Finite Group Databasebeta

← #6486800

Subgroups — #6486800

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic
3#33085C214yesmaximal, complemented
4#30013C214yesmaximal
5#29541C214yesmaximal, complemented
6#29416C214yesmaximal
7#29193C211yesmaximal, characteristic, complemented
8#29176C211yesmaximal, characteristic, complemented
9#28653C211yesmaximal, characteristic, complemented
10#28616C211yesmaximal, characteristic, complemented
11#27889C211yesmaximal, characteristic
12#27877C211yesmaximal, characteristic
13#25672C211yesmaximal, characteristic
14#25600C211yesmaximal, characteristic
15#22270C211yesmaximal, characteristic, complemented
16#22246C211yesmaximal, characteristic, complemented
17#22221C211yesmaximal, characteristic, complemented
18#22218C211yesmaximal, characteristic, complemented
19#20777C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#494C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25#2864—21no—
26#2863—21no—
27#2863C2212yes—
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2855—21no—
33#2855C2212yes—
34#2851—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×#125—21no—
42C4×#125—21no—
43C4×#125—21no—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8—21no—
47C4×C22⋊Q8C2212yes—
48C4×C22⋊Q8C2212yes—
49C4×C22⋊Q8C2212yes—
50#2296—21no—
51#2295—21no—
52#2295—21no—
53#2295C2212yes—
54#2295C2212yes—
55#2285C2211yescharacteristic
56#2283—21no—
57C42×Q8—21no—
58C42×Q8C2212yes—
59C4×#117—21no—
60C2×C43C2211yescharacteristic
61#2088C2214yes—
62#2075C2214yes—
63#2068C2214yes—
64#2066C2214yes—
65#2056C2214yes—
66#2035C2214yes—
67#2035C2214yes—
68#2001C2214yes—
69#1990C2211yescharacteristic
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1956C2211yescharacteristic
74#1956C2211yescharacteristic
75#1956C2211yescharacteristic
76#1956C2211yescharacteristic
77#1928C2214yes—
78#1927C2214yes—
79#1927C2214yes—
80#1927C2214yes—
81#1920C2214yes—
82#1920C2214yes—
83#1919C2214yes—
84#1918C2214yes—
85#1878C2214yes—
86#1878C2214yes—
87#1878C2214yes—
88#1878C2214yes—
89#1865C2214yes—
90#1865C2214yes—
91#1863C2214yes—
92#1863C2214yes—
93#1845C2211yescharacteristic
94#1844C2211yescharacteristic
95#1839C2212yes—
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1831C2212yes—
99#1817C2212yes—
100#1797C2212yes—
101C4×C4⋊C8C2211yescharacteristic
102C4×C4⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105#1744C2211yescharacteristic
106#1744C2211yescharacteristic
107C22×Q16C2314yes—
108#533—24no—
109#533—24no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×#125—21no—
115C2×#125—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×#117—21no—
129C2×#117—21no—
130C22×C42D411yescharacteristic
131C22×C42D411yescharacteristic
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#498—24no—
138#494—24no—
139#487—24no—
140#487—24no—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#106C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#389—21no—
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#389C2312yes—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#385—22no—
203#385—22no—
204#385—22no—
205#385—22no—
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#384C2311yescharacteristic
210#383—22no—
211#382—21no—
212#382—21no—
213#382—21no—
214#382—21no—
215#381—22no—
216#380—21no—
217#380—21no—
218#380—21no—
219#380—21no—
220#380C2312yes—
221#380C2312yes—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247#376—22no—
248#376—22no—
249C43—22no—
250C43D411yescharacteristic
251C43D411yescharacteristic
252C43D411yescharacteristic
253C43D411yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8C2412yes—
275C22×Q8C2412yes—
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2411yescharacteristic
283C2×Q16—28no—
284C2×Q16—44no—
285C2×Q16—44no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8#4114yes—
297C4⋊Q8#4114yes—
298#125—28no—
299#125—44no—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C4×Q8—28no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8—44no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×Q8—44no—
314#117—28no—
315#117—44no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C42—24no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—42no—
366C2×C42—42no—
367C2×C42—42no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
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×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396#107—44no—
397#106—28no—
398#106—44no—
399C4⋊C8—42no—
400C4⋊C8—42no—
401#103—28no—
402#103—28no—
403#103—44no—
404#103—44no—
405#103—44no—
406#103—44no—
407C22⋊C8—42no—
408C22⋊C8—42no—
409C4×C8—22no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8—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#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437C24C22×D411yescharacteristic
438C2×Q8—24no—
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—24no—
443C2×Q8—24no—
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8C2×#4114yes—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4#13514yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
496Q16—88no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
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—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4C2×#4114yes—
543C22⋊C4—28no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C42—28no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—21no—
563C42—21no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—41no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—44no—
575C42—44no—
576C42—44no—
577C42—44no—
578C42—44no—
579C42—44no—
580C42—44no—
581C42#13714yes—
582C42#13714yes—
583C42C2×#4112yes—
584C42C2×#4112yes—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23C2×#13512yes—
592C23C23×D411yescharacteristic
593Q8—28no—
594Q8—48no—
595Q8—44no—
596Q8—44no—
597Q8—44no—
598Q8—84no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—28no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—24no—
613C2×C4—24no—
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—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
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#52512yes—
660C2×C4#52512yes—
661C2×C4C2×#13512yes—
662C2×C4C2×#13512yes—
663C2×C4C2×#13712yes—
664C2×C4C2×#13712yes—
665C2×C4C2×#13712yes—
666C2×C4C2×#13712yes—
667C2×C4C2×#13712yes—
668C2×C4C2×#13712yes—
669C2×C4#57512yes—
670C2×C4C22×#4111yescharacteristic
671C2×C4C22×#4111yescharacteristic
672C2×C4#58512yes—
673C2×C4#58512yes—
674C8—42no—
675C8—42no—
676C8—42no—
677C8—42no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22C2×#52511yescharacteristic
686C22#344911yescharacteristic
687C22#344911yescharacteristic
688C22C22×#13511yescharacteristic
689C22C22×#13711yescharacteristic
690C22C22×#13711yescharacteristic
691C22C2×#57511yescharacteristic
692C4—22no—
693C4—22no—
694C4—22no—
695C4—22no—
696C4—22no—
697C4—41no—
698C4—41no—
699C4—41no—
700C4—41no—
701C4—41no—
702C4—44no—
703C4—41no—
704C4—44no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4#310814yes—
712C4#311614yes—
713C4#352112yes—
714C4#352412yes—
715C2—42no—
716C2#3521111yescharacteristic
717C2C2×#310811yescharacteristic
718C2C2×#311611yescharacteristic
719C2#6094411yescharacteristic
720C2#6094711yescharacteristic
721C2#6096611yescharacteristic
722C2#6097111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33085C2semidirect
5. #29541C2semidirect
7. #29193C2semidirect
8. #29176C2semidirect
9. #28653C2semidirect
10. #28616C2semidirect
15. #22270C2semidirect
16. #22246C2semidirect
17. #22221C2semidirect
18. #22218C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C4×#5332
#330852
#300132
#295412
#294162
#291932
#291762
#286532
#286162
#278892
#278772
#256722
#256002
#222702
#222462
#222212
#222182
#207772