Finite Group Databasebeta

← #4421758

Subgroups — #4421758

Order 512.

2346 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic
3C2×#2625C211yesmaximal, characteristic
4C2×#2346C211yesmaximal, characteristic
5#17070C212yesmaximal, complemented
6#16992C212yesmaximal, complemented
7#16565C212yesmaximal, complemented
8#16517C212yesmaximal, complemented
9#15929C214yesmaximal
10#15298C214yesmaximal
11#15156C212yesmaximal, complemented
12#15090C212yesmaximal, complemented
13#14903C212yesmaximal, complemented
14#14901C212yesmaximal, complemented
15#14660C211yesmaximal, characteristic
16#14355C211yesmaximal, characteristic
17#14339C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2625—28no—
22#2583—28no—
23#2484—28no—
24C2×#400C2211yescharacteristic
25C2×#400C2212yes—
26C2×#400C2211yescharacteristic
27C2×#398C2211yescharacteristic
28C2×#398C2211yescharacteristic
29C2×#398C2212yes—
30C2×#398C2211yescharacteristic
31C2×#398C2211yescharacteristic
32C2×#398C2214yes—
33C2×#397C2214yes—
34C2×#395C2211yescharacteristic
35C2×#395C2211yescharacteristic
36#2346—28no—
37C2×#391C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#387C2212yes—
44C2×#387C2212yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#382C2212yes—
49C2×#381C2211yescharacteristic
50C2×#381C2212yes—
51C2×#381C2212yes—
52C2×#380C2211yescharacteristic
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×#376C2212yes—
57#1446C2212yes—
58#1445C2212yes—
59#1445C2214yes—
60#1444C2212yes—
61#1444C2214yes—
62#1443C2214yes—
63#1443C2214yes—
64#1442C2212yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1441C2212yes—
77#1441C2214yes—
78#1440C2214yes—
79#1439C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4⋊#95C2214yes—
83#1434C2212yes—
84#1434C2214yes—
85#1434C2214yes—
86C22⋊#95C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117D418yes—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128#400—28no—
129#400—24no—
130#400—24no—
131#398—28no—
132#398—24no—
133#398—24no—
134#398—24no—
135#398—24no—
136#398—48no—
137#397—216no—
138#395—24no—
139#395—24no—
140#391—28no—
141#389—28no—
142#389—28no—
143#389—28no—
144#389—28no—
145#388—216no—
146#387—28no—
147#387—28no—
148#385—28no—
149#385—28no—
150#385—28no—
151#382—28no—
152#381—24no—
153#381—28no—
154#381—44no—
155#380—24no—
156#380—24no—
157C4×C4⋊C4—28no—
158C4×C22⋊C4D418yes—
159#376—28no—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C25C2411yescharacteristic
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—44no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—44no—
207C23×C4—41no—
208C23×C4—41no—
209C23×C4—44no—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C22⋊Q8—416no—
230#117—232no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
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—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—416no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4C2×D418yes—
300C2×C22⋊C4C2×D418yes—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—28no—
306C2×C42—44no—
307C2×C42C2×D418yes—
308C2×C42C2×D418yes—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—22no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
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—24no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—24no—
348C24C22×D412yes—
349C24C22×D412yes—
350C24C2×#4112yes—
351C24C2511yescharacteristic
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—84no—
355C2×Q8—84no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—44no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4#121116yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C22×C42−1+418yes—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—84no—
474C4⋊C4—88no—
475C4⋊C4—84no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C22⋊C4—216no—
479C22⋊C4—216no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C42—216no—
485C42—48no—
486C42—416no—
487C23—24no—
488C23—24no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—24no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23C2×#12114yes—
504C23C23×D411yescharacteristic
505C23C23×D411yescharacteristic
506C23C22×#4111yescharacteristic
507C23C2×H(5,𝔽2)11yescharacteristic
508C23C2×2−1+414yes—
509C23C2×2−1+411yescharacteristic
510C23#58514yes—
511C23#58511yescharacteristic
512C23#58511yescharacteristic
513Q8—88no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—88no—
602C2×C4C2×#12114yes—
603C2×C4C2×#12114yes—
604C2×C4#53618yes—
605C22—24no—
606C22—24no—
607C22—24no—
608C22—44no—
609C22—44no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—44no—
614C22C22×#12111yescharacteristic
615C22C2×#53612yes—
616C22#344912yes—
617C22#348512yes—
618C22#348811yescharacteristic
619C22#348811yescharacteristic
620C22#348911yescharacteristic
621C22#348911yescharacteristic
622C22#349112yes—
623C22#349211yescharacteristic
624C22#349311yescharacteristic
625C22#349912yes—
626C22#350912yes—
627C22#354714yes—
628C22#356314yes—
629C22#356414yes—
630C22#356514yes—
631C4—24no—
632C4—24no—
633C4—28no—
634C4—44no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—44no—
639C4—81no—
640C4—81no—
641C4—82no—
642C4—82no—
643C4—84no—
644C4—82no—
645C4—84no—
646C4—82no—
647C2—44no—
648C2#3508911yescharacteristic
649C2#3531611yescharacteristic
650C2#3551811yescharacteristic
651C2#4141112yes—
652C2#4231412yes—
653C2#4231612yes—
654C2#4268112yes—
655C2#4879912yes—
656C2#5018312yes—
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
5. #17070C2semidirect
6. #16992C2semidirect
7. #16565C2semidirect
8. #16517C2semidirect
11. #15156C2semidirect
12. #15090C2semidirect
13. #14903C2semidirect
14. #14901C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24842
C2×#26252
C2×#23462
#170702
#169922
#165652
#165172
#159292
#152982
#151562
#150902
#149032
#149012
#146602
#143552
#143392