Finite Group Databasebeta

← #6657544

Subgroups — #6657544

Order 512.

1616 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34971C211yesmaximal, characteristic
3#32954C212yesmaximal, complemented
4#32582C214yesmaximal, complemented
5#32468C212yesmaximal, complemented
6#32259C212yesmaximal, complemented
7#32008C214yesmaximal, complemented
8#31682C212yesmaximal, complemented
9#31263C212yesmaximal, complemented
10#29941C212yesmaximal, complemented
11#27631C212yesmaximal, complemented
12#24838C212yesmaximal, complemented
13#24710C212yesmaximal, complemented
14#24301C212yesmaximal, complemented
15#21851C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#475C2212yes—
20C2×#475C2212yes—
21C2×#432C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23#2857C2212yes—
24#2857C2212yes—
25#2628—21no—
26#2628—21no—
27#2599—21no—
28#2599—21no—
29#2581C2212yes—
30#2568—21no—
31#2567C2212yes—
32#2563C2212yes—
33#2542—21no—
34#2536C2212yes—
35#2487C2212yes—
36#2487C2212yes—
37#2486C2211yescharacteristic
38#2486C2211yescharacteristic
39#2484—21no—
40#2409—21no—
41#2409—21no—
42#2409C2211yescharacteristic
43#2409C2211yescharacteristic
44#2406—21no—
45#2399—21no—
46#2398—21no—
47#2317—21no—
48#2311—21no—
49C4×C22⋊Q8C2212yes—
50C4×C22⋊Q8C2212yes—
51C4×#123—21no—
52C4×#123—21no—
53#2285C2211yescharacteristic
54#2283C2211yescharacteristic
55#2101C2214yes—
56#2088C2214yes—
57#2085C2214yes—
58#2077C2214yes—
59#2077C2214yes—
60#2076C2214yes—
61#2075C2214yes—
62#2066C2214yes—
63#2058C2214yes—
64#2036C2214yes—
65#2035C2214yes—
66#2024C2214yes—
67#1988C2212yes—
68#1988C2212yes—
69#1985C2212yes—
70#1955C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1943C2214yes—
74#1935C2214yes—
75#1928C2214yes—
76#1919C2214yes—
77#1892C2214yes—
78#1872C2214yes—
79#1869C2212yes—
80#1869C2212yes—
81#1865C2214yes—
82#1865C2214yes—
83#1853C2214yes—
84#1853C2214yes—
85#1807C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2212yes—
89#1797C2212yes—
90#1787C2214yes—
91C22×#41D411yescharacteristic
92C23×Q8—21no—
93C23×Q8C2311yescharacteristic
94C23×C8C2312yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—42no—
98C2×C4⋊Q8—42no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2312yes—
101C2×#123—21no—
102C2×#123—21no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C4×#41—42no—
108C4×#41—42no—
109C2×C4×Q8—21no—
110C2×C4×Q8—21no—
111C2×C4×Q8—42no—
112C2×C4×Q8—42no—
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D411yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4C2311yescharacteristic
121C2×C4×D4C2311yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C2×#117D411yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C42—21no—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#475—44no—
136#475—44no—
137#432—42no—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8—42no—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2312yes—
153C2×#97C2312yes—
154#400—22no—
155#400—22no—
156#400—22no—
157#399—22no—
158#399—22no—
159#398—21no—
160#398—21no—
161#398—22no—
162#397—21no—
163#397—21no—
164#395—22no—
165#395—22no—
166#395—22no—
167#395—21no—
168#395—21no—
169#394—22no—
170#393—22no—
171#393—22no—
172#393—21no—
173#393—21no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391C2312yes—
178#391C2312yes—
179#390—22no—
180#389—22no—
181#389—22no—
182#389C2314yes—
183#389C2314yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#386—22no—
193#386—22no—
194#386—21no—
195#386—21no—
196#385—22no—
197#385—22no—
198#385—22no—
199#385—22no—
200#385—21no—
201#385—21no—
202#384—22no—
203#384C2312yes—
204#383—21no—
205#382—22no—
206#382—21no—
207#382C2312yes—
208#381—21no—
209#381—21no—
210#381—21no—
211#380—22no—
212#380—22no—
213#380—21no—
214#380C2314yes—
215#380C2314yes—
216#380C2312yes—
217#380C2312yes—
218#380C2312yes—
219#380C2311yescharacteristic
220#380C2311yescharacteristic
221C4×C4⋊C4—21no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—22no—
228C4×C22⋊C4—21no—
229#376—21no—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—24no—
241C2×#41—42no—
242C2×#41—42no—
243C22×Q8—21no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—24no—
247C22×Q8—41no—
248C22×Q8—41no—
249C22×Q8—42no—
250C22×Q8—44no—
251C22×Q8—84no—
252C22×Q8C2×D411yescharacteristic
253C22×Q8C2412yes—
254C22×D4C2×D411yescharacteristic
255C22×D4C2×D411yescharacteristic
256C22×D4C2×D411yescharacteristic
257C23×C4—21no—
258C23×C4—21no—
259C23×C4—22no—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D411yescharacteristic
264C23×C4C2×D411yescharacteristic
265C23×C4C2×D411yescharacteristic
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C22×C8—44no—
270C22×C8—42no—
271C22×C8C2414yes—
272C22×C8C2414yes—
273C4⋊Q8—24no—
274C4⋊Q8—24no—
275C4⋊Q8—42no—
276C4⋊Q8—42no—
277C4⋊Q8—44no—
278C4⋊Q8—44no—
279#123—44no—
280#123—44no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—42no—
284C22⋊Q8—42no—
285C22⋊Q8—84no—
286C22⋊Q8—84no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—42no—
290C4×Q8—41no—
291C4×Q8—41no—
292C4×Q8—42no—
293C4×Q8—42no—
294C4×Q8—44no—
295C4×Q8—44no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—42no—
299C4×D4—41no—
300C4×D4—41no—
301C4×D4—41no—
302C4×D4—41no—
303C4×D4—41no—
304C4×D4—41no—
305#117—42no—
306#117—42no—
307#117—42no—
308#117—44no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—41no—
324C2×C4⋊C4—41no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4C2×D411yescharacteristic
350C2×C22⋊C4C2×D411yescharacteristic
351C2×C22⋊C4C2×D411yescharacteristic
352C2×C22⋊C4C2×D411yescharacteristic
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C42—22no—
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—44no—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—44no—
377#106—44no—
378#106—44no—
379#106—44no—
380C4⋊C8—24no—
381C4⋊C8—24no—
382C4⋊C8—42no—
383#103—28no—
384#103—44no—
385#103—44no—
386#103—44no—
387C22⋊C8—44no—
388C22⋊C8—44no—
389C22⋊C8—42no—
390#97—42no—
391#95—21no—
392#95—21no—
393#95—22no—
394#95—22no—
395#95—22no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410C24C22×D411yescharacteristic
411C24C22×D411yescharacteristic
412C24C22×D411yescharacteristic
413#41—416no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—44no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—44no—
426C2×Q8—48no—
427C2×Q8—44no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×Q8—44no—
431C2×Q8—84no—
432C2×Q8—816no—
433C2×Q82−1+414yes—
434C2×D4—22no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4—41no—
441C2×D4—41no—
442C2×D4#13714yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—84no—
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×C4C2511yescharacteristic
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—24no—
501C2×C8—84no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—41no—
517C4⋊C4—41no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C42—21no—
559C42—44no—
560C42—44no—
561C42—44no—
562C42—44no—
563C42H(5,𝔽2)12yes—
564C42H(5,𝔽2)12yes—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—22no—
575C23—22no—
576C23—22no—
577C23C2×#13712yes—
578C23C2×#13712yes—
579C23#57512yes—
580C23C23×D411yescharacteristic
581Q8—24no—
582Q8—24no—
583Q8—44no—
584Q8—48no—
585Q8—42no—
586Q8—42no—
587Q8—88no—
588Q8—88no—
589D4—44no—
590D4—44no—
591D4—44no—
592C2×C4—28no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—48no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—84no—
649C2×C4—88no—
650C2×C4#53412yes—
651C2×C4#53412yes—
652C2×C4C2×#13712yes—
653C2×C4#57714yes—
654C2×C4C2×H(5,𝔽2)11yescharacteristic
655C2×C4C2×H(5,𝔽2)11yescharacteristic
656C2×C4C2×2−1+412yes—
657C2×C4#58512yes—
658C8—44no—
659C8—44no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22C2×#53411yescharacteristic
676C22#348912yes—
677C22C22×#13711yescharacteristic
678C22C2×#57511yescharacteristic
679C22C2×#57712yes—
680C4—22no—
681C4—22no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—44no—
691C4—84no—
692C4—82no—
693C4—82no—
694C4—84no—
695C4—82no—
696C4#320014yes—
697C4#353811yescharacteristic
698C4#354011yescharacteristic
699C4#354112yes—
700C2—42no—
701C2—42no—
702C2—42no—
703C2#3737411yescharacteristic
704C2C2×#320011yescharacteristic
705C2#6075111yescharacteristic
706C2#6212912yes—
707C2#6213112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32954C2semidirect
4. #32582C2semidirect
5. #32468C2semidirect
6. #32259C2semidirect
7. #32008C2semidirect
8. #31682C2semidirect
9. #31263C2semidirect
10. #29941C2semidirect
11. #27631C2semidirect
12. #24838C2semidirect
13. #24710C2semidirect
14. #24301C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#349712
#329542
#325822
#324682
#322592
#320082
#316822
#312632
#299412
#276312
#248382
#247102
#243012
#218512