Finite Group Databasebeta

← #6469722

Subgroups — #6469722

Order 512.

1846 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#29557C214yesmaximal, complemented
4#29212C211yesmaximal, characteristic, complemented
5#29150C211yesmaximal, characteristic, complemented
6#28754C211yesmaximal, characteristic, complemented
7#28451C211yesmaximal, characteristic, complemented
8#27895C212yesmaximal, complemented
9#25603C212yesmaximal, complemented
10#24653C212yesmaximal
11#24652C212yesmaximal, complemented
12#23578C214yesmaximal, complemented
13#21702C212yesmaximal, complemented
14#21655C212yesmaximal, complemented
15#21506C212yesmaximal, complemented
16#21505C212yesmaximal, complemented
17#20463C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2212yes—
20C2×#497C2212yes—
21C2×#489C2212yes—
22C2×#462C2212yes—
23C2×#460C2212yes—
24C2×#441C2214yes—
25C2×#440C2214yes—
26C2×#431C2212yes—
27#2861C2212yes—
28#2860—21no—
29#2860—21no—
30#2859C2211yescharacteristic
31#2858C2211yescharacteristic
32#2857C2211yescharacteristic
33#2856C2211yescharacteristic
34#2854—21no—
35#2854—21no—
36#2850C2212yes—
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C4×#125—21no—
44C4×#125—21no—
45C4×#124C2212yes—
46C4×C22⋊Q8—21no—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8C2212yes—
49C4×C22⋊Q8C2212yes—
50C4×#121—21no—
51C4×#121—21no—
52#2296—21no—
53#2295—21no—
54#2295C2212yes—
55#2293—21no—
56#2293—21no—
57#2285C2211yescharacteristic
58C42×Q8—21no—
59C42×Q8C2212yes—
60C4×#117—21no—
61C2×C43C2211yescharacteristic
62#2059C2214yes—
63#2042C2214yes—
64#2036C2214yes—
65#2034C2214yes—
66#2026C2214yes—
67#2015C2214yes—
68#2001C2214yes—
69#2000C2214yes—
70#1993C2212yes—
71#1982C2212yes—
72#1981C2212yes—
73#1969C2214yes—
74#1968C2214yes—
75#1960C2212yes—
76#1959C2212yes—
77#1959C2212yes—
78#1948C2212yes—
79#1947C2212yes—
80#1934C2214yes—
81#1933C2214yes—
82#1927C2214yes—
83#1925C2214yes—
84#1880C2214yes—
85#1878C2214yes—
86#1847C2212yes—
87#1836C2212yes—
88#1803C2212yes—
89#1803C2212yes—
90#1767C2212yes—
91#1767C2211yescharacteristic
92#1767C2211yescharacteristic
93#1763C2214yes—
94#1762C2214yes—
95#1749C2212yes—
96#1749C2212yes—
97#1744C2212yes—
98C4×#97C2212yes—
99C22×Q16C2314yes—
100C22×SD16C2314yes—
101#533—24no—
102#533—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107#529—48no—
108C2×#126—22no—
109C2×#125—21no—
110C2×#125C2312yes—
111C2×#124C2312yes—
112C2×#124C2312yes—
113C2×#123—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8C2312yes—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121—21no—
119C2×#121C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2312yes—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×#117—21no—
132C2×#117—21no—
133C2×#117—22no—
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#497—28no—
142#489—28no—
143#462—28no—
144#460—28no—
145#441—48no—
146#440—48no—
147#431—28no—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×C4⋊C8C2312yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2314yes—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×#97C2314yes—
164C2×C4×C8C2314yes—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390C2312yes—
180#390C2312yes—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389C2312yes—
196#389C2312yes—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#386—22no—
210#386—22no—
211#385—22no—
212#385—22no—
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#382—21no—
218#382C2312yes—
219#380—21no—
220#380—21no—
221#380—21no—
222#380C2312yes—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4C2312yes—
239C4×C4⋊C4C2312yes—
240C4×C4⋊C4C2312yes—
241C4×C22⋊C4—22no—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—22no—
245#376—22no—
246C43—22no—
247C43D411yescharacteristic
248C43D411yescharacteristic
249C43D411yescharacteristic
250C43D411yescharacteristic
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#336C2312yes—
260#336C2314yes—
261#336C2312yes—
262C22×Q8—21no—
263C22×Q8C2412yes—
264C22×Q8C2412yes—
265C22×D4—22no—
266C22×D4—21no—
267C22×D4—21no—
268C22×D4C2412yes—
269C23×C4—22no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C2×Q16—28no—
278C2×Q16—28no—
279C2×Q16—44no—
280C2×SD16—28no—
281C2×SD16—28no—
282C2×SD16—44no—
283C22×C8C2414yes—
284C22×C8C2414yes—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—24no—
288C4⋊Q8#4114yes—
289C4⋊Q8#4114yes—
290#127—24no—
291#126—48no—
292#125—28no—
293#125—44no—
294#124—24no—
295#124—24no—
296#124—44no—
297#123—48no—
298C22⋊Q8—28no—
299C22⋊Q8—24no—
300C22⋊Q8—24no—
301C22⋊Q8—44no—
302#121—44no—
303#121—44no—
304#121—44no—
305C4×Q8—28no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—28no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×D4—28no—
316C4×D4—44no—
317C4×D4—44no—
318C4×D4—48no—
319#117—28no—
320#117—48no—
321#117—44no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
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×C42—24no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—42no—
376C2×C42—42no—
377C2×C42—42no—
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×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#107—44no—
407#106—44no—
408C4⋊C8—24no—
409C4⋊C8#4118yes—
410#103—28no—
411#103—28no—
412#103—44no—
413#103—44no—
414#103—44no—
415#102—28no—
416#102—28no—
417#102—44no—
418C22⋊C8—28no—
419#97—24no—
420#97—24no—
421#97—28no—
422C4×C8—28no—
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—
437#95—22no—
438#95—22no—
439C24—22no—
440C24C22×D411yescharacteristic
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—24no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8C2×#4114yes—
454C2×D4—22no—
455C2×D4—24no—
456C2×D4—22no—
457C2×D4—24no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—44no—
462C2×D4—44no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
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—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4#13714yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C2511yescharacteristic
513Q16—416no—
514SD16—416no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8C2×#4118yes—
521C4⋊C4—28no—
522C4⋊C4—28no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4C2×#4114yes—
552C22⋊C4—28no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C42—28no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—21no—
584C42—21no—
585C42—22no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—41no—
591C42—41no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—44no—
596C42—44no—
597C42—44no—
598C42C2×#4112yes—
599C42C2×#4112yes—
600C42C2×#4114yes—
601C42H(5,𝔽2)14yes—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23C2×#13712yes—
616C23C23×D411yescharacteristic
617Q8—24no—
618Q8—24no—
619Q8—48no—
620Q8—42no—
621Q8—42no—
622Q8—44no—
623Q8—44no—
624D4—44no—
625D4—44no—
626D4—84no—
627D4—88no—
628D4—84no—
629C2×C4—28no—
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—21no—
645C2×C4—21no—
646C2×C4—28no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—88no—
687C2×C4#52512yes—
688C2×C4#52512yes—
689C2×C4#53212yes—
690C2×C4#53212yes—
691C2×C4#53512yes—
692C2×C4#53512yes—
693C2×C4C2×#13712yes—
694C2×C4C2×#13712yes—
695C2×C4#57512yes—
696C2×C4#57512yes—
697C2×C4#57512yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4112yes—
700C2×C4C22×#4111yescharacteristic
701C2×C4C2×H(5,𝔽2)12yes—
702C8—28no—
703C8—44no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22C2×#52511yescharacteristic
719C22C2×#53211yescharacteristic
720C22C2×#53511yescharacteristic
721C22C22×#13711yescharacteristic
722C22C2×#57511yescharacteristic
723C22C2×#57511yescharacteristic
724C22C2×#57511yescharacteristic
725C4—22no—
726C4—22no—
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—44no—
737C4—41no—
738C4—44no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4#318114yes—
745C4#348112yes—
746C4#348312yes—
747C2—42no—
748C2—162no—
749C2#3497711yescharacteristic
750C2#6055911yescharacteristic
751C2#6056011yescharacteristic
752C2C2×#318111yescharacteristic
753C2#6070211yescharacteristic
754C2#6080211yescharacteristic
755C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. C4×#533C2semidirect
3. #29557C2semidirect
4. #29212C2semidirect
5. #29150C2semidirect
6. #28754C2semidirect
7. #28451C2semidirect
8. #27895C2semidirect
9. #25603C2semidirect
11. #24652C2semidirect
12. #23578C2semidirect
13. #21702C2semidirect
14. #21655C2semidirect
15. #21506C2semidirect
16. #21505C2semidirect
17. #20463C2semidirect

Maximal subgroups

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

subgroupindex
#366312
C4×#5332
#295572
#292122
#291502
#287542
#284512
#278952
#256032
#246532
#246522
#235782
#217022
#216552
#215062
#215052
#204632