Finite Group Databasebeta

← #735421

Subgroups — #735421

Order 512.

2944 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic, complemented
2C2×#2501C211yesmaximal, characteristic
3C2×#2639C211yesmaximal, characteristic
4C2×#2368C211yesmaximal, characteristic, complemented
5#17063C212yesmaximal, complemented
6#16802C212yesmaximal, complemented
7#15739C214yesmaximal, complemented
8#15499C214yesmaximal, complemented
9#15152C212yesmaximal, complemented
10#15107C212yesmaximal, complemented
11#14912C214yesmaximal, complemented
12#14152C211yesmaximal, characteristic
13#13996C214yesmaximal, complemented
14C2×#1445C211yesmaximal, characteristic, complemented
15#13782C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#123C2212yes—
18C22×C22⋊Q8C2211yescharacteristic
19C22×#121C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2639—28no—
22#2503—28no—
23#2501—28no—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#400C2212yes—
27C2×#399C2214yes—
28C2×#397C2212yes—
29C2×#394C2212yes—
30C2×#393C2212yes—
31#2368—28no—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2212yes—
35C2×#387C2212yes—
36C2×#387C2212yes—
37C2×#386C2214yes—
38C2×#386C2214yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#384C2212yes—
42C2×#382C2212yes—
43C2×#381C2214yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50C22×#95C2214yes—
51#1445C2212yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2211yescharacteristic
55#1445C2211yescharacteristic
56#1445C2214yes—
57#1445C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2211yescharacteristic
69#1438C2211yescharacteristic
70#1437C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2212yes—
77#1434C2212yes—
78C22⋊#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C2×#123—28no—
87C2×#123—28no—
88C2×#123—216no—
89C2×#123—48no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8D418yes—
93C2×#121—24no—
94C2×#121—24no—
95C2×#121—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124#400—28no—
125#400—28no—
126#400—28no—
127#399—216no—
128#397—28no—
129#394—28no—
130#393—28no—
131#391—28no—
132#391—28no—
133#390—28no—
134#387—28no—
135#387—28no—
136#386—216no—
137#386—48no—
138#385—28no—
139#385—28no—
140#384—28no—
141#382—28no—
142#381—216no—
143C4×C4⋊C4—28no—
144C4×C22⋊C4—24no—
145C4×C22⋊C4—28no—
146C4×C22⋊C4D418yes—
147C2×#95—24no—
148C2×#95—24no—
149C2×#95—48no—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2412yes—
178C25C2411yescharacteristic
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—44no—
182C22×Q8C2×D414yes—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—44no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4#4114yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224#123—432no—
225#123—432no—
226C22⋊Q8—232no—
227#121—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—416no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D418yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
286C2×C22⋊C4C2×D414yes—
287C2×C22⋊C4C2×D414yes—
288C2×C22⋊C4#4118yes—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—44no—
294C2×C42—42no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D418yes—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95C2×D418yes—
325C24—22no—
326C24—22no—
327C24—24no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—24no—
333C24—24no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24C22×D412yes—
338C24C22×D412yes—
339C24C2×#4112yes—
340C24C2×#4114yes—
341C24H(5,𝔽2)14yes—
342C24C2511yescharacteristic
343C2×Q8—28no—
344C2×Q8—48no—
345C2×Q8—84no—
346C2×Q8—84no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—416no—
352C2×D4—416no—
353C2×D4—44no—
354C2×D4—84no—
355C2×D4—84no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
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—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—48no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—84no—
427C22×C4—88no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—88no—
431C22×C4#121116yes—
432C22×C4C22×D414yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C2×#4112yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4118yes—
443C22×C42−1+414yes—
444C4⋊C4—28no—
445C4⋊C4—216no—
446C4⋊C4—48no—
447C4⋊C4—416no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—416no—
452C4⋊C4—88no—
453C4⋊C4—84no—
454C22⋊C4—216no—
455C22⋊C4—28no—
456C22⋊C4—416no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—416no—
460C22⋊C4—88no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C22⋊C4—88no—
464C22⋊C4#124116yes—
465C42—48no—
466C42—48no—
467C42—48no—
468C42—84no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23C2×#12114yes—
502C23C2×#12414yes—
503C23#53514yes—
504C23#55018yes—
505C23#55918yes—
506C23C23×D411yescharacteristic
507C23C23×D411yescharacteristic
508C23C22×#4111yescharacteristic
509C23C22×#4112yes—
510C23C2×H(5,𝔽2)12yes—
511C23C2×2−1+412yes—
512C23#58512yes—
513C23#58514yes—
514Q8—88no—
515D4—88no—
516D4—816no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—28no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—416no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—44no—
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—816no—
585C2×C4—816no—
586C2×C4—816no—
587C2×C4C2×#12114yes—
588C2×C4C2×#12114yes—
589C2×C4C2×#12418yes—
590C22—216no—
591C22—24no—
592C22—28no—
593C22—28no—
594C22—28no—
595C22—28no—
596C22—24no—
597C22—84no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—82no—
606C22#254818yes—
607C22C22×#12111yescharacteristic
608C22C22×#12411yescharacteristic
609C22C2×#53511yescharacteristic
610C22#346112yes—
611C22C2×#55012yes—
612C22C2×#55912yes—
613C22#348512yes—
614C22#348512yes—
615C22#348912yes—
616C22#349112yes—
617C22#349911yescharacteristic
618C22#350011yescharacteristic
619C22#351514yes—
620C22#352114yes—
621C22#353314yes—
622C22#353514yes—
623C4—24no—
624C4—24no—
625C4—28no—
626C4—44no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—84no—
631C4—82no—
632C4—82no—
633C4—84no—
634C4—82no—
635C4—82no—
636C2—28no—
637C2—162no—
638C2C2×#254811yescharacteristic
639C2#3530811yescharacteristic
640C2#3544711yescharacteristic
641C2#3604212yes—
642C2#3926614yes—
643C2#4140912yes—
644C2#4349712yes—
645C2#4445112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2503C2semidirect
4. C2×#2368C2semidirect
5. #17063C2semidirect
6. #16802C2semidirect
7. #15739C2semidirect
8. #15499C2semidirect
9. #15152C2semidirect
10. #15107C2semidirect
11. #14912C2semidirect
13. #13996C2semidirect
14. C2×#1445C2semidirect
15. #13782C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#25012
C2×#26392
C2×#23682
#170632
#168022
#157392
#154992
#151522
#151072
#149122
#141522
#139962
C2×#14452
#137822