Finite Group Databasebeta

← #6644836

Subgroups — #6644836

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic
2C2×#2562C211yesmaximal, characteristic
3#33100C212yesmaximal, complemented
4#32265C214yesmaximal, complemented
5#31330C214yesmaximal, complemented
6#31326C214yesmaximal, complemented
7#31267C214yesmaximal, complemented
8#29643C212yesmaximal, complemented
9#27601C211yesmaximal, characteristic, complemented
10#27600C211yesmaximal, characteristic, complemented
11#24552C212yesmaximal, complemented
12#24551C212yesmaximal, complemented
13#24006C211yesmaximal, characteristic, complemented
14#24003C211yesmaximal, characteristic, complemented
15#21738C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#484C2212yes—
20C2×#483C2212yes—
21C22×C22⋊C8C2212yes—
22#2855C2212yes—
23#2848C2212yes—
24#2627—21no—
25#2626—21no—
26#2579—22no—
27#2564—22no—
28#2562—22no—
29#2562—22no—
30#2562C2214yes—
31#2562C2214yes—
32#2487C2212yes—
33#2417C2212yes—
34#2417C2212yes—
35#2409C2212yes—
36C2×#400—21no—
37C2×#394—21no—
38#2315—21no—
39#2308—21no—
40C2×#388—21no—
41C2×#388—21no—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×C4×C4⋊C4C2211yescharacteristic
48#2077C2214yescomplemented
49#2077C2214yescomplemented
50#2076C2214yescomplemented
51#2075C2214yescomplemented
52#2059C2214yescomplemented
53#2059C2214yescomplemented
54#2036C2214yescomplemented
55#2036C2214yescomplemented
56#2035C2214yescomplemented
57#2035C2214yescomplemented
58#2026C2214yescomplemented
59#2026C2214yescomplemented
60#1928C2214yescomplemented
61#1927C2214yescomplemented
62#1919C2214yescomplemented
63#1918C2214yescomplemented
64#1892C2214yescomplemented
65#1892C2214yescomplemented
66#1872C2212yes—
67#1872C2212yes—
68#1869C2214yes—
69#1869C2214yes—
70#1853C2212yes—
71#1853C2212yescomplemented
72#1853C2212yescomplemented
73#1852C2212yes—
74#1852C2212yescomplemented
75#1852C2212yescomplemented
76#1824C2212yes—
77#1823C2212yes—
78#1823C2212yescomplemented
79#1823C2212yescomplemented
80#1817C2212yes—
81#1808C2214yescomplemented
82#1807C2214yescomplemented
83#1787C2212yescomplemented
84#1787C2212yescomplemented
85#1779C2211yescharacteristic
86#1779C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×D4D411yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C23×C8C2314yes—
92C2×#125—22no—
93C2×#125—44no—
94C2×#125C2314yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—44no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C4×D4—44no—
103C2×C4×D4D412yes—
104C2×C4×D4D412yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2312yes—
107C22×C4⋊C4—21no—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4D411yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4D411yescharacteristic
125C22×C22⋊C4D411yescharacteristic
126C22×C42—21no—
127C22×C42C2311yescharacteristic
128#484—44no—
129#483—44no—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×C22⋊C8—44no—
139C2×C22⋊C8C2312yes—
140C2×C22⋊C8C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146#400—24no—
147#400—24no—
148#400—22no—
149#399—22no—
150#398—22no—
151#398—22no—
152#396—22no—
153#396—22no—
154#395—22no—
155#394—24no—
156#394—24no—
157#393—22no—
158#391C2312yes—
159#391C2312yes—
160#389C2312yes—
161#389C2312yes—
162#388—24no—
163#388—24no—
164#388—24no—
165#388—24no—
166#387—22no—
167#387—22no—
168#386—22no—
169#386—22no—
170#386—22no—
171#386—22no—
172#385—22no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#385—22no—
178#381—22no—
179#380—21no—
180#380—21no—
181#380—22no—
182#380—21no—
183#380—21no—
184#380—22no—
185#380—21no—
186#380—21no—
187#380—22no—
188#380—22no—
189#380C2314yes—
190#380C2314yes—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198#380C2312yes—
199#379—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4C2314yes—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95—21no—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#336C2314yes—
217#336C2314yes—
218C25C2×D411yescharacteristic
219C25C2×D411yescharacteristic
220C22×Q8—21no—
221C22×Q8—42no—
222C22×Q8—42no—
223C22×Q8—42no—
224C22×Q8C2412yes—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—42no—
228C22×D4C2×D412yes—
229C22×D4C2×D412yes—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—21no—
233C23×C4—21no—
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—21no—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—42no—
243C23×C4—42no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2×D411yescharacteristic
248C23×C4C2×D411yescharacteristic
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C23×C4C2411yescharacteristic
255C22×C8—44no—
256C22×C8—44no—
257C22×C8—44no—
258C22×C8C2414yes—
259C22×C8C2414yes—
260#125—44no—
261#125—44no—
262#125—48no—
263C22⋊Q8—24no—
264C22⋊Q8—24no—
265C22⋊Q8—42no—
266C22⋊Q8—42no—
267C22⋊Q8—44no—
268C22⋊Q8—44no—
269C22⋊Q8—44no—
270C22⋊Q8—44no—
271C4×D4—44no—
272C4×D4—44no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—42no—
296C2×C4⋊C4—42no—
297C2×C4⋊C4—42no—
298C2×C4⋊C4—42no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D412yes—
304C2×C4⋊C4C2×D412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2412yes—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—22no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4—42no—
327C2×C22⋊C4—42no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—82no—
333C2×C22⋊C4—82no—
334C2×C22⋊C4C2×D412yes—
335C2×C22⋊C4C2×D412yes—
336C2×C22⋊C4C2×D412yes—
337C2×C22⋊C4C2×D412yes—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—82no—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343#107—44no—
344#107—44no—
345#106—44no—
346#106—44no—
347#103—44no—
348#103—44no—
349#103—44no—
350#103—44no—
351C22⋊C8—24no—
352C22⋊C8—24no—
353C22⋊C8—44no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357C22⋊C8—42no—
358#95—24no—
359#95—22no—
360#95—24no—
361#95—24no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—41no—
380C24—41no—
381C24—41no—
382C24—41no—
383C24—41no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24#13514yes—
388C24#13714yes—
389C24C22×D411yescharacteristic
390C24C22×D411yescharacteristic
391C24C22×D412yes—
392C24C22×D411yescharacteristic
393C2×Q8—22no—
394C2×Q8—22no—
395C2×Q8—24no—
396C2×Q8—24no—
397C2×Q8—42no—
398C2×Q8—81no—
399C2×Q8—81no—
400C2×Q8—88no—
401C2×D4—24no—
402C2×D4—24no—
403C2×D4—24no—
404C2×D4—24no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—416no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—42no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4—81no—
468C22×C4—81no—
469C22×C4—81no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4—81no—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4H(5,𝔽2)14yes—
482C22×C4C2511yescharacteristic
483C2×C8—24no—
484C2×C8—24no—
485C2×C8—24no—
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—88no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—44no—
503C4⋊C4—42no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—44no—
509C4⋊C4—82no—
510C4⋊C4—82no—
511C4⋊C4—81no—
512C4⋊C4—81no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C22⋊C4—42no—
516C22⋊C4—42no—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—44no—
525C22⋊C4—48no—
526C22⋊C4—82no—
527C22⋊C4—82no—
528C22⋊C4—84no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4—82no—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—84no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—22no—
542C23—21no—
543C23—21no—
544C23—22no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—21no—
558C23—21no—
559C23—28no—
560C23—28no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—82no—
566C23—82no—
567C23—82no—
568C23—82no—
569C23—82no—
570C23—82no—
571C23—82no—
572C23—82no—
573C23#53412yes—
574C23C2×#13512yes—
575C23C2×#13512yes—
576C23C2×#13512yes—
577C23C2×#13714yes—
578C23C2×#13712yes—
579C23C23×D411yescharacteristic
580Q8—44no—
581Q8—44no—
582Q8—84no—
583D4—48no—
584D4—48no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—82no—
634C2×C4—84no—
635C2×C4—82no—
636C2×C4—82no—
637C2×C4—82no—
638C2×C4—84no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4—84no—
644C2×C4—162no—
645C2×C4—162no—
646C2×C4#53412yes—
647C2×C4C2×H(5,𝔽2)12yes—
648C2×C4#58512yes—
649C2×C4#58512yes—
650C8—44no—
651C8—44no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—24no—
663C22—24no—
664C22—22no—
665C22—22no—
666C22—24no—
667C22—42no—
668C22—42no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—42no—
675C22—41no—
676C22—41no—
677C22—42no—
678C22—41no—
679C22—88no—
680C22—88no—
681C22#319014yes—
682C22C2×#53411yescharacteristic
683C22#348511yescharacteristic
684C22#348511yescharacteristic
685C22C22×#13511yescharacteristic
686C22C22×#13711yescharacteristic
687C22#358511yescharacteristic
688C22#358611yescharacteristic
689C4—22no—
690C4—42no—
691C4—42no—
692C4—41no—
693C4—42no—
694C4—41no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#354514yes—
704C2—24no—
705C2—42no—
706C2—42no—
707C2—44no—
708C2#3711911yescharacteristic
709C2C2×#319011yescharacteristic
710C2#6075511yescharacteristic
711C2#6169611yescharacteristic
712C2#6169911yescharacteristic
713C2#6170511yescharacteristic
714C2#6170811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33100C2semidirect
4. #32265C2semidirect
5. #31330C2semidirect
6. #31326C2semidirect
7. #31267C2semidirect
8. #29643C2semidirect
9. #27601C2semidirect
10. #27600C2semidirect
11. #24552C2semidirect
12. #24551C2semidirect
13. #24006C2semidirect
14. #24003C2semidirect
48. #2077C22semidirect
49. #2077C22semidirect
50. #2076C22semidirect
51. #2075C22semidirect
52. #2059C22semidirect
53. #2059C22semidirect
54. #2036C22semidirect
55. #2036C22semidirect
56. #2035C22semidirect
57. #2035C22semidirect
58. #2026C22semidirect
59. #2026C22semidirect
60. #1928C22semidirect
61. #1927C22semidirect
62. #1919C22semidirect
63. #1918C22semidirect
64. #1892C22semidirect
65. #1892C22semidirect
71. #1853C22semidirect
72. #1853C22semidirect
74. #1852C22semidirect
75. #1852C22semidirect
78. #1823C22semidirect
79. #1823C22semidirect
81. #1808C22semidirect
82. #1807C22semidirect
83. #1787C22semidirect
84. #1787C22semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#25622
#331002
#322652
#313302
#313262
#312672
#296432
#276012
#276002
#245522
#245512
#240062
#240032
#217382