Finite Group Databasebeta

← #4416323

Subgroups — #4416323

Order 512.

2393 conjugacy classes of subgroups.

The subgroup lattice has 708 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2853C211yesmaximal, characteristic
3#17067C211yesmaximal, characteristic, complemented
4#17066C211yesmaximal, characteristic, complemented
5#16895C212yesmaximal
6#16892C212yesmaximal, complemented
7#16699C212yesmaximal, complemented
8#16560C211yesmaximal, characteristic, complemented
9#16511C211yesmaximal, characteristic, complemented
10#16360C212yesmaximal
11#15012C212yesmaximal, complemented
12#14972C212yesmaximal, complemented
13#14910C212yesmaximal, complemented
14#14908C212yesmaximal, complemented
15#14765C212yesmaximal
16#14763C212yesmaximal, complemented
17#14626C212yesmaximal, complemented
18#14625C212yesmaximal, complemented
19#13753C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×#123C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C23×C42C2211yescharacteristic
24#2857—28no—
25#2853—28no—
26C2×#400C2212yes—
27C2×#399C2212yes—
28C2×#395C2212yes—
29C2×#393C2212yes—
30C2×#391C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#388C2211yescharacteristic
35C2×#388C2211yescharacteristic
36C2×#388C2211yescharacteristic
37C2×#388C2211yescharacteristic
38C2×#388C2214yes—
39C2×#387C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#387C2211yescharacteristic
43C2×#386C2211yescharacteristic
44C2×#386C2211yescharacteristic
45C2×#385C2211yescharacteristic
46C2×#385C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#381C2211yescharacteristic
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×#376C2212yes—
53#1446C2212yes—
54#1446C2212yes—
55#1445C2214yes—
56#1445C2212yes—
57#1445C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2212yes—
62#1443C2214yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2212yes—
67#1442C2212yes—
68#1442C2212yes—
69#1442C2212yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2212yes—
73#1441C2212yes—
74#1441C2212yes—
75#1440C2214yes—
76#1440C2214yes—
77#1439C2214yes—
78#1439C2212yes—
79#1439C2214yes—
80#1439C2212yes—
81#1439C2212yes—
82#1439C2212yes—
83#1439C2214yes—
84#1438C2214yes—
85#1436C2214yes—
86#1436C2214yes—
87#1434C2212yes—
88#1434C2211yescharacteristic
89#1434C2211yescharacteristic
90C22⋊#95C2212yes—
91C22⋊#95C2212yes—
92C4×#95C2212yes—
93C23×Q8C2312yes—
94C23×D4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C2×C4⋊Q8—24no—
99C2×C4⋊Q8—44no—
100C2×#123—24no—
101C2×#123—44no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—48no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2314yes—
131C22×C42—44no—
132C22×C42D418yes—
133C22×C42C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#400—44no—
139#399—44no—
140#395—28no—
141#393—28no—
142#391—28no—
143#389—28no—
144#389—28no—
145#389—48no—
146#388—24no—
147#388—24no—
148#388—24no—
149#388—24no—
150#388—48no—
151#387—24no—
152#387—24no—
153#387—24no—
154#387—24no—
155#386—24no—
156#386—24no—
157#385—24no—
158#385—24no—
159#384—24no—
160#381—24no—
161#380—28no—
162#380—28no—
163C4×C22⋊C4—28no—
164#376—44no—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C25C2411yescharacteristic
200C25C2411yescharacteristic
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×D4—22no—
207C22×D4—24no—
208C22×D4—42no—
209C22×D4—42no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2414yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C4⋊Q8—416no—
241#123—416no—
242C22⋊Q8—432no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—28no—
311C2×C42—416no—
312C2×C42—44no—
313C2×C42—42no—
314C2×C42—42no—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319#95—28no—
320#95—28no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—44no—
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#95—44no—
351#95—44no—
352#95#4118yes—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—21no—
365C24—21no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24C22×D412yes—
370C242−1+414yes—
371C24C2511yescharacteristic
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—88no—
376C2×Q8—88no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—88no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—28no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
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—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C2×#4118yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)14yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C42−1+412yes—
478C22×C42−1+412yes—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—416no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—416no—
491C4⋊C4—44no—
492C4⋊C4—88no—
493C22⋊C4—416no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—416no—
498C22⋊C4—44no—
499C22⋊C4—416no—
500C22⋊C4—44no—
501C22⋊C4—44no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—88no—
506C42—28no—
507C42—28no—
508C42—44no—
509C42—44no—
510C42—416no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—28no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—44no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23#53614yes—
545C23C23×D411yescharacteristic
546C23C22×#4112yes—
547C23C2×H(5,𝔽2)12yes—
548C23C2×H(5,𝔽2)12yes—
549C23C2×2−1+412yes—
550C23C2×2−1+411yescharacteristic
551C23C2×2−1+411yescharacteristic
552C23#58512yes—
553C23#58512yes—
554Q8—816no—
555D4—88no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—416no—
579C2×C4—416no—
580C2×C4—416no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
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—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4#53214yes—
641C2×C4#53414yes—
642C2×C4#53514yes—
643C2×C4#53614yes—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—48no—
652C22—48no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—84no—
659C22—84no—
660C22—81no—
661C22—81no—
662C22C2×#53211yescharacteristic
663C22C2×#53411yescharacteristic
664C22C2×#53511yescharacteristic
665C22C2×#53611yescharacteristic
666C22C2×#53611yescharacteristic
667C22#344911yescharacteristic
668C22#344911yescharacteristic
669C22#349514yes—
670C22#351512yes—
671C22#351812yes—
672C22#351914yes—
673C22#352914yes—
674C22#353514yes—
675C22#354112yes—
676C22#354112yes—
677C22#355212yes—
678C22#355212yes—
679C4—24no—
680C4—24no—
681C4—24no—
682C4—24no—
683C4—42no—
684C4—42no—
685C4—48no—
686C4—81no—
687C4—84no—
688C4—81no—
689C4—81no—
690C4—84no—
691C4—81no—
692C4—81no—
693C4—84no—
694C4—81no—
695C4—81no—
696C2—44no—
697C2—161no—
698C2#3507811yescharacteristic
699C2#3520111yescharacteristic
700C2#3532611yescharacteristic
701C2#3668711yescharacteristic
702C2#3743811yescharacteristic
703C2#3744111yescharacteristic
704C2#3761711yescharacteristic
705C2#4602714yes—
706C2#4602914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2857C2semidirect
3. #17067C2semidirect
4. #17066C2semidirect
6. #16892C2semidirect
7. #16699C2semidirect
8. #16560C2semidirect
9. #16511C2semidirect
11. #15012C2semidirect
12. #14972C2semidirect
13. #14910C2semidirect
14. #14908C2semidirect
16. #14763C2semidirect
17. #14626C2semidirect
18. #14625C2semidirect
19. #13753C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#28532
#170672
#170662
#168952
#168922
#166992
#165602
#165112
#163602
#150122
#149722
#149102
#149082
#147652
#147632
#146262
#146252
#137532