Finite Group Databasebeta

← #4422454

Subgroups — #4422454

Order 512.

2490 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2599C211yesmaximal, characteristic
3C2×#2621C211yesmaximal, characteristic
4C2×#2343C211yesmaximal, characteristic
5#17061C212yesmaximal, complemented
6#16806C212yesmaximal, complemented
7#16565C212yesmaximal, complemented
8#16262C212yesmaximal
9#15874C214yesmaximal, complemented
10#15290C214yesmaximal
11#15150C212yesmaximal
12#15050C212yesmaximal
13#14901C212yesmaximal, complemented
14#14758C212yesmaximal
15#14663C211yesmaximal, characteristic, complemented
16#14356C211yesmaximal, characteristic, complemented
17#14346C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2621—28no—
23#2599—28no—
24#2567—28no—
25C2×#398C2212yes—
26C2×#398C2211yescharacteristic
27C2×#398C2212yes—
28C2×#397C2211yescharacteristic
29C2×#397C2212yes—
30C2×#397C2212yes—
31C2×#397C2214yes—
32C2×#395C2211yescharacteristic
33C2×#395C2212yes—
34C2×#395C2212yes—
35C2×#393C2211yescharacteristic
36#2343—28no—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#389C2214yes—
40C2×#388C2212yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#387C2212yes—
44C2×#387C2212yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#384C2212yes—
48C2×#383C2211yescharacteristic
49C2×#383C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#381C2211yescharacteristic
53C2×#380C2214yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56C2×#376C2211yescharacteristic
57C22×#95C2211yescharacteristic
58#1446C2212yes—
59#1446C2212yes—
60#1446C2212yes—
61#1445C2214yes—
62#1445C2214yes—
63#1445C2212yes—
64#1445C2212yes—
65#1445C2212yes—
66#1444C2214yes—
67#1444C2214yes—
68#1444C2212yes—
69#1443C2214yes—
70#1442C2214yes—
71#1442C2212yes—
72#1442C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2212yes—
76#1441C2212yes—
77#1441C2214yes—
78#1441C2214yes—
79#1441C2214yes—
80#1440C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83#1434C2214yes—
84#1434C2214yes—
85#1434C2212yes—
86C22⋊#95C2214yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—48no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128#398—24no—
129#398—28no—
130#398—28no—
131#397—24no—
132#397—28no—
133#397—28no—
134#397—48no—
135#395—24no—
136#395—28no—
137#395—28no—
138#393—24no—
139#391—28no—
140#391—28no—
141#389—216no—
142#388—28no—
143#387—28no—
144#387—28no—
145#387—28no—
146#387—28no—
147#385—28no—
148#385—28no—
149#384—28no—
150#383—24no—
151#383—24no—
152#382—24no—
153#382—24no—
154#381—24no—
155#380—48no—
156C4×C4⋊C4D418yes—
157C4×C22⋊C4—28no—
158#376—24no—
159C2×#95—24no—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C25C2411yescharacteristic
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—42no—
204C22×Q8—44no—
205C22×Q8—44no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—42no—
213C23×C4—44no—
214C23×C4—42no—
215C23×C4C2×D414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C4⋊Q8—416no—
233C22⋊Q8—432no—
234C4×Q8—416no—
235#117—232no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D418yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4C2×D414yes—
293C2×C22⋊C4C2×D414yes—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—44no—
302C2×C42C2×D414yes—
303C2×C42C2×D418yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42#4114yes—
307#95—22no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95C2×D414yes—
341#95C2×D414yes—
342C24—21no—
343C24—21no—
344C24—22no—
345C24—24no—
346C24—22no—
347C24—22no—
348C24C22×D412yes—
349C24H(5,𝔽2)14yes—
350C24C2511yescharacteristic
351C2×Q8—28no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—44no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—88no—
360C2×Q8—88no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—28no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
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—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—416no—
436C22×C4—48no—
437C22×C4—84no—
438C22×C4—88no—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C42−1+414yes—
454C22×C42−1+418yes—
455C4⋊C4—28no—
456C4⋊C4—28no—
457C4⋊C4—216no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—416no—
462C4⋊C4—48no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C4⋊C4—84no—
469C4⋊C4—88no—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C42—28no—
477C42—48no—
478C42—48no—
479C42—48no—
480C42#121116yes—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—28no—
488C23—28no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—44no—
496C23#53514yes—
497C23C23×D411yescharacteristic
498C23C23×D411yescharacteristic
499C23C22×#4111yescharacteristic
500C23C2×H(5,𝔽2)12yes—
501C23C2×2−1+414yes—
502C23C2×2−1+412yes—
503C23#58512yes—
504C23#58512yes—
505Q8—416no—
506Q8—816no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—416no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—816no—
589C2×C4C2×#12114yes—
590C2×C4C2×#12114yes—
591C2×C4C2×#12114yes—
592C2×C4#53618yes—
593C22—28no—
594C22—24no—
595C22—48no—
596C22—42no—
597C22—44no—
598C22—44no—
599C22—44no—
600C22—42no—
601C22C22×#12111yescharacteristic
602C22C2×#53511yescharacteristic
603C22C2×#53612yes—
604C22#344911yescharacteristic
605C22#348511yescharacteristic
606C22#348611yescharacteristic
607C22#348812yes—
608C22#348912yes—
609C22#349412yes—
610C22#349512yes—
611C22#349912yes—
612C22#350912yes—
613C22#353514yes—
614C22#354414yes—
615C22#355514yes—
616C22#356414yes—
617C4—24no—
618C4—24no—
619C4—28no—
620C4—24no—
621C4—42no—
622C4—44no—
623C4—42no—
624C4—44no—
625C4—42no—
626C4—84no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—84no—
632C4—82no—
633C2—44no—
634C2#3525211yescharacteristic
635C2#3531611yescharacteristic
636C2#3563311yescharacteristic
637C2#4141212yes—
638C2#4229712yes—
639C2#4231512yes—
640C2#4281312yes—
641C2#5148314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2567C2semidirect
5. #17061C2semidirect
6. #16806C2semidirect
7. #16565C2semidirect
9. #15874C2semidirect
13. #14901C2semidirect
15. #14663C2semidirect
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×#25672
C2×#25992
C2×#26212
C2×#23432
#170612
#168062
#165652
#162622
#158742
#152902
#151502
#150502
#149012
#147582
#146632
#143562
#143462