Finite Group Databasebeta

← #6657876

Subgroups — #6657876

Order 512.

1577 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#34979C211yesmaximal, characteristic
3#33154C212yesmaximal, complemented
4#32694C212yesmaximal, complemented
5#32645C212yesmaximal, complemented
6#32260C212yesmaximal, complemented
7#32029C212yesmaximal, complemented
8#31789C214yesmaximal, complemented
9#31329C214yesmaximal, complemented
10#29688C212yesmaximal, complemented
11#27630C212yesmaximal, complemented
12#24838C212yesmaximal, complemented
13#24710C212yesmaximal, complemented
14#24301C212yesmaximal, complemented
15#21851C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#475C2212yes—
20C2×#475C2212yes—
21C2×#432C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23#2857C2212yes—
24#2857C2212yes—
25#2628—21no—
26#2628—21no—
27#2611—21no—
28#2611—21no—
29#2568—21no—
30#2568C2211yescharacteristic
31#2567C2212yes—
32#2563C2212yes—
33#2547C2211yescharacteristic
34#2546C2211yescharacteristic
35#2542—21no—
36#2542C2211yescharacteristic
37#2499—21no—
38#2487C2212yes—
39#2487C2212yes—
40#2432—21no—
41#2415C2212yes—
42#2415C2212yes—
43#2409—21no—
44#2409—21no—
45#2395—21no—
46#2395—21no—
47#2317—21no—
48#2311—21no—
49C4×C4⋊Q8—21no—
50C4×#124—21no—
51C4×C22⋊Q8C2212yes—
52C4×C22⋊Q8C2212yes—
53#2285C2211yescharacteristic
54C4×#117C2211yescharacteristic
55#2088C2214yes—
56#2077C2214yes—
57#2077C2214yes—
58#2066C2214yes—
59#2058C2214yes—
60#2058C2214yes—
61#2043C2214yes—
62#2036C2214yes—
63#2036C2214yes—
64#2035C2214yes—
65#2035C2214yes—
66#2024C2214yes—
67#1988C2212yes—
68#1988C2212yes—
69#1985C2212yes—
70#1955C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1943C2214yes—
74#1935C2214yes—
75#1927C2214yes—
76#1921C2214yes—
77#1920C2214yes—
78#1918C2214yes—
79#1892C2214yes—
80#1872C2214yes—
81#1869C2212yes—
82#1869C2212yes—
83#1854C2214yes—
84#1853C2214yes—
85#1825C2214yes—
86#1807C2214yes—
87#1803C2212yes—
88#1803C2212yes—
89#1797C2212yes—
90#1797C2212yes—
91C22×#41D411yescharacteristic
92C23×Q8—21no—
93C23×C8C2312yes—
94#533—42no—
95#533—42no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—42no—
100C2×C4⋊Q8—42no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#125—21no—
106C2×#124—21no—
107C2×C22⋊Q8—21no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C22⋊Q8C2312yes—
113C4×#41—42no—
114C4×#41—42no—
115C2×C4×Q8—21no—
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2311yescharacteristic
119C2×C4×D4D411yescharacteristic
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×C4×D4C2311yescharacteristic
124C2×C4×D4C2311yescharacteristic
125C2×#117—21no—
126C2×#117—21no—
127C2×#117D411yescharacteristic
128C2×#117D411yescharacteristic
129C2×#117D411yescharacteristic
130C2×#117C2312yes—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#475—44no—
137#475—44no—
138#432—42no—
139C2×#107C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142C2×#106C2314yes—
143C2×C4⋊C8—42no—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×C4⋊C8C2312yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2312yes—
155#400—22no—
156#400—22no—
157#400—21no—
158#400—21no—
159#399—21no—
160#399—21no—
161#399—22no—
162#399—22no—
163#398—21no—
164#398—21no—
165#397—21no—
166#397—21no—
167#397—22no—
168#397—22no—
169#395—22no—
170#395—22no—
171#394—22no—
172#394—22no—
173#393—22no—
174#393—22no—
175#391—21no—
176#391—22no—
177#391—21no—
178#391—21no—
179#391—22no—
180#391—22no—
181#391C2312yes—
182#391C2312yes—
183#390—22no—
184#390—21no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389—22no—
189#389C2314yes—
190#389C2314yes—
191#389C2312yes—
192#389C2312yes—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#386—22no—
198#386—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—21no—
204#384—22no—
205#384C2312yes—
206#382—22no—
207#382—21no—
208#382—21no—
209#380—22no—
210#380—22no—
211#380—21no—
212#380C2314yes—
213#380C2314yes—
214#380C2312yes—
215#380C2312yes—
216#380C2312yes—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—21no—
228#376—21no—
229C43—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—44no—
247C22×Q8—84no—
248C22×Q8C2×D411yescharacteristic
249C22×Q8C2412yes—
250C22×D4C2×D411yescharacteristic
251C22×D4C2×D411yescharacteristic
252C22×D4C2×D411yescharacteristic
253C23×C4—22no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C22×C8—44no—
264C22×C8—42no—
265C22×C8C2414yes—
266C22×C8C2414yes—
267C4⋊Q8—24no—
268C4⋊Q8—24no—
269C4⋊Q8—42no—
270C4⋊Q8—42no—
271C4⋊Q8—41no—
272C4⋊Q8—41no—
273C4⋊Q8—41no—
274C4⋊Q8—41no—
275C4⋊Q8—44no—
276C4⋊Q8—44no—
277C4⋊Q8—44no—
278#125—44no—
279#124—44no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—42no—
283C22⋊Q8—42no—
284C22⋊Q8—42no—
285C22⋊Q8—42no—
286C22⋊Q8—44no—
287C22⋊Q8—84no—
288C22⋊Q8—84no—
289C4×Q8—24no—
290C4×Q8—24no—
291C4×Q8—42no—
292C4×Q8—41no—
293C4×Q8—41no—
294C4×Q8—44no—
295C4×D4—42no—
296C4×D4—42no—
297C4×D4—42no—
298C4×D4—41no—
299C4×D4—41no—
300C4×D4—41no—
301C4×D4—41no—
302C4×D4—41no—
303C4×D4—41no—
304#117—24no—
305#117—42no—
306#117—42no—
307#117—42no—
308#117—42no—
309#117—44no—
310#117—44no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—42no—
328C2×C4⋊C4—42no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4C2×D411yescharacteristic
351C2×C22⋊C4C2×D411yescharacteristic
352C2×C22⋊C4C2×D411yescharacteristic
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C22⋊C4C2×D411yescharacteristic
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—21no—
366C2×C42—21no—
367C2×C42—22no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—28no—
379#106—44no—
380#106—44no—
381#106—44no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384C4⋊C8—42no—
385#103—44no—
386#103—44no—
387#103—44no—
388#103—44no—
389C22⋊C8—44no—
390C22⋊C8—44no—
391C22⋊C8—42no—
392#97—42no—
393#95—22no—
394#95—21no—
395#95—21no—
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—22no—
421C2×Q8—22no—
422C2×Q8—41no—
423C2×Q8—42no—
424C2×Q8—44no—
425C2×Q8—42no—
426C2×Q8—44no—
427C2×Q8—42no—
428C2×Q8—44no—
429C2×Q8—44no—
430C2×Q8—84no—
431C2×Q8—816no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—41no—
436C2×D4—41no—
437C2×D4—41no—
438C2×D4—41no—
439C2×D4—41no—
440C2×D4#13714yes—
441C22×C4—22no—
442C22×C4—22no—
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—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—44no—
475C22×C4—84no—
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—84no—
495C4⋊C4—22no—
496C4⋊C4—24no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—24no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—41no—
510C4⋊C4—44no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C42−1+414yes—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—44no—
555C42—42no—
556C42—44no—
557C42—42no—
558C42—44no—
559C42—44no—
560C42H(5,𝔽2)12yes—
561C42H(5,𝔽2)12yes—
562C23—21no—
563C23—21no—
564C23—22no—
565C23—22no—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—22no—
570C23—21no—
571C23—22no—
572C23—21no—
573C23—21no—
574C23—22no—
575C23C2×#13712yes—
576C23C2×#13712yes—
577C23C23×D411yescharacteristic
578Q8—44no—
579Q8—42no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—84no—
584Q8—88no—
585Q8—88no—
586D4—44no—
587D4—44no—
588D4—44no—
589C2×C4—28no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
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—22no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
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—44no—
642C2×C4—44no—
643C2×C4—88no—
644C2×C4#53412yes—
645C2×C4#53412yes—
646C2×C4C2×#13712yes—
647C2×C4#57512yes—
648C2×C4#57714yes—
649C2×C4C2×H(5,𝔽2)11yescharacteristic
650C2×C4C2×H(5,𝔽2)11yescharacteristic
651C2×C4C2×2−1+412yes—
652C2×C4#58512yes—
653C8—44no—
654C8—44no—
655C22—22no—
656C22—22no—
657C22—24no—
658C22—24no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—42no—
664C22—42no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—42no—
669C22—42no—
670C22C2×#53411yescharacteristic
671C22#348912yes—
672C22C22×#13711yescharacteristic
673C22C2×#57511yescharacteristic
674C22C2×#57712yes—
675C4—22no—
676C4—22no—
677C4—22no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—44no—
686C4—82no—
687C4—84no—
688C4—82no—
689C4—82no—
690C4—84no—
691C4#320014yes—
692C4#354712yes—
693C4#355011yescharacteristic
694C4#355111yescharacteristic
695C2—42no—
696C2—42no—
697C2—81no—
698C2#3737611yescharacteristic
699C2C2×#320011yescharacteristic
700C2#6075111yescharacteristic
701C2#6213112yes—
702C2#6213412yes—
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. #33154C2semidirect
4. #32694C2semidirect
5. #32645C2semidirect
6. #32260C2semidirect
7. #32029C2semidirect
8. #31789C2semidirect
9. #31329C2semidirect
10. #29688C2semidirect
11. #27630C2semidirect
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
#349792
#331542
#326942
#326452
#322602
#320292
#317892
#313292
#296882
#276302
#248382
#247102
#243012
#218512