Finite Group Databasebeta

← #726566

Subgroups — #726566

Order 512.

2410 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic, complemented
2C2×#2326C211yesmaximal, characteristic
3#17046C212yesmaximal, complemented
4#17029C212yesmaximal, complemented
5#15960C214yesmaximal, complemented
6#15957C214yesmaximal, complemented
7#15217C212yesmaximal
8#15186C212yesmaximal, complemented
9#15110C212yesmaximal, complemented
10#15079C212yesmaximal, complemented
11#14277C211yesmaximal, characteristic, complemented
12#14141C212yesmaximal
13#13942C211yesmaximal, characteristic, complemented
14#13941C211yesmaximal, characteristic
15#13906C211yesmaximal, characteristic
16#13884C212yesmaximal, complemented
17#13848C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2355—28no—
22#2326—28no—
23C2×#391C2211yescharacteristic
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389C2212yes—
30C2×#388C2212yes—
31C2×#388C2212yes—
32C2×#388C2214yes—
33C2×#387C2214yes—
34C2×#387C2214yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#382C2212yes—
40C2×#382C2211yescharacteristic
41C2×#380C2214yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×#376C2211yescharacteristic
48#1445C2214yes—
49#1444C2211yescharacteristic
50#1444C2211yescharacteristic
51#1444C2212yes—
52#1444C2212yes—
53#1443C2214yes—
54#1443C2214yes—
55#1442C2211yescharacteristic
56#1442C2212yes—
57#1442C2211yescharacteristic
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2212yes—
75#1437C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#124—24no—
89C2×#124—24no—
90C2×#124—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8D418yes—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128#391—24no—
129#391—28no—
130#391—44no—
131#390—28no—
132#390—24no—
133#390—44no—
134#389—44no—
135#388—28no—
136#388—28no—
137#388—48no—
138#387—216no—
139#387—48no—
140#385—28no—
141#385—28no—
142#384—24no—
143#384—24no—
144#382—28no—
145#382—24no—
146#380—48no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152#376—24no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C25C2412yes—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8C2×D414yes—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4#4114yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#124—232no—
221C4×Q8—232no—
222C4×D4—416no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
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—24no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
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×D418yes—
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—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—28no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42C2×D418yes—
304C2×C42C2×D414yes—
305C2×C42#4114yes—
306C2×C42#4114yes—
307C2×C42#4114yes—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—28no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—44no—
329#95—44no—
330#95—42no—
331#95—44no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—24no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24C2×#4114yes—
350C24C2511yescharacteristic
351C2×Q8—28no—
352C2×Q8—28no—
353C2×Q8—28no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×D4—28no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—88no—
445C22×C4—84no—
446C22×C4—84no—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C42−1+414yes—
465C22×C42−1+412yes—
466C4⋊C4—28no—
467C4⋊C4—216no—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C4⋊C4—84no—
480C4⋊C4—88no—
481C22⋊C4—216no—
482C22⋊C4—216no—
483C22⋊C4—416no—
484C22⋊C4—48no—
485C22⋊C4—84no—
486C22⋊C4—84no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C42—216no—
491C42—28no—
492C42—48no—
493C42—44no—
494C42—48no—
495C42—88no—
496C42#123116yes—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—42no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23C23×D411yescharacteristic
521C23C22×#4111yescharacteristic
522C23C22×#4111yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4111yescharacteristic
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)12yes—
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+412yes—
529C23#58512yes—
530C23#58512yes—
531Q8—416no—
532D4—416no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—816no—
621C2×C4C2×#12314yes—
622C2×C4C2×#12314yes—
623C2×C4C2×#12314yes—
624C2×C4#52514yes—
625C2×C4#53714yes—
626C2×C4#55214yes—
627C2×C4#55314yes—
628C2×C4#55514yes—
629C2×C4#55614yes—
630C22—48no—
631C22—48no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—82no—
636C22—84no—
637C22—84no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22C22×#12311yescharacteristic
642C22C2×#52511yescharacteristic
643C22C2×#53711yescharacteristic
644C22#346112yes—
645C22C2×#55211yescharacteristic
646C22C2×#55311yescharacteristic
647C22C2×#55511yescharacteristic
648C22C2×#55611yescharacteristic
649C22#348611yescharacteristic
650C22#348711yescharacteristic
651C22#349212yes—
652C22#349312yes—
653C22#349412yes—
654C22#349811yescharacteristic
655C22#350011yescharacteristic
656C22#350612yes—
657C22#350712yes—
658C22#352012yes—
659C22#352112yes—
660C22#355412yes—
661C22#355412yes—
662C22#355512yes—
663C22#355812yes—
664C4—24no—
665C4—24no—
666C4—44no—
667C4—42no—
668C4—44no—
669C4—42no—
670C4—44no—
671C4—44no—
672C4—81no—
673C4—82no—
674C4—84no—
675C4—81no—
676C4—84no—
677C4—82no—
678C4—82no—
679C4#248518yes—
680C2—84no—
681C2C2×#248511yescharacteristic
682C2#3518211yescharacteristic
683C2#3553111yescharacteristic
684C2#3570111yescharacteristic
685C2#3601911yescharacteristic
686C2#3795212yes—
687C2#3822412yes—
688C2#4279511yescharacteristic
689C2#4288111yescharacteristic
690C2#4515512yes—
691C2#4537012yes—
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×#2355C2semidirect
3. #17046C2semidirect
4. #17029C2semidirect
5. #15960C2semidirect
6. #15957C2semidirect
8. #15186C2semidirect
9. #15110C2semidirect
10. #15079C2semidirect
11. #14277C2semidirect
13. #13942C2semidirect
16. #13884C2semidirect
17. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#23262
#170462
#170292
#159602
#159572
#152172
#151862
#151102
#150792
#142772
#141412
#139422
#139412
#139062
#138842
#138482