Finite Group Databasebeta

← #4389631

Subgroups — #4389631

Order 512.

2568 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic, complemented
3C2×#2639C211yesmaximal, characteristic
4C2×#2332C211yesmaximal, characteristic, complemented
5#16802C212yesmaximal, complemented
6#16705C212yesmaximal, complemented
7#15851C214yesmaximal, complemented
8#15774C214yesmaximal, complemented
9#15577C214yesmaximal, complemented
10#15404C214yesmaximal, complemented
11#15163C212yesmaximal, complemented
12#15152C212yesmaximal, complemented
13#14659C211yesmaximal, characteristic
14#13965C211yesmaximal, characteristic, complemented
15#13958C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×#117C2211yescharacteristic
19#2639—28no—
20#2625—28no—
21#2500—28no—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#397C2214yes—
27C2×#396C2214yes—
28C2×#394C2212yes—
29C2×#394C2214yes—
30C2×#393C2212yes—
31#2332—28no—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#389C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#387C2212yes—
42C2×#387C2212yes—
43C2×#386C2214yes—
44C2×#385C2214yes—
45C2×#384C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C22⋊C4C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1444C2214yes—
56#1443C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2212yes—
69#1438C2212yes—
70#1438C2212yes—
71#1437C2214yes—
72C4⋊#95C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C4⋊#95C2212yes—
76#1434C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2312yes—
81C23×D4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8D418yes—
87C2×#121—24no—
88C2×#121—24no—
89C2×#121—24no—
90C2×#117—24no—
91C2×#117—24no—
92C2×#117—24no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#400—28no—
124#400—28no—
125#398—28no—
126#397—28no—
127#397—216no—
128#396—48no—
129#394—28no—
130#394—216no—
131#393—28no—
132#391—28no—
133#391—28no—
134#390—24no—
135#390—24no—
136#389—28no—
137#388—28no—
138#388—28no—
139#388—216no—
140#388—48no—
141#387—28no—
142#387—28no—
143#386—216no—
144#385—48no—
145#384—24no—
146#382—24no—
147#381—24no—
148#380—24no—
149C4×C4⋊C4—28no—
150C4×C22⋊C4D418yes—
151C2×#95—24no—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C25C2412yes—
177C25C2411yescharacteristic
178C22×Q8—22no—
179C22×Q8—22no—
180C22×Q8—42no—
181C22×Q8—42no—
182C22×Q8C2×D414yes—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—42no—
190C22×D4—42no—
191C22×D4—44no—
192C22×D4—44no—
193C23×C4—24no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4—44no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4C2×D414yes—
201C23×C4C2×D414yes—
202C23×C4#4114yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C22⋊Q8—232no—
217#121—416no—
218#117—416no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D418yes—
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—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—44no—
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—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4C2×D418yes—
289C2×C22⋊C4C2×D418yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95C2×D418yes—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—22no—
332C24—22no—
333C24—22no—
334C24—24no—
335C24—24no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—24no—
340C24C22×D412yes—
341C24C22×D412yes—
342C24C2×#4112yes—
343C24C2511yescharacteristic
344C2×Q8—28no—
345C2×Q8—48no—
346C2×Q8—82no—
347C2×Q8—82no—
348C2×Q8—84no—
349C2×D4—44no—
350C2×D4—44no—
351C2×D4—416no—
352C2×D4—48no—
353C2×D4—44no—
354C2×D4—84no—
355C2×D4—84no—
356C2×D4—88no—
357C2×D4—84no—
358C2×D4—84no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
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—24no—
417C22×C4—24no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4#121116yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D414yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C22×C4H(5,𝔽2)14yes—
439C22×C42−1+414yes—
440C4⋊C4—28no—
441C4⋊C4—216no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—48no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C4⋊C4—84no—
450C4⋊C4—88no—
451C4⋊C4—88no—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—416no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—82no—
462C22⋊C4—82no—
463C22⋊C4—84no—
464C22⋊C4—88no—
465C42—44no—
466C42—44no—
467C42—48no—
468C42—48no—
469C42—84no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—44no—
485C23—42no—
486C23—44no—
487C23—44no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—42no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23C2×#12114yes—
507C23C23×D411yescharacteristic
508C23C23×D411yescharacteristic
509C23C22×#4111yescharacteristic
510C23C2×H(5,𝔽2)12yes—
511C23C2×2−1+412yes—
512C23#58512yes—
513C23#58512yes—
514C23#58514yes—
515Q8—88no—
516D4—88no—
517D4—816no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—416no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
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—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—816no—
588C2×C4C2×#12114yes—
589C2×C4C2×#12114yes—
590C2×C4#53718yes—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—44no—
595C22—44no—
596C22—44no—
597C22—44no—
598C22—48no—
599C22—82no—
600C22—84no—
601C22—82no—
602C22—84no—
603C22—84no—
604C22—82no—
605C22—84no—
606C22—84no—
607C22—84no—
608C22C22×#12111yescharacteristic
609C22C2×#53712yes—
610C22#348512yes—
611C22#348511yescharacteristic
612C22#348511yescharacteristic
613C22#348712yes—
614C22#349112yes—
615C22#349811yescharacteristic
616C22#349911yescharacteristic
617C22#350012yes—
618C22#351012yes—
619C22#351112yes—
620C22#354014yes—
621C22#355314yes—
622C22#356014yes—
623C22#356514yes—
624C4—24no—
625C4—24no—
626C4—28no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—82no—
631C4—82no—
632C4—84no—
633C4—84no—
634C4—82no—
635C4—82no—
636C4—82no—
637C2—44no—
638C2—162no—
639C2#3527311yescharacteristic
640C2#3544711yescharacteristic
641C2#3567711yescharacteristic
642C2#4137212yes—
643C2#4265012yes—
644C2#4681912yes—
645C2#4721212yes—
646C2#4888514yes—
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
2. C2×#2625C2semidirect
4. C2×#2332C2semidirect
5. #16802C2semidirect
6. #16705C2semidirect
7. #15851C2semidirect
8. #15774C2semidirect
9. #15577C2semidirect
10. #15404C2semidirect
11. #15163C2semidirect
12. #15152C2semidirect
14. #13965C2semidirect
15. #13958C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
C2×#26252
C2×#26392
C2×#23322
#168022
#167052
#158512
#157742
#155772
#154042
#151632
#151522
#146592
#139652
#139582