Finite Group Databasebeta

← #770150

Subgroups — #770150

Order 512.

3094 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C2×#2502C211yesmaximal, characteristic, complemented
3C22×#397C211yesmaximal, characteristic
4C2×#2329C211yesmaximal, characteristic, complemented
5#15648C214yesmaximal, complemented
6#15621C214yesmaximal, complemented
7#15583C214yesmaximal, complemented
8#15504C214yesmaximal, complemented
9#14935C212yesmaximal, complemented
10#14726C212yesmaximal
11#13992C212yesmaximal, complemented
12#13991C212yesmaximal
13#13924C211yesmaximal, characteristic, complemented
14#13903C211yesmaximal, characteristic
15#13778C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21#2561—28no—
22#2502—28no—
23C2×#398C2214yescomplemented
24C2×#397—28no—
25C2×#397C2214yes—
26C2×#397C2212yes—
27C2×#397C2214yes—
28C2×#396C2214yes—
29C2×#393C2212yes—
30#2329—28no—
31C2×#391C2212yescomplemented
32C2×#391C2212yescomplemented
33C2×#389C2212yescomplemented
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#387C2212yescomplemented
37C2×#387C2212yescomplemented
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#384C2212yescomplemented
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×#379C2212yes—
47C2×#379C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C22×#95C2211yescharacteristic
52C22×#95C2211yescharacteristic
53C22×#95C2211yescharacteristic
54#1445C2214yescomplemented
55#1443C2214yescomplemented
56#1443C2214yescomplemented
57#1442C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1440C2214yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1436C2214yescomplemented
71#1436C2214yescomplemented
72#1436C2214yescomplemented
73C4⋊#95C2212yescomplemented
74C4⋊#95C2212yescomplemented
75#1434C2214yescomplemented
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C26C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8D418yes—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127#398—216no—
128#397—28no—
129#397—216no—
130#397—216no—
131#396—48no—
132#393—28no—
133#391—28no—
134#391—28no—
135#389—28no—
136#388—216no—
137#388—48no—
138#387—28no—
139#387—28no—
140#385—28no—
141#385—28no—
142#385—28no—
143#384—28no—
144#381—28no—
145#381—28no—
146#380—24no—
147#380—28no—
148#379—28no—
149#379—28no—
150C4×C22⋊C4—24no—
151C4×C22⋊C4—28no—
152C4×C22⋊C4—24no—
153C2×#95—24no—
154C2×#95—24no—
155C2×#95D418yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C25—22no—
181C25—24no—
182C25—22no—
183C25—22no—
184C25—24no—
185C25—24no—
186C25—42no—
187C25—42no—
188C25—42no—
189C25C2412yes—
190C25C2411yescharacteristic
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8C2×D414yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4#4114yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C22⋊Q8—232no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D418yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—416no—
299C2×C22⋊C4—416no—
300C2×C22⋊C4—416no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4#4118yes—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—42no—
312C2×C42—42no—
313C2×C42—44no—
314C2×C42—42no—
315C2×C42—42no—
316C2×C42—44no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C2×D418yes—
340#95C2×D418yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—28no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—48no—
357C24—48no—
358C24—416no—
359C24—416no—
360C24—44no—
361C24—44no—
362C24—44no—
363C24—44no—
364C24—44no—
365C24—44no—
366C24—48no—
367C24—48no—
368C24—84no—
369C24—84no—
370C24—88no—
371C24—88no—
372C24C22×D412yes—
373C24C22×D412yes—
374C24C2×#4112yes—
375C24C2×#4114yes—
376C24H(5,𝔽2)14yes—
377C242−1+414yes—
378C24C2511yescharacteristic
379C2×Q8—28no—
380C2×Q8—48no—
381C2×Q8—82no—
382C2×Q8—82no—
383C2×Q8—84no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—48no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—84no—
465C22×C4—84no—
466C22×C4—84no—
467C22×C4—84no—
468C22×C4#121116yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C2×#4118yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C4⋊C4—28no—
481C4⋊C4—216no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—82no—
485C4⋊C4—82no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C22⋊C4—216no—
492C22⋊C4—216no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C42—48no—
504C42—84no—
505C42—84no—
506C42—84no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—28no—
524C23—416no—
525C23—416no—
526C23—416no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—42no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—832no—
546C23—832no—
547C23—88no—
548C23—88no—
549C23—88no—
550C23—88no—
551C23C2×#12114yes—
552C23#53214yes—
553C23#53614yes—
554C23#54818yes—
555C23#55618yes—
556C23C23×D411yescharacteristic
557C23C23×D411yescharacteristic
558C23C22×#4111yescharacteristic
559C23C22×#4112yes—
560C23C2×H(5,𝔽2)12yes—
561C23C2×2−1+412yes—
562C23#58512yes—
563C23#58512yes—
564C23#58512yes—
565Q8—88no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4C2×#12114yes—
639C2×C4C2×#12114yes—
640C22—216no—
641C22—24no—
642C22—216no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—48no—
649C22—44no—
650C22—48no—
651C22—44no—
652C22—44no—
653C22—832no—
654C22—84no—
655C22—84no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22#254518yes—
661C22C22×#12111yescharacteristic
662C22C2×#53211yescharacteristic
663C22C2×#53611yescharacteristic
664C22C2×#54812yes—
665C22C2×#55612yes—
666C22#348411yescharacteristic
667C22#348512yes—
668C22#348511yescharacteristic
669C22#348511yescharacteristic
670C22#348711yescharacteristic
671C22#348812yes—
672C22#349112yes—
673C22#349412yes—
674C22#351314yes—
675C22#352114yes—
676C22#353514yes—
677C22#356414yes—
678C4—24no—
679C4—24no—
680C4—48no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—84no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4—82no—
691C2—28no—
692C2—84no—
693C2C2×#254511yescharacteristic
694C2#3539411yescharacteristic
695C2#3544711yescharacteristic
696C2#3546911yescharacteristic
697C2#3556311yescharacteristic
698C2#3672011yescharacteristic
699C2#3696211yescharacteristic
700C2#4353314yes—
701C2#4359914yes—
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
2. C2×#2502C2semidirect
4. C2×#2329C2semidirect
5. #15648C2semidirect
6. #15621C2semidirect
7. #15583C2semidirect
8. #15504C2semidirect
9. #14935C2semidirect
11. #13992C2semidirect
13. #13924C2semidirect
15. #13778C2semidirect
23. C2×#398C22semidirect
31. C2×#391C22semidirect
32. C2×#391C22semidirect
33. C2×#389C22semidirect
36. C2×#387C22semidirect
37. C2×#387C22semidirect
41. C2×#384C22semidirect
54. #1445C22semidirect
55. #1443C22semidirect
56. #1443C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
70. #1436C22semidirect
71. #1436C22semidirect
72. #1436C22semidirect
73. C4⋊#95C22semidirect
74. C4⋊#95C22semidirect
75. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#25022
C22×#3972
C2×#23292
#156482
#156212
#155832
#155042
#149352
#147262
#139922
#139912
#139242
#139032
#137782