Finite Group Databasebeta

← #4420167

Subgroups — #4420167

Order 512.

2228 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2611C211yesmaximal, characteristic
2C2×#2642C211yesmaximal, characteristic, complemented
3C2×#2628C211yesmaximal, characteristic
4C2×#2346C211yesmaximal, characteristic
5#16805C212yesmaximal
6#16803C212yesmaximal
7#16567C212yesmaximal, complemented
8#16400C212yesmaximal
9#15758C214yesmaximal
10#15678C214yesmaximal, complemented
11#15164C212yesmaximal, complemented
12#15097C212yesmaximal, complemented
13#14908C212yesmaximal, complemented
14#14841C212yesmaximal
15#14664C211yesmaximal, characteristic, complemented
16#14282C211yesmaximal, characteristic, complemented
17#13963C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#117C2211yescharacteristic
21#2642—28no—
22#2628—28no—
23#2611—28no—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#400C2211yescharacteristic
27C2×#400C2212yes—
28C2×#400C2214yes—
29C2×#399C2212yes—
30C2×#399C2211yescharacteristic
31C2×#398C2211yescharacteristic
32C2×#397C2211yescharacteristic
33C2×#395C2212yes—
34C2×#393C2212yes—
35#2346—28no—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#388C2212yes—
44C2×#387C2212yes—
45C2×#385C2212yes—
46C2×#382C2211yescharacteristic
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#380C2211yescharacteristic
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×#376C2211yescharacteristic
54C2×#376C2211yescharacteristic
55C2×#376C2211yescharacteristic
56#1446C2212yes—
57#1446C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1444C2212yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2212yes—
75#1440C2214yes—
76#1440C2214yes—
77#1440C2214yes—
78#1439C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83C4⋊#95C2212yes—
84C4⋊#95C2214yes—
85#1434C2214yes—
86#1434C2212yes—
87C23×Q8C2312yes—
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117D418yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#400—24no—
128#400—28no—
129#400—28no—
130#400—28no—
131#400—216no—
132#399—24no—
133#399—28no—
134#398—24no—
135#397—24no—
136#395—28no—
137#393—28no—
138#391—28no—
139#391—28no—
140#389—28no—
141#389—28no—
142#389—28no—
143#388—28no—
144#388—28no—
145#388—28no—
146#387—28no—
147#385—28no—
148#382—24no—
149#382—24no—
150#382—24no—
151#380—28no—
152#380—24no—
153C4×C4⋊C4—28no—
154C4×C22⋊C4—24no—
155#376—24no—
156#376—24no—
157#376D418yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C25C2411yescharacteristic
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—44no—
197C22×Q8—44no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4C2×D414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C4⋊Q8—416no—
220#125—416no—
221#117—232no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—44no—
292C2×C42—42no—
293C2×C42—42no—
294C2×C42—44no—
295C2×C42C2×D414yes—
296C2×C42C2×D418yes—
297C2×C42C2×D414yes—
298C2×C42C2×D414yes—
299C2×C42#4114yes—
300#95—22no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95C2×D414yes—
332#95C2×D414yes—
333#95C2×D418yes—
334C24—21no—
335C24—21no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24C22×D412yes—
342C24C2511yescharacteristic
343C2×Q8—48no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—88no—
347C2×Q8—88no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—24no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—28no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
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—24no—
418C22×C4—44no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4C22×D412yes—
422C22×C4C22×D414yes—
423C22×C4C22×D414yes—
424C22×C4C22×D412yes—
425C22×C4C22×D412yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C2×#4112yes—
433C22×C4C2×#4112yes—
434C22×C4C2×#4112yes—
435C22×C42−1+414yes—
436C22×C42−1+414yes—
437C4⋊C4—28no—
438C4⋊C4—28no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4—88no—
452C4⋊C4—88no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—84no—
459C22⋊C4—88no—
460C42—28no—
461C42—48no—
462C42—48no—
463C42—48no—
464C42—48no—
465C42#121116yes—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—24no—
470C23—24no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23C23×D411yescharacteristic
483C23C23×D411yescharacteristic
484C23C22×#4111yescharacteristic
485C23C2×2−1+412yes—
486C23C2×2−1+412yes—
487C23#58512yes—
488C23#58514yes—
489C23#58512yes—
490Q8—816no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—416no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4C2×#12114yes—
573C2×C4C2×#12114yes—
574C2×C4C2×#12114yes—
575C2×C4#53614yes—
576C22—42no—
577C22—44no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—82no—
582C22—82no—
583C22—84no—
584C22C22×#12111yescharacteristic
585C22C2×#53611yescharacteristic
586C22#344912yes—
587C22#344911yescharacteristic
588C22#348912yes—
589C22#348912yes—
590C22#349512yes—
591C22#349911yescharacteristic
592C22#350011yescharacteristic
593C22#350012yes—
594C22#350712yes—
595C22#351012yes—
596C22#356314yes—
597C22#356514yes—
598C22#356514yes—
599C22#356614yes—
600C4—24no—
601C4—24no—
602C4—24no—
603C4—24no—
604C4—42no—
605C4—44no—
606C4—44no—
607C4—44no—
608C4—42no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—84no—
614C4—82no—
615C4—84no—
616C2—82no—
617C2#3530611yescharacteristic
618C2#3532711yescharacteristic
619C2#3562111yescharacteristic
620C2#4215712yes—
621C2#4230712yes—
622C2#4231612yes—
623C2#4297812yes—
624C2#5161814yes—
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
2. C2×#2642C2semidirect
7. #16567C2semidirect
10. #15678C2semidirect
11. #15164C2semidirect
12. #15097C2semidirect
13. #14908C2semidirect
15. #14664C2semidirect
16. #14282C2semidirect
17. #13963C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26112
C2×#26422
C2×#26282
C2×#23462
#168052
#168032
#165672
#164002
#157582
#156782
#151642
#150972
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