2212yes—
65#1948C2212yes—
66#1947C2212yes—
67#1947C2214yes—
68#1928C2214yes—
69#1928C2214yes—
70#1925C2214yes—
71#1925C2214yes—
72#1921C2214yes—
73#1921C2214yes—
74#1918C2214yes—
75#1918C2214yes—
76#1874C2211yescharacteristic
77#1874C2212yes—
78#1874C2211yescharacteristic
79#1872C2211yescharacteristic
80#1872C2212yes—
81#1872C2211yescharacteristic
82#1853C2212yes—
83#1852C2212yes—
84#1852C2212yes—
85#1852C2212yes—
86#1845C2212yes—
87#1834C2212yes—
88#1817C2212yes—
89#1817C2212yes—
90#1803C2212yes—
91#1802C2212yes—
92#1776C2214yes—
93C4×#103C2212yes—
94C4×#102C2212yes—
95C23×Q8D411yescharacteristic
96#529—24no—
97#529—24no—
98C2×#126—21no—
99C2×#125C2311yescharacteristic
100C2×#124C2311yescharacteristic
101C2×#123—21no—
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2311yescharacteristic
106C2×#121C2311yescharacteristic
107C2×C4×Q8—24no—
108C2×C4×Q8—24no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8D412yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×D4—21no—
117C2×C4×D4C2311yescharacteristic
118C2×#117—21no—
119C2×#117—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4D412yes—
123C22×C4⋊C4D411yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C42—21no—
128C22×C42—22no—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#432—24no—
132#431—24no—
133C2×#107C2314yes—
134C2×#107C2314yes—
135C2×#107C2314yes—
136C2×#106C2314yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2312yes—
139C2×#103C2312yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#102C2312yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×C22⋊C8C2311yescharacteristic
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2312yes—
148C2×#97C2312yes—
149C2×C4×C8C2312yes—
150#398—22no—
151#398—22no—
152#398—22no—
153#397—22no—
154#396—22no—
155#396—22no—
156#395—22no—
157#393—22no—
158#391—21no—
159#391—21no—
160#391—22no—
161#391—22no—
162#391—24no—
163#391C2312yes—
164#391C2312yes—
165#390C2312yes—
166#390C2312yes—
167#389—21no—
168#389—21no—
169#389—22no—
170#389—22no—
171#389C2312yes—
172#389C2312yes—
173#389C2312yes—
174#389C2312yes—
175#388—22no—
176#388—22no—
177#388—22no—
178#388—22no—
179#387—24no—
180#387—24no—
181#387—24no—
182#387—24no—
183#387—22no—
184#387—22no—
185#386—22no—
186#385—22no—
187#384—21no—
188#384—21no—
189#384C2312yes—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#382—22no—
194#382—22no—
195#382—21no—
196#382C2314yes—
197#382C2314yes—
198#381—22no—
199#380—21no—
200#380—21no—
201#380—21no—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—24no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C2×#95—22no—
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#340C2312yes—
230#336C2314yes—
231#336C2314yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—24no—
235C22×Q8—42no—
236C22×Q8—42no—
237C22×Q8C2×D411yescharacteristic
238C22×Q8C2×D411yescharacteristic
239C22×Q8C2×D412yes—
240C22×Q8C2411yescharacteristic
241C22×D4—21no—
242C22×D4C2411yescharacteristic
243C23×C4—21no—
244C23×C4—22no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4C2×D412yes—
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C22×C8C2412yes—
253C22×C8C2412yes—
254C22×C8C2414yes—
255#126—44no—
256#125—22no—
257#125—22no—
258#124—22no—
259#124—22no—
260#123—44no—
261#123—84no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—44no—
265C22⋊Q8—44no—
266#121—22no—
267#121—22no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—28no—
271C4×Q8—22no—
272C4×Q8—22no—
273C4×Q8—24no—
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—24no—
277C4×Q8—22no—
278C4×Q8—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—44no—
282#117—44no—
283#117—82no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4—44no—
309C2×C4⋊C4—82no—
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2×D412yes—
313C2×C4⋊C4C2×D412yes—
314C2×C4⋊C4C2×D412yes—
315C2×C4⋊C4C2×D412yes—
316C2×C4⋊C4C2×D412yes—
317C2×C4⋊C4C2412yes—
318C2×C4⋊C4C2414yes—
319C2×C4⋊C4C2411yescharacteristic
320C2×C4⋊C4C2411yescharacteristic
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—21no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—22no—
340C2×C42—24no—
341C2×C42—24no—
342C2×C42—44no—
343C2×C42—84no—
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42#4114yes—
349C2×C42#4114yes—
350C2×C42C2412yes—
351C2×C42C2412yes—
352C2×C42C2411yescharacteristic
353C2×C42C2411yescharacteristic
354#107—28no—
355#107—44no—
356#107—44no—
357#106—44no—
358C4⋊C8—24no—
359C4⋊C8—24no—
360#103—24no—
361#103—42no—
362#103—44no—
363#102—24no—
364#102—42no—
365#102—44no—
366C22⋊C8—22no—
367C22⋊C8—22no—
368C22⋊C8—24no—
369#97—24no—
370C4×C8—24no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—24no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389C24—21no—
390C24C22×D411yescharacteristic
391C2×Q8—24no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—22no—
401C2×Q8—22no—
402C2×Q8—42no—
403C2×Q8—42no—
404C2×Q8—416no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—42no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4#12314yes—
445C22×C4#12314yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2511yescharacteristic
458C2×C8—22no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—22no—
466C2×C8—22no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—22no—
471C4⋊C4—24no—
472C4⋊C4—24no—
473C4⋊C4—24no—
474C4⋊C4—24no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—24no—
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—22no—
485C4⋊C4—24no—
486C4⋊C4—42no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—44no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—82no—
499C4⋊C4—82no—
500C4⋊C4—81no—
501C4⋊C4—81no—
502C4⋊C4—82no—
503C4⋊C4—82no—
504C4⋊C42−1+414yes—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C42—24no—
516C42—24no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—22no—
521C42—44no—
522C42—42no—
523C42—44no—
524C42—84no—
525C42—84no—
526C42—82no—
527C42#12314yes—
528C42#12314yes—
529C42C2×#4112yes—
530C42C2×#4112yes—
531C42C2×#4112yes—
532C42C2×#4112yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23C2×#12312yes—
545C23#57612yes—
546C23#57812yes—
547C23C23×D411yescharacteristic
548Q8—48no—
549Q8—44no—
550Q8—44no—
551Q8—42no—
552Q8—42no—
553D4—42no—
554D4—42no—
555D4—84no—
556C2×C4—28no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—48no—
582C2×C4—48no—
583C2×C4—48no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—88no—
608C2×C4—84no—
609C2×C4—84no—
610C2×C4C2×#12312yes—
611C2×C4C2×#12312yes—
612C2×C4C2×#12312yes—
613C2×C4C2×#12312yes—
614C2×C4C2×#12312yes—
615C2×C4#57512yes—
616C2×C4#57512yes—
617C2×C4C22×#4111yescharacteristic
618C2×C4C22×#4111yescharacteristic
619C2×C4C2×2−1+412yes—
620C2×C4#58512yes—
621C8—44no—
622C8—42no—
623C8—42no—
624C22—22no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22#309814yes—
638C22C22×#12311yescharacteristic
639C22#348611yescharacteristic
640C22#348711yescharacteristic
641C22C2×#57511yescharacteristic
642C22C2×#57511yescharacteristic
643C22C2×#57611yescharacteristic
644C22C2×#57811yescharacteristic
645C4—22no—
646C4—22no—
647C4—22no—
648C4—22no—
649C4—42no—
650C4—42no—
651C4—42no—
652C4—42no—
653C4—44no—
654C4—81no—
655C4—81no—
656C4—81no—
657C4—82no—
658C4—82no—
659C4—84no—
660C4—82no—
661C4#309314yes—
662C4#346112yes—
663C4#350712yes—
664C2—24no—
665C2—161no—
666C2#3518211yescharacteristic
667C2C2×#309311yescharacteristic
668C2C2×#309811yescharacteristic
669C2#6183511yescharacteristic
670C2#6184211yescharacteristic
671C2#6189811yescharacteristic
672C2#6190511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2355C2semidirect
3. #33099C2semidirect
5. #29913C2semidirect
7. #26329C2semidirect
8. #26291C2semidirect
9. #25063C2semidirect
10. #25057C2semidirect
11. #24995C2semidirect
12. #24993C2semidirect
13. #21982C2semidirect
15. #21820C2semidirect
17. #21365C2semidirect

Maximal subgroups

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

subgroupindex
#352472
C2×#23552
#330992
#330942
#299132
#299122
#263292
#262912
#250632
#250572
#249952
#249932
#219822
#218672
#218202
#213882
#213652