Finite Group Databasebeta

← #6469724

Subgroups — #6469724

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#29555C212yesmaximal, complemented
4#29553C212yesmaximal, complemented
5#29211C211yesmaximal, characteristic, complemented
6#29147C211yesmaximal, characteristic, complemented
7#28753C211yesmaximal, characteristic, complemented
8#28447C211yesmaximal, characteristic, complemented
9#27894C212yesmaximal, complemented
10#25603C212yesmaximal, complemented
11#24655C214yesmaximal, complemented
12#23577C212yesmaximal, complemented
13#23576C212yesmaximal
14#21702C212yesmaximal, complemented
15#21655C212yesmaximal, complemented
16#21508C214yesmaximal, complemented
17#20463C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#532C2211yescharacteristic
21C2×#497C2212yes—
22C2×#489C2212yes—
23C2×#464C2214yes—
24C2×#441C2214yes—
25C2×#440C2214yes—
26C2×#431C2212yes—
27#2862C2211yescharacteristic
28#2862C2211yescharacteristic
29#2861—21no—
30#2861—21no—
31#2858C2211yescharacteristic
32#2858C2211yescharacteristic
33#2850—21no—
34#2850—21no—
35#2850C2212yes—
36#2850C2212yes—
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×#125—21no—
42C4×#125—21no—
43C4×#124—21no—
44C4×#124—21no—
45C4×#124C2212yes—
46C4×#124C2212yes—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2212yes—
51#2297—21no—
52#2297—21no—
53#2296—21no—
54#2295—21no—
55#2295C2212yes—
56#2285C2211yescharacteristic
57C42×Q8—21no—
58C42×Q8C2212yes—
59C4×#117—21no—
60C2×C43C2211yescharacteristic
61#2058C2212yes—
62#2057C2212yes—
63#2043C2212yes—
64#2041C2212yes—
65#2035C2214yes—
66#2033C2214yes—
67#2024C2212yes—
68#2024C2212yes—
69#2016C2212yes—
70#2014C2212yes—
71#2001C2214yes—
72#2000C2214yes—
73#1993C2212yes—
74#1983C2214yes—
75#1969C2214yes—
76#1968C2214yes—
77#1960C2212yes—
78#1959C2212yes—
79#1959C2212yes—
80#1949C2214yes—
81#1934C2214yes—
82#1933C2214yes—
83#1928C2214yes—
84#1926C2214yes—
85#1880C2214yes—
86#1878C2214yes—
87#1847C2212yes—
88#1836C2212yes—
89#1805C2214yes—
90#1767C2212yes—
91#1767C2211yescharacteristic
92#1767C2211yescharacteristic
93#1763C2214yes—
94#1762C2214yes—
95#1749C2212yes—
96#1749C2212yes—
97#1744C2212yes—
98C4×#97C2212yes—
99C22×#41—21no—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102#533—24no—
103#533—24no—
104#533—44no—
105#532—44no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×C4⋊Q8C2311yescharacteristic
111C2×#127C2311yescharacteristic
112C2×#125—21no—
113C2×#125—21no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#124—21no—
117C2×#124C2312yes—
118C2×#124C2312yes—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×#121—21no—
124C2×#121C2312yes—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×Q8C2312yes—
133C2×C4×D4—21no—
134C2×C4×D4C2312yes—
135C2×#117—21no—
136C2×#117—21no—
137C2×#117—21no—
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42D411yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#497—28no—
146#489—28no—
147#464—216no—
148#441—48no—
149#440—48no—
150#431—28no—
151C2×#107C2314yes—
152C2×#106C2314yes—
153C2×C4⋊C8C2312yes—
154C2×C4⋊C8C2312yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#102C2314yes—
161C2×#102C2314yes—
162C2×#102C2314yes—
163C2×C22⋊C8C2314yes—
164C2×#97C2312yes—
165C2×#97C2312yes—
166C2×#97C2314yes—
167C2×C4×C8C2314yes—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391C2312yes—
177#391C2312yes—
178#391C2312yes—
179#391C2312yes—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390—21no—
185#390—21no—
186#390C2312yes—
187#390C2312yes—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389C2312yes—
195#389C2312yes—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#387—22no—
208#386—22no—
209#386—22no—
210#386—22no—
211#386—22no—
212#385—22no—
213#385—22no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#382—21no—
217#382—21no—
218#382—21no—
219#382C2312yes—
220#382C2312yes—
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—22no—
235C4×C4⋊C4—22no—
236C4×C4⋊C4C2312yes—
237C4×C4⋊C4C2312yes—
238C4×C4⋊C4C2312yes—
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
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—
262C2×#41—44no—
263C2×#41—44no—
264C2×#41—44no—
265C22×Q8—21no—
266C22×Q8—21no—
267C22×Q8—21no—
268C22×Q8C2412yes—
269C22×Q8C2412yes—
270C22×D4—22no—
271C22×D4—21no—
272C22×D4—21no—
273C22×D4C2412yes—
274C23×C4—22no—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2411yescharacteristic
282C2×Q16—28no—
283C2×Q16—28no—
284C2×Q16—44no—
285C2×SD16—28no—
286C2×SD16—28no—
287C2×SD16—44no—
288C22×C8C2414yes—
289C22×C8C2414yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—42no—
293C4⋊Q8—42no—
294C4⋊Q8—42no—
295C4⋊Q8#4114yescomplemented
296C4⋊Q8#4114yescomplemented
297#127—42no—
298#125—28no—
299#125—44no—
300#124—28no—
301#124—24no—
302#124—24no—
303#124—44no—
304#124—44no—
305#124—44no—
306C22⋊Q8—28no—
307C22⋊Q8—24no—
308C22⋊Q8—24no—
309C22⋊Q8—44no—
310C22⋊Q8—44no—
311#121—44no—
312#121—44no—
313C4×Q8—28no—
314C4×Q8—24no—
315C4×Q8—24no—
316C4×Q8—28no—
317C4×Q8—28no—
318C4×Q8—28no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×D4—28no—
323C4×D4—44no—
324#117—28no—
325#117—44no—
326#117—44no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—24no—
369C2×C42—24no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—42no—
377C2×C42—42no—
378C2×C42—42no—
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×C42C2×D411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407#107—44no—
408#106—44no—
409C4⋊C8—24no—
410C4⋊C8#4118yescomplemented
411#103—28no—
412#103—28no—
413#103—44no—
414#103—44no—
415#103—44no—
416#102—28no—
417#102—28no—
418#102—44no—
419C22⋊C8—28no—
420#97—24no—
421#97—24no—
422#97—28no—
423C4×C8—28no—
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—
439#95—22no—
440#95—22no—
441#95—22no—
442C24—22no—
443C24C22×D411yescharacteristic
444#41—816no—
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—24no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8C2×#4114yes—
462C2×D4—22no—
463C2×D4—24no—
464C2×D4—22no—
465C2×D4—24no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4—42no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C2511yescharacteristic
521Q16—416no—
522SD16—416no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8C2×#4118yes—
529C4⋊C4—28no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4C2×#4114yes—
554C22⋊C4—28no—
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—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C42—28no—
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—21no—
583C42—21no—
584C42—22no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—41no—
591C42—41no—
592C42—41no—
593C42—44no—
594C42—44no—
595C42—44no—
596C42—44no—
597C42—44no—
598C42—44no—
599C42#13714yes—
600C42C2×#4112yes—
601C42C2×#4112yes—
602C42C2×#4114yes—
603C42H(5,𝔽2)14yes—
604C23—21no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23#57512yes—
618C23C23×D411yescharacteristic
619Q8—28no—
620Q8—48no—
621Q8—44no—
622Q8—44no—
623Q8—44no—
624Q8—84no—
625Q8—84no—
626D4—44no—
627D4—44no—
628D4—84no—
629D4—84no—
630D4—88no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—28no—
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—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—24no—
659C2×C4—24no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
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—42no—
687C2×C4—42no—
688C2×C4—88no—
689C2×C4#52512yes—
690C2×C4#52512yes—
691C2×C4#53212yes—
692C2×C4#53212yes—
693C2×C4#53512yes—
694C2×C4#53512yes—
695C2×C4C2×#13712yes—
696C2×C4C2×#13712yes—
697C2×C4C2×#13712yes—
698C2×C4#57512yes—
699C2×C4#57512yes—
700C2×C4C22×#4111yescharacteristic
701C2×C4C22×#4112yes—
702C2×C4C22×#4111yescharacteristic
703C2×C4C2×H(5,𝔽2)12yes—
704C8—28no—
705C8—44no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C2×#52511yescharacteristic
721C22C2×#53211yescharacteristic
722C22C2×#53511yescharacteristic
723C22C22×#13711yescharacteristic
724C22C2×#57511yescharacteristic
725C22C2×#57511yescharacteristic
726C22C2×#57511yescharacteristic
727C4—22no—
728C4—22no—
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—41no—
739C4—44no—
740C4—41no—
741C4—44no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4#348112yes—
748C4#348312yes—
749C2—42no—
750C2—162no—
751C2#3497711yescharacteristic
752C2#6055911yescharacteristic
753C2#6056011yescharacteristic
754C2#6068811yescharacteristic
755C2#6070111yescharacteristic
756C2#6079911yescharacteristic
757C2#6080411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #29555C2semidirect
4. #29553C2semidirect
5. #29211C2semidirect
6. #29147C2semidirect
7. #28753C2semidirect
8. #28447C2semidirect
9. #27894C2semidirect
10. #25603C2semidirect
11. #24655C2semidirect
12. #23577C2semidirect
14. #21702C2semidirect
15. #21655C2semidirect
16. #21508C2semidirect
17. #20463C2semidirect
295. C4⋊Q8#41semidirect
296. C4⋊Q8#41semidirect
410. C4⋊C8#41semidirect

Maximal subgroups

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

subgroupindex
#366392
C4×#5332
#295552
#295532
#292112
#291472
#287532
#284472
#278942
#256032
#246552
#235772
#235762
#217022
#216552
#215082
#204632