Finite Group Databasebeta

← #4456855

Subgroups — #4456855

Order 512.

2646 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2561C211yesmaximal, characteristic
2C2×#2482C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic, complemented
4C2×#2312C211yesmaximal, characteristic
5#17060C212yesmaximal, complemented
6#16805C212yesmaximal, complemented
7#16803C212yesmaximal, complemented
8#16701C212yesmaximal, complemented
9#16396C214yesmaximal, complemented
10#16391C214yesmaximal
11#14988C214yesmaximal, complemented
12#14970C214yesmaximal, complemented
13#14292C211yesmaximal, characteristic, complemented
14#13953C211yesmaximal, characteristic, complemented
15#13853C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2561—28no—
20#2499—28no—
21#2482—28no—
22C2×#399C2212yes—
23C2×#397C2214yes—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#393C2212yes—
27#2312—28no—
28C2×#391C2211yescharacteristic, complemented
29C2×#391C2211yescharacteristic, complemented
30C2×#388C2214yes—
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#388C2214yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#387C2214yescomplemented
37C2×#385C2212yes—
38C2×#384C2211yescharacteristic, complemented
39C2×#382C2211yescharacteristic, complemented
40C2×#381C2212yes—
41C2×#381C2211yescharacteristic
42C2×#380C2212yes—
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×#379C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#1446C2212yescomplemented
50#1445C2212yescomplemented
51#1445C2212yescomplemented
52#1445C2212yescomplemented
53#1443C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1436C2212yescomplemented
73#1436C2214yescomplemented
74#1436C2212yescomplemented
75C4⋊#95C2212yescomplemented
76#1434C2212yescomplemented
77C4×#95C2212yescomplemented
78C4×#95C2212yescomplemented
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×#117—24no—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2311yescharacteristic
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2314yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4—24no—
99C22×C22⋊C4—24no—
100C22×C22⋊C4D418yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#399—28no—
124#397—28no—
125#397—28no—
126#397—216no—
127#393—28no—
128#391—24no—
129#391—24no—
130#388—216no—
131#388—216no—
132#388—28no—
133#388—28no—
134#388—28no—
135#388—28no—
136#387—216no—
137#385—28no—
138#384—24no—
139#382—24no—
140#381—28no—
141#381—24no—
142#380—28no—
143#379—28no—
144#379—28no—
145#379—28no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4D418yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C25—22no—
174C25—22no—
175C25—42no—
176C25—42no—
177C25—42no—
178C25—42no—
179C25—44no—
180C25C2×D414yes—
181C25C2412yes—
182C25C2411yescharacteristic
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—42no—
187C22×Q8—42no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—42no—
191C23×C4—42no—
192C23×C4—44no—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2×D414yes—
196C23×C4#4114yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2414yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C22⋊Q8—416no—
211#117—416no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—28no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
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×C22⋊C4—232no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—42no—
263C2×C22⋊C4—42no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4C2×D418yes—
279C2×C22⋊C4C2×D418yes—
280C2×C22⋊C4C2×D418yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—42no—
288C2×C42—42no—
289C2×C42—42no—
290C2×C42—42no—
291C2×C42—42no—
292C2×C42—42no—
293C2×C42C2×D414yes—
294C2×C42C2×D414yes—
295#95—24no—
296#95—24no—
297#95—24no—
298#95—24no—
299#95—24no—
300#95—28no—
301#95—28no—
302#95—24no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95—44no—
318#95C2×D418yes—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—24no—
324C24—22no—
325C24—22no—
326C24—22no—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—24no—
331C24—24no—
332C24—24no—
333C24—24no—
334C24—28no—
335C24—48no—
336C24—48no—
337C24—48no—
338C24—48no—
339C24—44no—
340C24—44no—
341C24—82no—
342C24—82no—
343C24—84no—
344C24—84no—
345C24—84no—
346C24—88no—
347C24—88no—
348C24—88no—
349C24—88no—
350C24C22×D414yes—
351C24C22×D412yes—
352C24C22×D412yes—
353C24C2×#4112yes—
354C24C2511yescharacteristic
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—84no—
359C2×Q8—84no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
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—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
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—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—48no—
426C22×C4—84no—
427C22×C4—84no—
428C22×C4#121116yes—
429C22×C4C22×D414yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C2×#4112yes—
437C22×C4C2×#4112yes—
438C4⋊C4—416no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—88no—
444C4⋊C4—84no—
445C4⋊C4—84no—
446C4⋊C4—88no—
447C22⋊C4—216no—
448C22⋊C4—216no—
449C22⋊C4—48no—
450C22⋊C4—48no—
451C22⋊C4—416no—
452C22⋊C4—416no—
453C22⋊C4—48no—
454C22⋊C4—48no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—88no—
462C22⋊C4—88no—
463C42—44no—
464C42—44no—
465C42—48no—
466C42—82no—
467C42—82no—
468C42—84no—
469C23—24no—
470C23—24no—
471C23—22no—
472C23—22no—
473C23—24no—
474C23—24no—
475C23—22no—
476C23—22no—
477C23—24no—
478C23—24no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—28no—
484C23—28no—
485C23—416no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—42no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—42no—
498C23—42no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—88no—
509C23—88no—
510C23—832no—
511C23—816no—
512C23—816no—
513C23—816no—
514C23—816no—
515C23C2×#12114yes—
516C23C23×D411yescharacteristic
517C23C23×D411yescharacteristic
518C23C22×#4111yescharacteristic
519C23#58512yes—
520C23#58512yes—
521C23#58512yes—
522C23#58512yes—
523C23#58514yes—
524Q8—88no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—416no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—88no—
594C2×C4C2×#12114yes—
595C2×C4C2×#12114yes—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—44no—
600C22—44no—
601C22—44no—
602C22—44no—
603C22—48no—
604C22—44no—
605C22—44no—
606C22—44no—
607C22—48no—
608C22—48no—
609C22—832no—
610C22—84no—
611C22—84no—
612C22—82no—
613C22—82no—
614C22—82no—
615C22—82no—
616C22C22×#12111yescharacteristic
617C22#344912yes—
618C22#344912yes—
619C22#345511yescharacteristic
620C22#345511yescharacteristic
621C22#348512yes—
622C22#348511yescharacteristic
623C22#348511yescharacteristic
624C22#348512yes—
625C22#349112yes—
626C22#350012yes—
627C22#351012yes—
628C22#356614yes—
629C22#356614yes—
630C22#356614yes—
631C22#356814yes—
632C4—24no—
633C4—24no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—82no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—84no—
644C4—84no—
645C2—44no—
646C2—84no—
647C2#3463311yescharacteristic
648C2#3544711yescharacteristic
649C2#3561711yescharacteristic
650C2#3701412yes—
651C2#3702012yes—
652C2#4138012yes—
653C2#4138712yes—
654C2#4141414yes—
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
3. C2×#2499C2semidirect
5. #17060C2semidirect
6. #16805C2semidirect
7. #16803C2semidirect
8. #16701C2semidirect
9. #16396C2semidirect
11. #14988C2semidirect
12. #14970C2semidirect
13. #14292C2semidirect
14. #13953C2semidirect
15. #13853C2semidirect
28. C2×#391C22semidirect
29. C2×#391C22semidirect
36. C2×#387C22semidirect
38. C2×#384C22semidirect
39. C2×#382C22semidirect
49. #1446C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
72. #1436C22semidirect
73. #1436C22semidirect
74. #1436C22semidirect
75. C4⋊#95C22semidirect
76. #1434C22semidirect
77. C4×#95C22semidirect
78. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25612
C2×#24822
C2×#24992
C2×#23122
#170602
#168052
#168032
#167012
#163962
#163912
#149882
#149702
#142922
#139532
#138532