Finite Group Databasebeta

← #6485270

Subgroups — #6485270

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#33033C212yesmaximal, complemented
4#33031C212yesmaximal, complemented
5#30114C214yesmaximal, complemented
6#29374C214yesmaximal, complemented
7#29224C211yesmaximal, characteristic, complemented
8#28754C211yesmaximal, characteristic, complemented
9#28522C211yesmaximal, characteristic, complemented
10#28451C211yesmaximal, characteristic, complemented
11#27895C212yesmaximal, complemented
12#25603C212yesmaximal, complemented
13#24155C212yesmaximal, complemented
14#24154C212yesmaximal
15#22273C212yesmaximal, complemented
16#22224C212yesmaximal, complemented
17#20778C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#532C2211yescharacteristic
21C2×#497C2212yes—
22C2×#489C2212yes—
23C2×#477C2212yes—
24C2×#475C2212yes—
25C2×#462C2212yes—
26C2×#460C2212yes—
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#2101C2214yes—
62#2097C2214yes—
63#2042C2214yes—
64#2036C2214yes—
65#2034C2214yes—
66#2015C2214yes—
67#2001C2214yes—
68#2000C2214yes—
69#1993C2212yes—
70#1988C2212yes—
71#1987C2212yes—
72#1982C2212yes—
73#1981C2212yes—
74#1960C2212yes—
75#1959C2212yes—
76#1959C2212yes—
77#1927C2214yes—
78#1927C2214yes—
79#1925C2214yes—
80#1925C2214yes—
81#1880C2214yes—
82#1880C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1878C2214yes—
86#1877C2214yes—
87#1864C2214yes—
88#1863C2214yes—
89#1847C2212yes—
90#1839C2212yes—
91#1836C2212yes—
92#1831C2212yes—
93#1817C2212yes—
94#1797C2212yes—
95#1767C2211yescharacteristic
96#1767C2211yescharacteristic
97#1749C2212yes—
98#1745C2212yes—
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#477—44no—
148#475—44no—
149#462—44no—
150#460—44no—
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—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390—21no—
183#390—21no—
184#390C2312yes—
185#390C2312yes—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389C2312yes—
193#389C2312yes—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#386—22no—
207#386—22no—
208#386—22no—
209#386—22no—
210#385—22no—
211#385—22no—
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#382—21no—
215#382—21no—
216#382—21no—
217#382C2312yes—
218#382C2312yes—
219#380—21no—
220#380C2312yes—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
234C4×C4⋊C4C2312yes—
235C4×C4⋊C4C2312yes—
236C4×C4⋊C4C2312yes—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
241C4×C22⋊C4—22no—
242C4×C22⋊C4—22no—
243#376—22no—
244C43—22no—
245C43D411yescharacteristic
246C43D411yescharacteristic
247C43D411yescharacteristic
248C43D411yescharacteristic
249C43C2311yescharacteristic
250C43C2311yescharacteristic
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253#340C2314yes—
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#336C2314yes—
258#336C2314yes—
259#336C2312yes—
260#336C2312yes—
261#336C2314yes—
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—44no—
284C2×Q16—44no—
285C2×SD16—28no—
286C2×SD16—44no—
287C2×SD16—44no—
288C22×C8C2414yes—
289C22×C8C2414yes—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—24no—
295C4⋊Q8#4114yescomplemented
296C4⋊Q8#4114yescomplemented
297#127—24no—
298#125—28no—
299#125—44no—
300#124—24no—
301#124—24no—
302#124—44no—
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—24no—
317C4×Q8—24no—
318C4×Q8—42no—
319C4×Q8—42no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×D4—44no—
325C4×D4—44no—
326#117—28no—
327#117—44no—
328#117—44no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
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×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—24no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—42no—
378C2×C42—42no—
379C2×C42—42no—
380C2×C42—42no—
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×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409#107—44no—
410#106—44no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413#103—28no—
414#103—44no—
415#103—44no—
416#103—44no—
417#103—44no—
418#102—28no—
419#102—44no—
420#102—44no—
421C22⋊C8—44no—
422#97—24no—
423#97—24no—
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—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—24no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—24no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—42no—
463C2×Q8C2×#4114yes—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—44no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
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—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4#13714yes—
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×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C2511yescharacteristic
523Q16—88no—
524SD16—88no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—24no—
531C4⋊C4—28no—
532C4⋊C4—28no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—44no—
556C4⋊C4—42no—
557C4⋊C4C2×#4114yes—
558C22⋊C4—28no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C42—28no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—21no—
587C42—21no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—41no—
592C42—41no—
593C42—41no—
594C42—41no—
595C42—41no—
596C42—41no—
597C42—41no—
598C42—41no—
599C42—44no—
600C42—44no—
601C42—44no—
602C42—44no—
603C42—44no—
604C42—44no—
605C42C2×#4112yes—
606C42C2×#4112yes—
607C42H(5,𝔽2)14yes—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23C2×#13712yes—
622C23C23×D411yescharacteristic
623Q8—24no—
624Q8—24no—
625Q8—48no—
626Q8—42no—
627Q8—42no—
628Q8—44no—
629Q8—44no—
630Q8—84no—
631Q8—84no—
632D4—44no—
633D4—44no—
634D4—84no—
635D4—88no—
636D4—84no—
637C2×C4—28no—
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—28no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
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—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—88no—
697C2×C4#52512yes—
698C2×C4#52512yes—
699C2×C4#53512yes—
700C2×C4#53512yes—
701C2×C4C2×#13712yes—
702C2×C4C2×#13712yes—
703C2×C4#57512yes—
704C2×C4#57512yes—
705C2×C4#57512yes—
706C2×C4C22×#4111yescharacteristic
707C2×C4C22×#4111yescharacteristic
708C2×C4C2×H(5,𝔽2)12yes—
709C2×C4#58512yes—
710C8—44no—
711C8—44no—
712C22—22no—
713C22—22no—
714C22—22no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—41no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22C2×#52511yescharacteristic
727C22C2×#53511yescharacteristic
728C22#344911yescharacteristic
729C22C22×#13711yescharacteristic
730C22C2×#57511yescharacteristic
731C22C2×#57511yescharacteristic
732C22C2×#57511yescharacteristic
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—44no—
744C4—41no—
745C4—41no—
746C4—44no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4—82no—
751C4—82no—
752C4#321514yes—
753C4#351412yes—
754C4#351712yes—
755C2—42no—
756C2—162no—
757C2#3519011yescharacteristic
758C2#6055911yescharacteristic
759C2#6056011yescharacteristic
760C2C2×#321511yescharacteristic
761C2#6080211yescharacteristic
762C2#6095311yescharacteristic
763C2#6096311yescharacteristic
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. #33033C2semidirect
4. #33031C2semidirect
5. #30114C2semidirect
6. #29374C2semidirect
7. #29224C2semidirect
8. #28754C2semidirect
9. #28522C2semidirect
10. #28451C2semidirect
11. #27895C2semidirect
12. #25603C2semidirect
13. #24155C2semidirect
15. #22273C2semidirect
16. #22224C2semidirect
17. #20778C2semidirect
295. C4⋊Q8#41semidirect
296. C4⋊Q8#41semidirect

Maximal subgroups

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

subgroupindex
#366392
C4×#5332
#330332
#330312
#301142
#293742
#292242
#287542
#285222
#284512
#278952
#256032
#241552
#241542
#222732
#222242
#207782