Finite Group Databasebeta

← #4431144

Subgroups — #4431144

Order 512.

2766 conjugacy classes of subgroups.

The subgroup lattice has 659 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic, complemented
3C2×#2626C211yesmaximal, characteristic
4C2×#2346C211yesmaximal, characteristic, complemented
5#16595C214yesmaximal, complemented
6#16566C212yesmaximal, complemented
7#16564C212yesmaximal, complemented
8#15398C214yesmaximal, complemented
9#15282C214yesmaximal
10#14910C212yesmaximal, complemented
11#14902C212yesmaximal, complemented
12#14788C214yesmaximal, complemented
13#14675C211yesmaximal, characteristic, complemented
14#14292C211yesmaximal, characteristic, complemented
15#13903C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2626—28no—
20#2625—28no—
21#2482—28no—
22C2×#400C2212yescomplemented
23C2×#399C2212yes—
24C2×#399C2212yes—
25C2×#398C2212yescomplemented
26C2×#397C2214yes—
27C2×#397C2212yes—
28C2×#396C2214yes—
29C2×#393C2212yes—
30#2346—28no—
31C2×#391C2212yescomplemented
32C2×#389C2212yescomplemented
33C2×#389C2212yescomplemented
34C2×#389C2214yescomplemented
35C2×#388C2214yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#382C2212yescomplemented
42C2×#381C2211yescharacteristic
43C2×#381C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×#379C2212yes—
47C2×#379C2212yes—
48C2×#379C2212yes—
49C2×C4×C4⋊C4C2212yescomplemented
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×#376C2212yescomplemented
52#1446C2211yescharacteristic, complemented
53#1446C2211yescharacteristic, complemented
54#1445C2214yescomplemented
55#1445C2211yescharacteristic, complemented
56#1445C2211yescharacteristic, complemented
57#1444C2214yescomplemented
58#1444C2212yescomplemented
59#1443C2214yescomplemented
60#1442C2214yescomplemented
61#1442C2212yescomplemented
62#1441C2214yescomplemented
63#1441C2214yescomplemented
64#1441C2214yescomplemented
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yescomplemented
78#1436C2214yescomplemented
79C26C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C2×C22⋊Q8—24no—
86C2×#117—24no—
87C2×#117—24no—
88C2×#117D418yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2314yes—
121C22×C42C2312yes—
122C22×C42C2312yes—
123#400—28no—
124#399—28no—
125#399—28no—
126#398—28no—
127#397—28no—
128#397—216no—
129#396—216no—
130#393—28no—
131#391—28no—
132#389—28no—
133#389—28no—
134#389—48no—
135#388—216no—
136#388—28no—
137#388—28no—
138#388—48no—
139#385—28no—
140#385—28no—
141#382—28no—
142#381—24no—
143#381—44no—
144#380—24no—
145#380D418yes—
146#379—28no—
147#379—28no—
148#379—44no—
149C4×C4⋊C4—28no—
150C4×C22⋊C4—24no—
151#376—28no—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C25—22no—
177C25—24no—
178C25—22no—
179C25—22no—
180C25—42no—
181C25—44no—
182C25—42no—
183C25—42no—
184C25C2412yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—44no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—42no—
196C23×C4C2×D414yes—
197C23×C4C2×D414yes—
198C23×C4#4114yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C22⋊Q8—416no—
213#117—232no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4C2×D414yes—
244C2×C4⋊C4C2×D414yes—
245C2×C4⋊C4C2×D418yes—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—28no—
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—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—416no—
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—42no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D418yes—
290C2×C22⋊C4C2×D418yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42C2×D418yes—
299#95—22no—
300#95—28no—
301#95—22no—
302#95—28no—
303#95—24no—
304#95—22no—
305#95—22no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95C2×D418yes—
323C24—22no—
324C24—22no—
325C24—22no—
326C24—24no—
327C24—22no—
328C24—28no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—24no—
338C24—24no—
339C24—48no—
340C24—48no—
341C24—44no—
342C24—48no—
343C24—48no—
344C24—44no—
345C24—84no—
346C24—84no—
347C24—84no—
348C24—82no—
349C24—82no—
350C24—88no—
351C24—88no—
352C24—88no—
353C24—88no—
354C24C22×D412yes—
355C24C22×D412yes—
356C24C2×#4112yes—
357C242−1+414yes—
358C24C2511yescharacteristic
359C2×Q8—48no—
360C2×Q8—44no—
361C2×Q8—84no—
362C2×Q8—84no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
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—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—44no—
427C22×C4—84no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4#121116yes—
431C22×C4C22×D414yes—
432C22×C4C22×D418yes—
433C22×C4C22×D414yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4112yes—
439C22×C4C2×#4112yes—
440C22×C4H(5,𝔽2)14yes—
441C4⋊C4—28no—
442C4⋊C4—28no—
443C4⋊C4—48no—
444C4⋊C4—48no—
445C4⋊C4—416no—
446C4⋊C4—44no—
447C4⋊C4—48no—
448C4⋊C4—88no—
449C4⋊C4—88no—
450C4⋊C4—88no—
451C4⋊C4—84no—
452C4⋊C4—84no—
453C22⋊C4—216no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—416no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—48no—
462C22⋊C4—48no—
463C22⋊C4—88no—
464C22⋊C4—88no—
465C22⋊C4—84no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C22⋊C4—88no—
469C42—216no—
470C42—416no—
471C42—48no—
472C23—24no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—22no—
482C23—22no—
483C23—24no—
484C23—24no—
485C23—24no—
486C23—28no—
487C23—28no—
488C23—416no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—44no—
500C23—42no—
501C23—44no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—88no—
512C23—88no—
513C23—816no—
514C23—832no—
515C23—816no—
516C23—816no—
517C23—816no—
518C23C2×#12114yes—
519C23C23×D411yescharacteristic
520C23C23×D411yescharacteristic
521C23C22×#4111yescharacteristic
522C23C2×H(5,𝔽2)12yes—
523C23C2×2−1+412yes—
524C23#58514yes—
525C23#58512yes—
526C23#58512yes—
527Q8—88no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—28no—
545C2×C4—416no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—88no—
597C2×C4C2×#12114yes—
598C2×C4C2×#12114yes—
599C22—24no—
600C22—24no—
601C22—24no—
602C22—44no—
603C22—48no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—44no—
608C22—48no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—832no—
613C22—82no—
614C22—84no—
615C22—82no—
616C22—82no—
617C22—82no—
618C22—84no—
619C22C22×#12111yescharacteristic
620C22#344912yes—
621C22#346012yes—
622C22#348512yes—
623C22#348511yescharacteristic
624C22#348611yescharacteristic
625C22#348812yes—
626C22#348912yes—
627C22#349112yes—
628C22#349512yes—
629C22#349811yescharacteristic
630C22#350011yescharacteristic
631C22#353514yes—
632C22#354014yes—
633C22#356414yes—
634C22#356614yes—
635C4—24no—
636C4—24no—
637C4—44no—
638C4—44no—
639C4—44no—
640C4—48no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4—82no—
645C4—82no—
646C4—84no—
647C4—82no—
648C2—44no—
649C2—84no—
650C2#3508911yescharacteristic
651C2#3525611yescharacteristic
652C2#3563311yescharacteristic
653C2#4162414yes—
654C2#4196012yes—
655C2#4239212yes—
656C2#4877912yes—
657C2#5017112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2346C2semidirect
5. #16595C2semidirect
6. #16566C2semidirect
7. #16564C2semidirect
8. #15398C2semidirect
10. #14910C2semidirect
11. #14902C2semidirect
12. #14788C2semidirect
13. #14675C2semidirect
14. #14292C2semidirect
22. C2×#400C22semidirect
25. C2×#398C22semidirect
31. C2×#391C22semidirect
32. C2×#389C22semidirect
33. C2×#389C22semidirect
34. C2×#389C22semidirect
41. C2×#382C22semidirect
49. C2×C4×C4⋊C4C22semidirect
51. C2×#376C22semidirect
52. #1446C22semidirect
53. #1446C22semidirect
54. #1445C22semidirect
55. #1445C22semidirect
56. #1445C22semidirect
57. #1444C22semidirect
58. #1444C22semidirect
59. #1443C22semidirect
60. #1442C22semidirect
61. #1442C22semidirect
62. #1441C22semidirect
63. #1441C22semidirect
64. #1441C22semidirect
77. #1436C22semidirect
78. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C2×#26252
C2×#26262
C2×#23462
#165952
#165662
#165642
#153982
#152822
#149102
#149022
#147882
#146752
#142922
#139032