Finite Group Databasebeta

← #6499019

Subgroups — #6499019

Order 512.

1440 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic, complemented
3#33297C211yesmaximal, characteristic, complemented
4#33252C211yesmaximal, characteristic, complemented
5#33043C211yesmaximal, characteristic, complemented
6#33042C211yesmaximal, characteristic
7#30206C211yesmaximal, characteristic, complemented
8#30203C211yesmaximal, characteristic, complemented
9#29878C211yesmaximal, characteristic, complemented
10#29807C211yesmaximal, characteristic
11#29085C214yesmaximal, complemented
12#29084C214yesmaximal, complemented
13#28989C214yesmaximal, complemented
14#28988C214yesmaximal, complemented
15#22268C211yesmaximal, characteristic, complemented
16#22247C211yesmaximal, characteristic, complemented
17#21647C211yesmaximal, characteristic, complemented
18#21646C211yesmaximal, characteristic, complemented
19#20473C211yesmaximal, characteristic, complemented
20C2×#500C2211yescharacteristic
21C2×#492C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#486C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C22×C4×C8C2211yescharacteristic
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2858C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2856C2211yescharacteristic
34#2647—21no—
35#2644—21no—
36#2628—21no—
37#2624—21no—
38#2513C2212yes—
39#2511—21no—
40#2511C2212yes—
41#2492—21no—
42#2492C2212yes—
43#2490C2212yes—
44#2454C2212yes—
45#2451—21no—
46#2451C2212yes—
47#2448—21no—
48#2447—21no—
49#2427—21no—
50#2426—21no—
51#2423—21no—
52#2423C2212yes—
53#2420C2212yes—
54#2331—21no—
55#2324—21no—
56#2319—21no—
57#2314—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2087C2214yes—
64#2076C2214yes—
65#2074C2214yes—
66#2071C2214yes—
67#2055C2214yes—
68#2026C2214yes—
69#2023C2214yes—
70#2000C2214yes—
71#1993C2214yes—
72#1992C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1973C2214yes—
78#1973C2214yes—
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1936C2214yes—
84#1935C2214yes—
85#1934C2214yes—
86#1933C2214yes—
87#1918C2214yes—
88#1918C2214yes—
89#1893C2214yes—
90#1892C2214yes—
91#1880C2214yes—
92#1877C2214yes—
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1770C2212yes—
98#1770C2212yes—
99#1763C2214yes—
100#1762C2214yes—
101#1752C2212yes—
102#1752C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C4×#97C2211yescharacteristic
107C22×SD16C2314yes—
108C23×C8C2312yes—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#127C2311yescharacteristic
116C2×#125C2312yes—
117C2×#124—21no—
118C2×#124—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2312yes—
122C2×#121—21no—
123C2×#121—21no—
124C2×#121C2312yes—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8C2312yes—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×#117D411yescharacteristic
132C2×#117D411yescharacteristic
133C22×C42C2311yescharacteristic
134#500—24no—
135#492—24no—
136#487—24no—
137#486—24no—
138#431—24no—
139C2×#107C2314yes—
140C2×C4⋊C8C2312yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×C22⋊C8C2312yes—
148C2×#97C2312yes—
149C2×#97C2312yes—
150C2×#97C2312yes—
151C2×#97C2312yes—
152C2×C4×C8—24no—
153C2×C4×C8C2312yes—
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158#400—22no—
159#400—22no—
160#400—22no—
161#399—22no—
162#399—22no—
163#398—22no—
164#398—22no—
165#397—22no—
166#397—22no—
167#397—22no—
168#397—22no—
169#396—22no—
170#395—22no—
171#394—22no—
172#394—22no—
173#393—22no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—22no—
181#391—22no—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#390—22no—
187#390—21no—
188#390—21no—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390C2312yes—
194#390C2312yes—
195#390C2312yes—
196#390C2312yes—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#389C2312yes—
213#388—22no—
214#388—22no—
215#388—22no—
216#388—22no—
217#388—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#387—22no—
222#386—22no—
223#385—22no—
224#384—21no—
225#384—21no—
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2312yes—
231#382—21no—
232#382—21no—
233#382C2312yes—
234#381—22no—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380—22no—
240#380C2312yes—
241#380C2312yes—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2311yescharacteristic
251C4×C4⋊C4C2311yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4C2311yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256#376D411yescharacteristic
257#376D411yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—22no—
275C22×Q8C2412yes—
276C22×D4—22no—
277C22×D4—21no—
278C22×D4—21no—
279C22×D4C2412yes—
280C23×C4—22no—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2411yescharacteristic
284C2×SD16—28no—
285C2×SD16—44no—
286C2×SD16—44no—
287C22×C8—24no—
288C22×C8—42no—
289C22×C8—42no—
290C22×C8C2412yes—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—44no—
301C4⋊Q8—44no—
302C4⋊Q8—82no—
303#127—22no—
304#127—22no—
305#125—44no—
306#124—82no—
307#124—82no—
308C22⋊Q8—44no—
309C22⋊Q8—82no—
310C22⋊Q8—82no—
311#121—44no—
312#121—44no—
313#121—82no—
314C4×Q8—44no—
315C4×Q8—82no—
316C4×Q8—82no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—82no—
320#117—24no—
321#117—42no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2×D411yescharacteristic
348C2×C4⋊C4C2×D411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393#107—28no—
394C4⋊C8—24no—
395#103—28no—
396#103—44no—
397#103—44no—
398#102—28no—
399#102—44no—
400#102—44no—
401C22⋊C8—24no—
402#97—42no—
403#97—42no—
404#97—42no—
405#97—42no—
406C4×C8—24no—
407C4×C8—22no—
408C4×C8—22no—
409C4×C8#4114yes—
410C4×C8#4114yes—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437C24—22no—
438C24C22×D411yescharacteristic
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—24no—
442C2×Q8—24no—
443C2×Q8—44no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—82no—
451C2×Q8—82no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—24no—
455C2×D4—24no—
456C2×D4—44no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—42no—
462C2×D4—42no—
463C2×D4—82no—
464C2×D4—82no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—44no—
482C22×C4—82no—
483C22×C4—82no—
484C22×C4#13714yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C2511yescharacteristic
499SD16—88no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—48no—
511C2×C8C2×#4114yes—
512C2×C8C2×#4114yes—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—41no—
524C4⋊C4—41no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—82no—
539C4⋊C4—82no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C4⋊C4—82no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—41no—
551C22⋊C4—41no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C22⋊C4—82no—
564C22⋊C4—82no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—21no—
576C42—21no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—82no—
582C42—82no—
583C42—82no—
584C42C2×#4112yes—
585C42C2×#4112yes—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23C2×#13712yes—
600C23C23×D411yescharacteristic
601Q8—44no—
602Q8—44no—
603Q8—84no—
604Q8—84no—
605Q8—164no—
606D4—44no—
607D4—44no—
608D4—84no—
609D4—84no—
610D4—164no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—28no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—164no—
669C2×C4#53212yes—
670C2×C4#53212yes—
671C2×C4C2×#13712yes—
672C2×C4C2×#13712yes—
673C2×C4#57512yes—
674C2×C4#57612yes—
675C2×C4#57712yes—
676C2×C4C22×#4111yescharacteristic
677C2×C4C22×#4111yescharacteristic
678C2×C4#58512yes—
679C2×C4#58512yes—
680C8—24no—
681C8—24no—
682C8—42no—
683C8—42no—
684C22—22no—
685C22—22no—
686C22—22no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22C2×#53211yescharacteristic
699C22#350011yescharacteristic
700C22#350011yescharacteristic
701C22C22×#13711yescharacteristic
702C22C2×#57511yescharacteristic
703C22C2×#57611yescharacteristic
704C22C2×#57711yescharacteristic
705C4—22no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—41no—
710C4—41no—
711C4—41no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4#317914yes—
724C4#318114yes—
725C4#345512yes—
726C4#351012yes—
727C2—42no—
728C2—162no—
729C2#3533111yescharacteristic
730C2C2×#317911yescharacteristic
731C2C2×#318111yescharacteristic
732C2#6253711yescharacteristic
733C2#6254011yescharacteristic
734C2#6254411yescharacteristic
735C2#6254611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35283C2semidirect
3. #33297C2semidirect
4. #33252C2semidirect
5. #33043C2semidirect
7. #30206C2semidirect
8. #30203C2semidirect
9. #29878C2semidirect
11. #29085C2semidirect
12. #29084C2semidirect
13. #28989C2semidirect
14. #28988C2semidirect
15. #22268C2semidirect
16. #22247C2semidirect
17. #21647C2semidirect
18. #21646C2semidirect
19. #20473C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#352832
#332972
#332522
#330432
#330422
#302062
#302032
#298782
#298072
#290852
#290842
#289892
#289882
#222682
#222472
#216472
#216462
#204732