Finite Group Databasebeta

← #4422103

Subgroups — #4422103

Order 512.

2328 conjugacy classes of subgroups.

The subgroup lattice has 667 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2592C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic, complemented
3C2×#2632C211yesmaximal, characteristic
4C2×#2356C211yesmaximal, characteristic, complemented
5#17023C212yesmaximal, complemented
6#16901C212yesmaximal, complemented
7#16679C212yesmaximal, complemented
8#16665C212yesmaximal, complemented
9#15902C214yesmaximal
10#15867C214yesmaximal, complemented
11#15175C212yesmaximal, complemented
12#15103C212yesmaximal, complemented
13#14902C212yesmaximal, complemented
14#14894C212yesmaximal, complemented
15#14662C211yesmaximal, characteristic
16#14356C211yesmaximal, characteristic, complemented
17#14346C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2632—28no—
22#2592—28no—
23#2581—28no—
24C2×#400C2212yes—
25C2×#399C2211yescharacteristic
26C2×#399C2211yescharacteristic
27C2×#397C2211yescharacteristic
28C2×#397C2211yescharacteristic
29C2×#395C2212yes—
30C2×#395C2212yes—
31C2×#394C2212yes—
32C2×#393C2211yescharacteristic
33C2×#393C2211yescharacteristic
34C2×#393C2211yescharacteristic
35C2×#393C2211yescharacteristic
36#2356—28no—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#388C2212yes—
45C2×#388C2212yes—
46C2×#387C2214yes—
47C2×#387C2214yes—
48C2×#383C2211yescharacteristic
49C2×#383C2212yes—
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×#376C2212yes—
55#1446C2212yes—
56#1446C2212yes—
57#1445C2212yes—
58#1444C2212yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2212yes—
79#1438C2212yes—
80#1437C2214yes—
81#1437C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C2×#124—24no—
92C2×#124—24no—
93C2×#124—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8D418yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
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⋊C4C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#400—28no—
131#399—24no—
132#399—24no—
133#397—24no—
134#397—24no—
135#395—28no—
136#395—28no—
137#394—28no—
138#393—24no—
139#393—24no—
140#393—24no—
141#393—24no—
142#391—28no—
143#391—28no—
144#391—28no—
145#390—28no—
146#389—28no—
147#388—28no—
148#388—28no—
149#388—28no—
150#388—28no—
151#387—216no—
152#387—216no—
153#383—24no—
154#383—28no—
155#382—24no—
156#382—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4D418yes—
159#376—28no—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C25C2411yescharacteristic
192C25C2411yescharacteristic
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—44no—
206C22×Q8C2×D414yes—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—44no—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226#124—416no—
227C4×Q8—232no—
228C4×Q8—416no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4C2×D418yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4C2×D418yes—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C2×D418yes—
305C2×C42C2×D418yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42#4114yes—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—24no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95C2×D414yes—
339#95C2×D414yes—
340C24—21no—
341C24—21no—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—44no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—48no—
366C2×Q8—48no—
367C2×Q8—44no—
368C2×Q8—44no—
369C2×Q8—84no—
370C2×Q8—84no—
371C2×D4—48no—
372C2×D4—44no—
373C2×D4—84no—
374C2×D4—84no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
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—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
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—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4C22×D414yes—
442C22×C4C22×D414yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C42−1+414yes—
455C4⋊C4—28no—
456C4⋊C4—216no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—416no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4—88no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C22⋊C4—48no—
469C22⋊C4—48no—
470C22⋊C4—44no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—84no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C22⋊C4—84no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C42—216no—
481C42—48no—
482C42—416no—
483C42—48no—
484C42#121116yes—
485C23—41no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—41no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C22×#4111yescharacteristic
520C23C2×2−1+412yes—
521C23#58512yes—
522C23#58512yes—
523C23#58514yes—
524C23#58512yes—
525Q8—416no—
526Q8—416no—
527Q8—88no—
528D4—88no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
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—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
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—44no—
606C2×C4C2×#12114yes—
607C2×C4C2×#12114yes—
608C2×C4C2×#12114yes—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—82no—
619C22—82no—
620C22—82no—
621C22—84no—
622C22—84no—
623C22—81no—
624C22—81no—
625C22C22×#12111yescharacteristic
626C22#344912yes—
627C22#344912yes—
628C22#348511yescharacteristic
629C22#348611yescharacteristic
630C22#348711yescharacteristic
631C22#348711yescharacteristic
632C22#348912yes—
633C22#349312yes—
634C22#349512yes—
635C22#350012yes—
636C22#351012yes—
637C22#355714yes—
638C22#356514yes—
639C22#356614yes—
640C22#356814yes—
641C4—24no—
642C4—24no—
643C4—24no—
644C4—44no—
645C4—44no—
646C4—44no—
647C4—44no—
648C4—44no—
649C4—82no—
650C4—82no—
651C4—84no—
652C4—82no—
653C4—84no—
654C4—81no—
655C4—81no—
656C2—161no—
657C2—161no—
658C2#3515011yescharacteristic
659C2#3532711yescharacteristic
660C2#3566111yescharacteristic
661C2#4142112yes—
662C2#4215312yes—
663C2#4215412yes—
664C2#4217012yes—
665C2#5143014yes—
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
2. C2×#2581C2semidirect
4. C2×#2356C2semidirect
5. #17023C2semidirect
6. #16901C2semidirect
7. #16679C2semidirect
8. #16665C2semidirect
10. #15867C2semidirect
11. #15175C2semidirect
12. #15103C2semidirect
13. #14902C2semidirect
14. #14894C2semidirect
16. #14356C2semidirect
17. #14346C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25922
C2×#25812
C2×#26322
C2×#23562
#170232
#169012
#166792
#166652
#159022
#158672
#151752
#151032
#149022
#148942
#146622
#143562
#143462