Finite Group Databasebeta

← #4928891

Subgroups — #4928891

Order 512.

2858 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic, complemented
2C2×#2633C211yesmaximal, characteristic
3C2×#2398C211yesmaximal, characteristic
4C2×#2275C211yesmaximal, characteristic
5#15161C212yesmaximal, complemented
6#15150C212yesmaximal, complemented
7#15050C212yesmaximal, complemented
8#15016C212yesmaximal, complemented
9#14844C214yesmaximal, complemented
10#14824C214yesmaximal
11#14202C214yesmaximal, complemented
12#14170C211yesmaximal, characteristic, complemented
13#14047C214yesmaximal, complemented
14#13964C211yesmaximal, characteristic, complemented
15#13961C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20#2633—28no—
21#2565—28no—
22#2398—28no—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#397C2212yes—
28C2×#397C2214yes—
29C2×#397C2212yes—
30C2×#394C2212yes—
31C2×#394C2212yes—
32C2×#393C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#387C2214yes—
36C2×#385C2214yes—
37C2×#384C2211yescharacteristic
38C2×#383C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#380C2214yes—
41#2275—28no—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2214yes—
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2211yescharacteristic
49C2×#376C2211yescharacteristic
50C22×#95C2212yes—
51#1446C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1445C2214yes—
55#1445C2212yes—
56#1443C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1436C2214yes—
64#1436C2212yes—
65#1436C2212yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2214yes—
72#1434C2212yes—
73#1434C2214yes—
74#1434C2212yes—
75#1434C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C2×#125—24no—
85C2×#125—24no—
86C2×#125—24no—
87C2×C4×Q8—24no—
88C2×C4×Q8—24no—
89C2×C4×Q8—24no—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C42C2312yes—
115C22×C42C2312yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122#400—28no—
123#399—28no—
124#398—216no—
125#398—28no—
126#397—28no—
127#397—28no—
128#397—216no—
129#394—28no—
130#394—28no—
131#393—28no—
132#391—24no—
133#391—24no—
134#387—216no—
135#385—216no—
136#384—24no—
137#383—24no—
138#382—24no—
139#380—216no—
140C4×C4⋊C4—28no—
141C4×C4⋊C4D418yes—
142C4×C22⋊C4—216no—
143C4×C22⋊C4—28no—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—28no—
146#376—24no—
147#376—24no—
148C2×#95—28no—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C25C2412yes—
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—22no—
181C22×Q8—24no—
182C22×Q8—42no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4C2×D414yes—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4C2×D418yes—
202C23×C4#4118yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217#125—416no—
218C4×Q8—416no—
219C4×D4—264no—
220C4×D4—232no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
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—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D418yes—
253C2×C4⋊C4C2×D414yes—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4C2×D418yes—
274C2×C22⋊C4#4118yes—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42—24no—
278C2×C42—24no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—42no—
285C2×C42—42no—
286C2×C42—44no—
287C2×C42—44no—
288C2×C42—44no—
289C2×C42C2×D414yes—
290C2×C42C2×D414yes—
291C2×C42C2×D414yes—
292C2×C42C2×D414yes—
293C2×C42#4114yes—
294#95—22no—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—22no—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
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#95C2×D418yes—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—24no—
328C24—24no—
329C24—24no—
330C24C22×D414yes—
331C24C2×#4114yes—
332C24C2×#4114yes—
333C24C2511yescharacteristic
334C2×Q8—28no—
335C2×Q8—48no—
336C2×Q8—48no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×D4—216no—
340C2×D4—28no—
341C2×D4—28no—
342C2×D4—216no—
343C2×D4—28no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—216no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—84no—
408C22×C4—84no—
409C22×C4—84no—
410C22×C4—84no—
411C22×C4—88no—
412C22×C4—88no—
413C22×C4—84no—
414C22×C4—88no—
415C22×C4—84no—
416C22×C4—88no—
417C22×C4C22×D414yes—
418C22×C4C22×D414yes—
419C22×C4C22×D412yes—
420C22×C4C22×D412yes—
421C22×C4C22×D412yes—
422C22×C4C22×D412yes—
423C22×C4C22×D412yes—
424C22×C4C22×D412yes—
425C22×C4C22×D414yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4114yes—
428C22×C4C2×#4112yes—
429C22×C4C2×#4112yes—
430C22×C4C2×#4112yes—
431C22×C4C2×#4114yes—
432C22×C4C2×#4114yes—
433C22×C42−1+414yes—
434C4⋊C4—28no—
435C4⋊C4—216no—
436C4⋊C4—48no—
437C4⋊C4—44no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—416no—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C22⋊C4—216no—
446C22⋊C4—216no—
447C22⋊C4—216no—
448C22⋊C4—416no—
449C22⋊C4—416no—
450C22⋊C4—416no—
451C22⋊C4—88no—
452C42—216no—
453C42—44no—
454C42—44no—
455C42—48no—
456C42—48no—
457C42—84no—
458C42—84no—
459C42#121116yes—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—28no—
467C23—28no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—44no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23#53218yes—
483C23C23×D411yescharacteristic
484C23C23×D411yescharacteristic
485C23C22×#4111yescharacteristic
486C23C22×#4112yes—
487C23C22×#4112yes—
488C23C2×2−1+412yes—
489C23#58512yes—
490C23#58514yes—
491Q8—416no—
492D4—232no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—28no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—816no—
565C2×C4—816no—
566C2×C4—816no—
567C2×C4C2×#12114yes—
568C2×C4C2×#12114yes—
569C2×C4C2×#12114yes—
570C2×C4#52518yes—
571C22—28no—
572C22—216no—
573C22—44no—
574C22—44no—
575C22—44no—
576C22—44no—
577C22—48no—
578C22—44no—
579C22—44no—
580C22—48no—
581C22—48no—
582C22C22×#12111yescharacteristic
583C22C2×#52512yes—
584C22C2×#53212yes—
585C22#344411yescharacteristic
586C22#344411yescharacteristic
587C22#348512yes—
588C22#348511yescharacteristic
589C22#348611yescharacteristic
590C22#348912yes—
591C22#349114yes—
592C22#349512yes—
593C22#350012yes—
594C22#350714yes—
595C22#351012yes—
596C22#351514yes—
597C22#352114yes—
598C4—24no—
599C4—24no—
600C4—24no—
601C4—28no—
602C4—44no—
603C4—44no—
604C4—44no—
605C4—44no—
606C4—44no—
607C4—82no—
608C4—82no—
609C4—82no—
610C4—84no—
611C4—84no—
612C2—48no—
613C2#3546414yes—
614C2#3561711yescharacteristic
615C2#3563311yescharacteristic
616C2#3571212yes—
617C2#3576012yes—
618C2#3601012yes—
619C2#3607212yes—
620C2#3662411yescharacteristic
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
1. C2×#2565C2semidirect
5. #15161C2semidirect
6. #15150C2semidirect
7. #15050C2semidirect
8. #15016C2semidirect
9. #14844C2semidirect
11. #14202C2semidirect
12. #14170C2semidirect
13. #14047C2semidirect
14. #13964C2semidirect
15. #13961C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25652
C2×#26332
C2×#23982
C2×#22752
#151612
#151502
#150502
#150162
#148442
#148242
#142022
#141702
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