Finite Group Databasebeta

← #4421760

Subgroups — #4421760

Order 512.

2702 conjugacy classes of subgroups.

The subgroup lattice has 735 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C2×#2482C211yesmaximal, characteristic
3C2×#2625C211yesmaximal, characteristic, complemented
4C2×#2346C211yesmaximal, characteristic, complemented
5#17040C212yesmaximal, complemented
6#16821C212yesmaximal, complemented
7#16566C212yesmaximal, complemented
8#16518C212yesmaximal, complemented
9#15560C214yesmaximal, complemented
10#15297C214yesmaximal
11#15175C212yesmaximal, complemented
12#15103C212yesmaximal, complemented
13#14902C212yesmaximal, complemented
14#14894C212yesmaximal, complemented
15#14660C211yesmaximal, characteristic, complemented
16#14355C211yesmaximal, characteristic, complemented
17#14338C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21#2625—28no—
22#2561—28no—
23#2482—28no—
24C2×#400C2212yescomplemented
25C2×#399C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#398C2212yescomplemented
28C2×#397C2211yescharacteristic
29C2×#397C2211yescharacteristic
30C2×#397C2214yes—
31C2×#397C2211yescharacteristic
32C2×#397C2211yescharacteristic
33C2×#396C2214yes—
34C2×#393C2211yescharacteristic
35C2×#393C2211yescharacteristic
36#2346—28no—
37C2×#391C2212yescomplemented
38C2×#389C2212yescomplemented
39C2×#389C2212yescomplemented
40C2×#388C2212yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#388C2214yes—
45C2×#385C2212yes—
46C2×#382C2212yescomplemented
47C2×#381C2211yescharacteristic
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#379C2212yes—
52C2×#379C2212yes—
53C2×#379C2212yes—
54C2×C4×C4⋊C4C2212yescomplemented
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×#376C2212yescomplemented
57#1446C2212yescomplemented
58#1445C2212yescomplemented
59#1444C2212yescomplemented
60#1443C2214yescomplemented
61#1442C2212yescomplemented
62#1442C2214yescomplemented
63#1442C2214yescomplemented
64#1441C2212yescomplemented
65#1441C2212yescomplemented
66#1441C2214yescomplemented
67#1441C2214yescomplemented
68#1441C2214yescomplemented
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2214yes—
80#1437C2214yes—
81#1437C2214yes—
82#1436C2214yescomplemented
83#1436C2214yescomplemented
84C4⋊#95C2212yescomplemented
85#1434C2212yescomplemented
86C22⋊#95C2212yes—
87C26C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117D418yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134#400—28no—
135#399—24no—
136#399—24no—
137#398—28no—
138#397—24no—
139#397—24no—
140#397—216no—
141#397—24no—
142#397—24no—
143#396—48no—
144#393—24no—
145#393—24no—
146#391—28no—
147#389—28no—
148#389—28no—
149#388—216no—
150#388—28no—
151#388—28no—
152#388—28no—
153#388—28no—
154#385—28no—
155#382—28no—
156#381—24no—
157#381—28no—
158#380—24no—
159#380—24no—
160#379—28no—
161#379—28no—
162#379—44no—
163C4×C4⋊C4—28no—
164C4×C22⋊C4D418yes—
165#376—28no—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2311yescharacteristic
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C25—22no—
194C25—22no—
195C25—22no—
196C25—22no—
197C25—22no—
198C25—41no—
199C25—41no—
200C25—44no—
201C25—44no—
202C25C2411yescharacteristic
203C25C2411yescharacteristic
204C25C2411yescharacteristic
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—44no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4#4114yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C22⋊Q8—416no—
232#117—232no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—416no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4C2×D418yes—
316C2×C22⋊C4C2×D418yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42C2×D418yes—
324C2×C42C2×D418yes—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95C2×D414yes—
351#95C2×D414yes—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—24no—
357C24—24no—
358C24—28no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—44no—
376C24—44no—
377C24—44no—
378C24—44no—
379C24—44no—
380C24—48no—
381C24—48no—
382C24—44no—
383C24—84no—
384C24—84no—
385C24—84no—
386C24—84no—
387C24—88no—
388C24—88no—
389C24—88no—
390C24—88no—
391C24C22×D412yes—
392C24C22×D412yes—
393C24C2×#4112yes—
394C24H(5,𝔽2)12yes—
395C242−1+412yes—
396C24C2511yescharacteristic
397C2×Q8—48no—
398C2×Q8—44no—
399C2×Q8—84no—
400C2×Q8—84no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—84no—
471C22×C4—84no—
472C22×C4#121116yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C4⋊C4—28no—
484C4⋊C4—28no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4—84no—
492C4⋊C4—84no—
493C4⋊C4—88no—
494C22⋊C4—216no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—84no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—216no—
515C42—48no—
516C42—416no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—28no—
536C23—28no—
537C23—416no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—42no—
555C23—42no—
556C23—41no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—88no—
569C23—88no—
570C23—816no—
571C23—816no—
572C23—816no—
573C23—816no—
574C23—816no—
575C23—816no—
576C23C2×#12114yes—
577C23C23×D411yescharacteristic
578C23C23×D411yescharacteristic
579C23C22×#4111yescharacteristic
580C23C2×H(5,𝔽2)11yescharacteristic
581C23C2×2−1+411yescharacteristic
582C23#58514yes—
583C23#58514yes—
584C23#58511yescharacteristic
585C23#58511yescharacteristic
586Q8—88no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—416no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—88no—
661C2×C4C2×#12114yes—
662C2×C4C2×#12114yes—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—42no—
678C22—42no—
679C22—44no—
680C22—832no—
681C22—82no—
682C22—82no—
683C22—81no—
684C22—81no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—81no—
690C22—81no—
691C22C22×#12111yescharacteristic
692C22#344912yes—
693C22#344912yes—
694C22#348512yes—
695C22#348811yescharacteristic
696C22#348811yescharacteristic
697C22#348911yescharacteristic
698C22#348911yescharacteristic
699C22#349112yes—
700C22#349211yescharacteristic
701C22#349311yescharacteristic
702C22#350012yes—
703C22#351012yes—
704C22#354514yes—
705C22#356414yes—
706C22#356614yes—
707C22#356614yes—
708C4—24no—
709C4—24no—
710C4—44no—
711C4—44no—
712C4—44no—
713C4—44no—
714C4—82no—
715C4—82no—
716C4—84no—
717C4—82no—
718C4—82no—
719C4—84no—
720C4—81no—
721C4—81no—
722C2—44no—
723C2—82no—
724C2—82no—
725C2#3508911yescharacteristic
726C2#3532711yescharacteristic
727C2#3551811yescharacteristic
728C2#4142312yes—
729C2#4217312yes—
730C2#4239012yes—
731C2#4239212yes—
732C2#4879712yes—
733C2#5018912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2625C2semidirect
4. C2×#2346C2semidirect
5. #17040C2semidirect
6. #16821C2semidirect
7. #16566C2semidirect
8. #16518C2semidirect
9. #15560C2semidirect
11. #15175C2semidirect
12. #15103C2semidirect
13. #14902C2semidirect
14. #14894C2semidirect
15. #14660C2semidirect
16. #14355C2semidirect
24. C2×#400C22semidirect
27. C2×#398C22semidirect
37. C2×#391C22semidirect
38. C2×#389C22semidirect
39. C2×#389C22semidirect
46. C2×#382C22semidirect
54. C2×C4×C4⋊C4C22semidirect
56. C2×#376C22semidirect
57. #1446C22semidirect
58. #1445C22semidirect
59. #1444C22semidirect
60. #1443C22semidirect
61. #1442C22semidirect
62. #1442C22semidirect
63. #1442C22semidirect
64. #1441C22semidirect
65. #1441C22semidirect
66. #1441C22semidirect
67. #1441C22semidirect
68. #1441C22semidirect
82. #1436C22semidirect
83. #1436C22semidirect
84. C4⋊#95C22semidirect
85. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#24822
C2×#26252
C2×#23462
#170402
#168212
#165662
#165182
#155602
#152972
#151752
#151032
#149022
#148942
#146602
#143552
#143382