Finite Group Databasebeta

← #6528583

Subgroups — #6528583

Order 512.

1479 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35241C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#29336C212yesmaximal
4#29329C212yesmaximal, complemented
5#27983C212yesmaximal, complemented
6#27977C212yesmaximal
7#27761C214yesmaximal, complemented
8#27757C212yesmaximal, complemented
9#27748C212yesmaximal, complemented
10#27745C214yesmaximal, complemented
11#27736C212yesmaximal, complemented
12#27730C212yesmaximal, complemented
13#21867C211yesmaximal, characteristic
14#21860C211yesmaximal, characteristic, complemented
15#21394C211yesmaximal, characteristic, complemented
16#21388C211yesmaximal, characteristic
17#20535C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2857C2212yes—
23#2850C2212yes—
24#2664—22no—
25#2571—21no—
26#2571—21no—
27#2571—22no—
28#2571C2212yes—
29#2571C2212yes—
30#2571C2214yes—
31#2568C2211yescharacteristic
32#2565C2212yes—
33#2547C2211yescharacteristic
34#2546C2211yescharacteristic
35#2545C2212yes—
36#2542C2211yescharacteristic
37#2498—22no—
38#2417—21no—
39#2417—21no—
40C2×#400—22no—
41#2323—21no—
42#2323—21no—
43C2×#389—21no—
44C2×#389C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382—21no—
47C2×#382C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2211yescharacteristic
50#2101C2214yes—
51#2100C2214yes—
52#2074C2214yes—
53#2074C2214yes—
54#2074C2214yes—
55#2073C2214yes—
56#2043C2214yes—
57#2042C2214yes—
58#1992C2212yes—
59#1990C2212yes—
60#1989C2212yes—
61#1988C2212yes—
62#1983C2212yes—
63#1981C2212yes—
64#1960C2214yes—
65#1959C2212yes—
66#1958C2214yes—
67#1956C2212yes—
68#1936C2214yes—
69#1935C2214yes—
70#1933C2214yes—
71#1933C2214yes—
72#1874C2211yescharacteristic
73#1874C2212yes—
74#1874C2211yescharacteristic
75#1872C2212yes—
76#1872C2212yes—
77#1872C2211yescharacteristic
78#1872C2212yes—
79#1872C2212yes—
80#1872C2211yescharacteristic
81#1872C2212yes—
82#1836C2214yes—
83#1835C2214yes—
84#1803C2212yes—
85#1803C2214yes—
86#1802C2212yes—
87#1771C2214yes—
88#1770C2212yes—
89C8×C4⋊C4C2212yes—
90#1763C2214yes—
91C4×#103C2212yes—
92C4×#102C2212yes—
93#1749C2212yes—
94C4×C22⋊C8C2212yes—
95#533—24no—
96#533—24no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×C4⋊Q8C2311yescharacteristic
100#529—24no—
101#529—24no—
102C2×#126—21no—
103C2×#125—21no—
104C2×#125C2311yescharacteristic
105C2×#124C2311yescharacteristic
106C2×#124C2312yes—
107C2×#123—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2311yescharacteristic
113C2×#121C2311yescharacteristic
114C2×C4×Q8—21no—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4D411yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C42—21no—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#432—24no—
133#431—24no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2314yes—
143C2×#102C2314yes—
144C2×#102C2312yes—
145C2×#102C2312yes—
146C2×C22⋊C8C2311yescharacteristic
147C2×C22⋊C8C2311yescharacteristic
148C2×C22⋊C8C2312yes—
149C2×#97C2312yes—
150C2×#97C2314yes—
151C2×C4×C8C2312yes—
152C2×C4×C8C2314yes—
153#400—22no—
154#400—24no—
155#400—24no—
156#400—24no—
157#400—24no—
158#399—22no—
159#398—22no—
160#398—22no—
161#398—22no—
162#397—22no—
163#396—22no—
164#396—22no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#390C2312yes—
171#389—22no—
172#389—22no—
173#389—24no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389—22no—
180#389C2314yes—
181#389C2314yes—
182#388—22no—
183#388—22no—
184#388—22no—
185#388—22no—
186#387—22no—
187#387—22no—
188#386—22no—
189#386—22no—
190#384—21no—
191#384—21no—
192#384C2312yes—
193#384C2312yes—
194#384C2312yes—
195#382—22no—
196#382—22no—
197#382—24no—
198#382—22no—
199#382—22no—
200#382C2312yes—
201#382C2314yes—
202#382C2314yes—
203#380—22no—
204#380—22no—
205#380—21no—
206#380—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D412yes—
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C2×#95—21no—
218C2×#95—22no—
219C2×#95—22no—
220C2×#95—22no—
221C2×#95D412yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2312yes—
229#340C2312yes—
230#340C2314yes—
231#340C2312yes—
232#336C2314yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8C2411yescharacteristic
236C22×Q8C2412yes—
237C22×D4—21no—
238C22×D4C2411yescharacteristic
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4C2×D411yescharacteristic
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2411yescharacteristic
248C22×C8C2412yes—
249C22×C8C2412yes—
250C22×C8C2414yes—
251C4⋊Q8—22no—
252C4⋊Q8—22no—
253C4⋊Q8—22no—
254C4⋊Q8—22no—
255C4⋊Q8—44no—
256#126—44no—
257#125—22no—
258#125—22no—
259#125—44no—
260#124—22no—
261#124—22no—
262#124—44no—
263#123—44no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266C22⋊Q8—24no—
267C22⋊Q8—24no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271#121—22no—
272#121—22no—
273C4×Q8—22no—
274C4×Q8—22no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—44no—
281#117—44no—
282#117—44no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—42no—
306C2×C4⋊C4—42no—
307C2×C4⋊C4—42no—
308C2×C4⋊C4—42no—
309C2×C4⋊C4—44no—
310C2×C4⋊C4—44no—
311C2×C4⋊C4—82no—
312C2×C4⋊C4—84no—
313C2×C4⋊C4C2×D411yescharacteristic
314C2×C4⋊C4C2×D411yescharacteristic
315C2×C4⋊C4C2×D411yescharacteristic
316C2×C4⋊C4C2×D411yescharacteristic
317C2×C4⋊C4C2×D412yes—
318C2×C4⋊C4C2×D412yes—
319C2×C4⋊C4C2412yes—
320C2×C4⋊C4C2412yes—
321C2×C4⋊C4C2414yes—
322C2×C4⋊C4C2411yescharacteristic
323C2×C4⋊C4C2411yescharacteristic
324C2×C4⋊C4C2412yes—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—21no—
327C2×C22⋊C4—21no—
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C42—21no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—22no—
339C2×C42—44no—
340C2×C42—44no—
341C2×C42C2×D412yes—
342C2×C42C2×D412yes—
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42#4114yes—
346C2×C42#4114yes—
347C2×C42C2412yes—
348C2×C42C2412yes—
349C2×C42C2411yescharacteristic
350C2×C42C2411yescharacteristic
351C4⋊C8—24no—
352C4⋊C8—24no—
353#103—42no—
354#103—44no—
355#103—42no—
356#103—44no—
357#103—44no—
358#103—44no—
359#103—44no—
360#102—42no—
361#102—44no—
362#102—42no—
363C22⋊C8—22no—
364C22⋊C8—22no—
365C22⋊C8—24no—
366#97—24no—
367#97—44no—
368C4×C8—24no—
369C4×C8—44no—
370#95—22no—
371#95—22no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—22no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95C2×D412yes—
387#95C2×D412yes—
388#95C2×D412yes—
389#95C2×D412yes—
390C24—21no—
391C24C22×D411yescharacteristic
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—24no—
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—42no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—42no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—44no—
434C22×C4—42no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4#12414yes—
443C22×C4#12414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2511yescharacteristic
456C2×C8—22no—
457C2×C8—22no—
458C2×C8—24no—
459C2×C8—22no—
460C2×C8—24no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—22no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—24no—
481C4⋊C4—42no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—42no—
490C4⋊C4—42no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—44no—
495C4⋊C4—42no—
496C4⋊C4—44no—
497C4⋊C4—42no—
498C4⋊C4—44no—
499C4⋊C4—82no—
500C4⋊C4—82no—
501C4⋊C4—84no—
502C4⋊C4—84no—
503C4⋊C4—82no—
504C4⋊C4—82no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C22⋊C4—42no—
511C22⋊C4—42no—
512C22⋊C4—42no—
513C22⋊C4—82no—
514C22⋊C4—82no—
515C42—24no—
516C42—24no—
517C42—42no—
518C42—42no—
519C42—44no—
520C42—42no—
521C42—44no—
522C42—44no—
523C42#12414yes—
524C42#12414yes—
525C42C2×#4112yes—
526C42C2×#4112yes—
527C42C2×#4112yes—
528C42C2×#4112yes—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23C2×#12412yes—
541C23#57712yes—
542C23#57812yes—
543C23C23×D411yescharacteristic
544Q8—42no—
545Q8—42no—
546Q8—44no—
547Q8—44no—
548Q8—84no—
549D4—42no—
550D4—42no—
551D4—84no—
552C2×C4—28no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—84no—
601C2×C4—84no—
602C2×C4—84no—
603C2×C4—88no—
604C2×C4—84no—
605C2×C4—84no—
606C2×C4C2×#12412yes—
607C2×C4C2×#12412yes—
608C2×C4C2×#12412yes—
609C2×C4C2×#12412yes—
610C2×C4C2×#12412yes—
611C2×C4#57512yes—
612C2×C4#57512yes—
613C2×C4C22×#4111yescharacteristic
614C2×C4C22×#4111yescharacteristic
615C2×C4#58512yes—
616C2×C4#58512yes—
617C8—42no—
618C8—42no—
619C8—44no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—22no—
624C22—22no—
625C22—22no—
626C22—81no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22#313714yes—
634C22C22×#12411yescharacteristic
635C22#348511yescharacteristic
636C22#348711yescharacteristic
637C22C2×#57511yescharacteristic
638C22C2×#57511yescharacteristic
639C22C2×#57711yescharacteristic
640C22C2×#57811yescharacteristic
641C4—22no—
642C4—22no—
643C4—22no—
644C4—22no—
645C4—42no—
646C4—42no—
647C4—42no—
648C4—42no—
649C4—82no—
650C4—81no—
651C4—82no—
652C4—81no—
653C4—81no—
654C4—82no—
655C4—84no—
656C4—82no—
657C4#314214yes—
658C4#351212yes—
659C4#351212yes—
660C2—24no—
661C2—161no—
662C2#3564911yescharacteristic
663C2C2×#313711yescharacteristic
664C2C2×#314211yescharacteristic
665C2#6173011yescharacteristic
666C2#6173111yescharacteristic
667C2#6190411yescharacteristic
668C2#6190511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2571C2semidirect
4. #29329C2semidirect
5. #27983C2semidirect
7. #27761C2semidirect
8. #27757C2semidirect
9. #27748C2semidirect
10. #27745C2semidirect
11. #27736C2semidirect
12. #27730C2semidirect
14. #21860C2semidirect
15. #21394C2semidirect
17. #20535C2semidirect

Maximal subgroups

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

subgroupindex
#352412
C2×#25712
#293362
#293292
#279832
#279772
#277612
#277572
#277482
#277452
#277362
#277302
#218672
#218602
#213942
#213882
#205352