Finite Group Databasebeta

← #767502

Subgroups — #767502

Order 512.

2918 conjugacy classes of subgroups.

The subgroup lattice has 673 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#397C211yesmaximal, characteristic
2C2×#2329C211yesmaximal, characteristic, complemented
3#15652C214yesmaximal, complemented
4#15637C214yesmaximal, complemented
5#15443C214yesmaximal, complemented
6#15350C214yesmaximal, complemented
7#15015C211yesmaximal, characteristic, complemented
8#14914C211yesmaximal, characteristic, complemented
9#14734C211yesmaximal, characteristic
10#14717C211yesmaximal, characteristic
11#14044C212yesmaximal, complemented
12#14039C212yesmaximal, complemented
13#13991C212yesmaximal
14#13971C212yesmaximal
15#13796C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C2×#399C2212yes—
22C2×#397—28no—
23C2×#397C2214yes—
24C2×#397C2214yes—
25C2×#393C2212yes—
26#2329—28no—
27C2×#391C2212yescomplemented
28C2×#389C2212yescomplemented
29C2×#388C2214yes—
30C2×#388C2214yes—
31C2×#387C2212yescomplemented
32C2×#387C2212yescomplemented
33C2×#387C2214yescomplemented
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#380C2211yescharacteristic
38C2×#379C2212yes—
39C2×#379C2212yes—
40C2×C4×C22⋊C4C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2212yescomplemented
46#1445C2212yescomplemented
47#1444C2212yescomplemented
48#1443C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2212yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2214yes—
56#1440C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2212yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67C4⋊#95C2214yescomplemented
68#1434C2214yescomplemented
69#1434C2214yescomplemented
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—28no—
87C22×C4⋊C4—24no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—44no—
102C22×C22⋊C4—44no—
103C22×C22⋊C4—44no—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C42C2314yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119#399—44no—
120#397—216no—
121#397—216no—
122#393—44no—
123#391—28no—
124#389—28no—
125#388—48no—
126#388—48no—
127#387—28no—
128#387—28no—
129#387—48no—
130#385—28no—
131#385—28no—
132#385—28no—
133#380—24no—
134#379—28no—
135#379—28no—
136C4×C22⋊C4—28no—
137C4×C22⋊C4—28no—
138C2×#95—24no—
139C2×#95—24no—
140C2×#95D418yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2312yes—
145C2×#95C2314yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C25—22no—
169C25—24no—
170C25—24no—
171C25—24no—
172C25—22no—
173C25—22no—
174C25—42no—
175C25—42no—
176C25—42no—
177C25C2412yes—
178C25C2411yescharacteristic
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—42no—
182C22×Q8—42no—
183C23×C4—22no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—42no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—42no—
200C23×C4—42no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4C2×D414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C22⋊Q8—232no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—416no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—42no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—416no—
277C2×C22⋊C4—416no—
278C2×C22⋊C4—416no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4#4118yes—
284C2×C22⋊C4#4118yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—28no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42—44no—
292#95—28no—
293#95—24no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—42no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—42no—
316#95—42no—
317#95C2×D418yes—
318#95C2×D418yes—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—28no—
324C24—22no—
325C24—24no—
326C24—24no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—416no—
335C24—416no—
336C24—48no—
337C24—48no—
338C24—44no—
339C24—44no—
340C24—44no—
341C24—44no—
342C24—44no—
343C24—44no—
344C24—48no—
345C24—48no—
346C24—88no—
347C24—88no—
348C24—84no—
349C24—84no—
350C24C22×D412yes—
351C24C2×#4114yes—
352C24C2×#4114yes—
353C24H(5,𝔽2)14yes—
354C242−1+412yes—
355C242−1+412yes—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—82no—
359C2×Q8—82no—
360C2×Q8—88no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—44no—
427C22×C4—44no—
428C22×C4—44no—
429C22×C4—44no—
430C22×C4—416no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—88no—
440C22×C4—88no—
441C22×C4—84no—
442C22×C4—84no—
443C22×C4—88no—
444C22×C4—84no—
445C22×C4—84no—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)14yes—
454C4⋊C4—28no—
455C4⋊C4—216no—
456C4⋊C4—416no—
457C4⋊C4—416no—
458C4⋊C4—84no—
459C4⋊C4—82no—
460C4⋊C4—84no—
461C4⋊C4—82no—
462C4⋊C4—88no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—416no—
468C22⋊C4—416no—
469C22⋊C4—416no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—88no—
474C22⋊C4—88no—
475C42—84no—
476C42—84no—
477C42—88no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—28no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—24no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—28no—
492C23—28no—
493C23—24no—
494C23—24no—
495C23—416no—
496C23—416no—
497C23—416no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—832no—
517C23—832no—
518C23—88no—
519C23—88no—
520C23—88no—
521C23—88no—
522C23#52514yes—
523C23#53214yes—
524C23#53614yes—
525C23#55218yes—
526C23#55618yes—
527C23C23×D411yescharacteristic
528C23C22×#4112yes—
529C23C22×#4112yes—
530C23C2×H(5,𝔽2)12yes—
531C23C2×H(5,𝔽2)12yes—
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+411yescharacteristic
534C23#58512yes—
535C23#58512yes—
536Q8—88no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—28no—
554C2×C4—28no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—416no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—88no—
606C2×C4—88no—
607C2×C4—88no—
608C2×C4—88no—
609C2×C4—88no—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—24no—
614C22—24no—
615C22—24no—
616C22—216no—
617C22—216no—
618C22—44no—
619C22—44no—
620C22—44no—
621C22—48no—
622C22—48no—
623C22—832no—
624C22—82no—
625C22—82no—
626C22—82no—
627C22—82no—
628C22—84no—
629C22—84no—
630C22#260418yes—
631C22C2×#52511yescharacteristic
632C22C2×#53211yescharacteristic
633C22C2×#53611yescharacteristic
634C22C2×#55212yes—
635C22C2×#55612yes—
636C22#348411yescharacteristic
637C22#348411yescharacteristic
638C22#348511yescharacteristic
639C22#348711yescharacteristic
640C22#351314yes—
641C22#351314yes—
642C22#351314yes—
643C22#351414yes—
644C22#353612yes—
645C22#353612yes—
646C22#356412yes—
647C22#356412yes—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—42no—
653C4—42no—
654C4—44no—
655C4—48no—
656C4—82no—
657C4—82no—
658C4—82no—
659C4—82no—
660C4—84no—
661C2—28no—
662C2—84no—
663C2C2×#260411yescharacteristic
664C2#3535611yescharacteristic
665C2#3539411yescharacteristic
666C2#3539611yescharacteristic
667C2#3545111yescharacteristic
668C2#3671811yescharacteristic
669C2#3696211yescharacteristic
670C2#4322514yes—
671C2#4324614yes—
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×#2329C2semidirect
3. #15652C2semidirect
4. #15637C2semidirect
5. #15443C2semidirect
6. #15350C2semidirect
7. #15015C2semidirect
8. #14914C2semidirect
11. #14044C2semidirect
12. #14039C2semidirect
15. #13796C2semidirect
27. C2×#391C22semidirect
28. C2×#389C22semidirect
31. C2×#387C22semidirect
32. C2×#387C22semidirect
33. C2×#387C22semidirect
45. #1446C22semidirect
46. #1445C22semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. C4⋊#95C22semidirect
68. #1434C22semidirect
69. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3972
C2×#23292
#156522
#156372
#154432
#153502
#150152
#149142
#147342
#147172
#140442
#140392
#139912
#139712
#137962