Finite Group Databasebeta

← #4456230

Subgroups — #4456230

Order 512.

2300 conjugacy classes of subgroups.

The subgroup lattice has 651 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2584C211yesmaximal, characteristic, complemented
2C2×#2611C211yesmaximal, characteristic
3C2×#2599C211yesmaximal, characteristic
4C2×#2320C211yesmaximal, characteristic, complemented
5#17070C212yesmaximal, complemented
6#17040C212yesmaximal
7#16993C212yesmaximal, complemented
8#16901C212yesmaximal
9#16679C212yesmaximal
10#16678C212yesmaximal, complemented
11#16564C212yesmaximal
12#16517C212yesmaximal, complemented
13#14967C214yesmaximal, complemented
14#14753C214yesmaximal
15#14289C211yesmaximal, characteristic, complemented
16#13952C211yesmaximal, characteristic, complemented
17#13852C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2611—28no—
22#2599—28no—
23#2584—28no—
24C2×#400C2211yescharacteristic
25C2×#399C2211yescharacteristic
26C2×#398C2211yescharacteristic
27C2×#398C2211yescharacteristic
28C2×#398C2212yes—
29C2×#398C2214yes—
30C2×#397C2211yescharacteristic
31C2×#397C2211yescharacteristic
32C2×#395C2212yes—
33C2×#395C2211yescharacteristic
34C2×#393C2211yescharacteristic
35#2320—28no—
36C2×#391C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#387C2214yes—
45C2×#387C2212yes—
46C2×#387C2212yes—
47C2×#387C2212yes—
48C2×#385C2212yes—
49C2×#385C2212yes—
50C2×#383C2212yes—
51C2×#383C2211yescharacteristic
52C2×#382C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56C2×#376C2211yescharacteristic
57#1445C2212yes—
58#1445C2212yes—
59#1444C2212yes—
60#1444C2214yes—
61#1443C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2212yes—
67#1441C2214yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2212yes—
74#1441C2214yes—
75#1441C2214yes—
76#1439C2212yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2212yes—
80#1439C2214yes—
81#1439C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#125—24no—
90C2×#125—24no—
91C2×#125—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8D418yes—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#400—24no—
130#399—24no—
131#398—28no—
132#398—24no—
133#398—24no—
134#398—48no—
135#397—24no—
136#397—24no—
137#395—28no—
138#395—24no—
139#393—24no—
140#391—28no—
141#389—216no—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—28no—
146#388—28no—
147#388—28no—
148#387—28no—
149#387—28no—
150#387—216no—
151#387—28no—
152#385—28no—
153#385—28no—
154#383—24no—
155#383—28no—
156#382—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4—24no—
159C4×C22⋊C4—44no—
160#376D418yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2312yes—
193C25C2411yescharacteristic
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—44no—
197C22×Q8C2×D414yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—41no—
204C23×C4—41no—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#125—416no—
222C4×Q8—232no—
223#117—416no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
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⋊C4C2×D418yes—
273C2×C4⋊C4C2×D414yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—42no—
305C2×C42C2×D418yes—
306C2×C42C2×D418yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42#4114yes—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95C2×D414yes—
340#95C2×D414yes—
341#95C2×D418yes—
342C24—21no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—24no—
347C24—24no—
348C24—21no—
349C24—21no—
350C242−1+412yes—
351C24C2511yescharacteristic
352C2×Q8—28no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C22×C4—21no—
357C22×C4—24no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4112yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C42−1+418yes—
451C4⋊C4—216no—
452C4⋊C4—28no—
453C4⋊C4—216no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—48no—
459C4⋊C4—416no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C4⋊C4—88no—
469C22⋊C4—48no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—416no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C42—216no—
477C42—48no—
478C42—44no—
479C42—84no—
480C42—84no—
481C42#121116yes—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—28no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23C23×D411yescharacteristic
500C23C23×D411yescharacteristic
501C23C22×#4111yescharacteristic
502C23C2×H(5,𝔽2)11yescharacteristic
503C23C2×2−1+414yes—
504C23C2×2−1+411yescharacteristic
505C23#58514yes—
506C23#58511yescharacteristic
507C23#58511yescharacteristic
508Q8—416no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—416no—
540C2×C4—41no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4C2×#12114yes—
596C2×C4C2×#12114yes—
597C2×C4C2×#12114yes—
598C22—44no—
599C22—44no—
600C22—42no—
601C22—44no—
602C22—82no—
603C22—81no—
604C22—81no—
605C22—82no—
606C22—82no—
607C22C22×#12111yescharacteristic
608C22#344912yes—
609C22#344912yes—
610C22#345512yes—
611C22#348612yes—
612C22#348811yescharacteristic
613C22#348811yescharacteristic
614C22#348911yescharacteristic
615C22#348911yescharacteristic
616C22#349211yescharacteristic
617C22#349311yescharacteristic
618C22#349912yes—
619C22#350912yes—
620C22#354114yes—
621C22#355714yes—
622C22#356314yes—
623C22#356414yes—
624C4—24no—
625C4—24no—
626C4—24no—
627C4—48no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—42no—
632C4—44no—
633C4—82no—
634C4—81no—
635C4—82no—
636C4—81no—
637C4—82no—
638C4—82no—
639C4—84no—
640C2—82no—
641C2#3463311yescharacteristic
642C2#3551811yescharacteristic
643C2#3566711yescharacteristic
644C2#3683412yes—
645C2#3702812yes—
646C2#4194712yes—
647C2#4198812yes—
648C2#4217112yes—
649C2#4230612yes—
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×#2584C2semidirect
4. C2×#2320C2semidirect
5. #17070C2semidirect
7. #16993C2semidirect
10. #16678C2semidirect
12. #16517C2semidirect
13. #14967C2semidirect
15. #14289C2semidirect
16. #13952C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25842
C2×#26112
C2×#25992
C2×#23202
#170702
#170402
#169932
#169012
#166792
#166782
#165642
#165172
#149672
#147532
#142892
#139522
#138522