Finite Group Databasebeta

← #7697254

Subgroups — #7697254

Order 512.

11925 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3488C214yesmaximal, complemented
2#59816C213yesmaximal, complemented
3#58997C216yesmaximal, complemented
4#52103C213yesmaximal, complemented
5#45103C216yesmaximal, complemented
6#44877C2112yesmaximal, complemented
7#39355C216yesmaximal, complemented
8#37648C2112yesmaximal, complemented
9#36465C211yesmaximal, characteristic, complemented
10#36428C212yesmaximal, complemented
11#36091C212yesmaximal, complemented
12#35831C214yesmaximal, complemented
13#34313C212yesmaximal, complemented
14C22×H(5,𝔽2)C2212yescomplemented
15C23×#41C2212yescomplemented
16#3488—264no—
17C2×#550C2212yes—
18C2×#550C2214yescomplemented
19C2×#550C2211yescharacteristic, complemented
20C2×D4×Q8C2211yescharacteristic, complemented
21C2×#548C2214yescomplemented
22C2×#548C2214yescomplemented
23C2×#546C22112yescomplemented
24C2×#546C22112yescomplemented
25C2×D42C22112yescomplemented
26C2×D42C22112yescomplemented
27C2×D42C2216yescomplemented
28C2×#537C22112yescomplemented
29C2×#535C22112yescomplemented
30C2×#525C22112yescomplemented
31C22×#121C22112yescomplemented
32C2×#520C2214yescomplemented
33#2858C2213yescomplemented
34#2848C2216yescomplemented
35#2683C2211yescharacteristic, complemented
36#2682C22112yescomplemented
37#2682C2216yescomplemented
38#2681C22112yescomplemented
39#2680C2216yescomplemented
40#2679C22112yescomplemented
41#2679C2213yescomplemented
42#2679C2216yescomplemented
43#2678C22112yescomplemented
44#2675C22112yescomplemented
45#2671C22124yescomplemented
46#2669C2216yescomplemented
47#2668C22112yescomplemented
48#2620C22112yescomplemented
49#2615C2216yescomplemented
50#2544C22112yescomplemented
51#2542C2213yescomplemented
52#2541C2213yescomplemented
53#2540C22112yescomplemented
54#2537C22112yescomplemented
55#2537C2216yescomplemented
56#2437C22112yescomplemented
57#2435C2216yescomplemented
58#2433C2216yescomplemented
59#2433C22112yescomplemented
60#2432C22112yescomplemented
61#2428C22112yescomplemented
62#2427C22112yescomplemented
63#2424C22124yescomplemented
64#2424C22124yescomplemented
65#2417C22112yescomplemented
66#2416C22112yescomplemented
67#2410C22124yescomplemented
68#2408C22124yescomplemented
69#2408C22112yescomplemented
70#2408C2216yescomplemented
71#2408C2216yescomplemented
72#2407C22112yescomplemented
73#2403C22112yescomplemented
74#2392C22112yescomplemented
75#2388C2216yescomplemented
76#2366C2216yescomplemented
77#2361C22112yescomplemented
78Q8×C4⋊C4C2212yescomplemented
79D4×C4⋊C4C2216yescomplemented
80#2341C2214yescomplemented
81#2341C2212yescomplemented
82Q8×C22⋊C4C2212yescomplemented
83#2324C22112yescomplemented
84C4×#127C2216yescomplemented
85C4×#127C2213yescomplemented
86C4×#120C2216yescomplemented
87#2293C2216yescomplemented
88C42×Q8C2211yescharacteristic, complemented
89C2×H(5,𝔽2)—232no—
90C2×H(5,𝔽2)—48no—
91C22×#41—232no—
92C22×#41—48no—
93C22×#41C2312yes—
94C22×#41C23112yes—
95C22×#41C2316yes—
96C22×#41C2314yes—
97C23×Q8C2312yes—
98C23×D4C2316yes—
99C23×D4C23112yes—
100C23×D4C2316yes—
101C23×D4C23124yes—
102C24×C4C2316yes—
103#550—232no—
104#550—44no—
105#550D4132yescomplemented
106D4×Q8—44no—
107#548—232no—
108#548—232no—
109#546—296no—
110#546—296no—
111D42—296no—
112D42—296no—
113D42—424no—
114#537—296no—
115#535—296no—
116C2×C4⋊Q8C2313yes—
117C2×#127C2313yes—
118C2×#127C23112yes—
119C2×#127C2313yes—
120C2×#127C2316yes—
121C2×#124C23112yes—
122C2×#124C23112yes—
123#525—296no—
124C2×#123C23112yes—
125C2×#123C23112yes—
126C2×C22⋊Q8C23112yes—
127C2×C22⋊Q8C2314yes—
128C2×C22⋊Q8C23112yes—
129C2×C22⋊Q8C23112yes—
130C2×C22⋊Q8C2316yes—
131C2×#121—296no—
132C2×#121C23112yes—
133C2×#121C23112yes—
134C2×#121C23112yes—
135C2×#121C23124yes—
136C2×#121C23112yes—
137C2×#121C23124yes—
138C2×#121C23112yes—
139C2×#121C23112yes—
140C2×#121C23124yes—
141C2×#121C2316yes—
142C2×#121C23124yes—
143C2×#121C23112yes—
144C2×#120C23124yes—
145C2×#120C23112yes—
146C2×#120C2316yes—
147C2×#120C23124yes—
148C2×#120C23112yes—
149C2×#120C23112yes—
150C2×#120C23124yes—
151#520—232no—
152C2×C4×Q8C2314yes—
153C2×C4×Q8C2311yescharacteristic
154C2×C4×Q8C2312yes—
155C2×C4×D4C23112yes—
156C2×C4×D4C23112yes—
157C2×C4×D4C23112yes—
158C2×C4×D4C23112yes—
159C2×C4×D4C23112yes—
160C2×C4×D4C23112yes—
161C2×C4×D4C2313yes—
162C2×C4×D4C2313yes—
163C2×C4×D4C23112yes—
164C2×C4×D4C2316yes—
165C2×#117C23112yes—
166C22×C4⋊C4C23112yes—
167C22×C22⋊C4C23124yes—
168#399C23112yes—
169#398C23112yes—
170#397C23112yes—
171#396C23112yes—
172#396C23124yes—
173#396C2314yes—
174#395C2312yes—
175#394C23124yes—
176#394C23124yes—
177#394C23112yes—
178#394C2316yes—
179#394C23112yes—
180#394C23112yes—
181#394C23112yes—
182#394C23124yes—
183#394C23112yes—
184#393C2316yes—
185#392C23112yes—
186#392C23124yes—
187#392C2316yes—
188#392C23112yes—
189#392C23112yes—
190#392C23112yes—
191#392C23112yes—
192#391C23112yes—
193#390C2311yescharacteristic
194#390C2316yes—
195#390C23112yes—
196#390C23112yes—
197#390C2313yes—
198#390C23112yes—
199#390C23112yes—
200#390C23112yes—
201#390C23112yes—
202#390C2316yes—
203#390C2316yes—
204#389C23112yes—
205#389C2316yes—
206#389C2312yes—
207#388C23112yes—
208#388C23112yes—
209#386C23124yes—
210#386C23124yes—
211#386C23112yes—
212#386C23124yes—
213#386C2312yes—
214#385C23124yes—
215#385C2316yes—
216#385C23124yes—
217#385C23112yes—
218#384C2316yes—
219#384C2313yes—
220#380C23112yes—
221#380C2316yes—
222#380C2316yes—
223#380C23124yes—
224#379C23112yes—
225#379C2316yes—
226#379C23112yes—
227C4×C4⋊C4C2316yes—
228C4×C4⋊C4C2316yes—
229C4×C4⋊C4C2313yes—
230C4×C22⋊C4C2316yes—
231C4×C22⋊C4C23112yes—
232C4×C22⋊C4C2316yes—
233C43C2313yes—
234C25C2416yes—
235C25C24112yes—
236H(5,𝔽2)—4128no—
237C2×#41—216no—
238C2×#41—216no—
239C2×#41—216no—
240C2×#41—28no—
241C2×#41—248no—
242C2×#41—248no—
243C2×#41—248no—
244C2×#41—412no—
245C2×#41—424no—
246C2×#41—4128no—
247C2×#41C2×D4132yescomplemented
248C22×Q8—216no—
249C22×Q8—28no—
250C22×Q8C2416yes—
251C22×Q8C2412yes—
252C22×Q8C2411yescharacteristic
253C22×D4—248no—
254C22×D4—248no—
255C22×D4—248no—
256C22×D4—296no—
257C22×D4—248no—
258C22×D4—412no—
259C22×D4—424no—
260C22×D4—412no—
261C22×D4—448no—
262C22×D4—824no—
263C22×D4C2416yes—
264C22×D4C24112yes—
265C22×D4C24112yes—
266C22×D4C24112yes—
267C22×D4C2416yes—
268C22×D4C24112yes—
269C22×D4C24112yes—
270C22×D4C24112yes—
271C22×D4C2413yes—
272C22×D4C24124yes—
273C22×D4C2416yes—
274C22×D4C2416yes—
275C22×D4C24112yes—
276C22×D4C2411yescharacteristic
277C23×C4—248no—
278C23×C4—412no—
279C23×C4C2416yes—
280C23×C4C2416yes—
281C23×C4C24112yes—
282C23×C4C2412yes—
283C23×C4C24112yes—
284C23×C4C24124yes—
285C4⋊Q8—46no—
286#127—248no—
287#127—212no—
288#127—46no—
289#127C2×D4148yescomplemented
290#124—248no—
291#124—248no—
292#123—248no—
293#123—248no—
294C22⋊Q8—248no—
295C22⋊Q8—216no—
296C22⋊Q8—248no—
297C22⋊Q8—248no—
298C22⋊Q8—412no—
299#121—296no—
300#121—248no—
301#121—248no—
302#121—296no—
303#121—248no—
304#121—248no—
305#121—248no—
306#121—248no—
307#121—296no—
308#121—448no—
309#121—412no—
310#121C2×D4196yescomplemented
311#120—248no—
312#120—296no—
313#120—296no—
314#120—224no—
315#120—248no—
316#120—424no—
317#120—824no—
318C4×Q8—216no—
319C4×Q8—24no—
320C4×Q8C2×D4116yescomplemented
321C4×D4—248no—
322C4×D4—248no—
323C4×D4—248no—
324C4×D4—248no—
325C4×D4—248no—
326C4×D4—424no—
327C4×D4—46no—
328C4×D4—46no—
329C4×D4—812no—
330C4×D4C2×D4148yescomplemented
331#117—248no—
332C2×C4⋊C4—248no—
333C2×C4⋊C4C24112yes—
334C2×C4⋊C4C24112yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2414yes—
337C2×C4⋊C4C24112yes—
338C2×C4⋊C4C24112yes—
339C2×C4⋊C4C2413yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2416yes—
342C2×C4⋊C4C24112yes—
343C2×C4⋊C4C2413yes—
344C2×C4⋊C4C2416yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2416yes—
347C2×C4⋊C4C2412yes—
348C2×C22⋊C4—296no—
349C2×C22⋊C4C24112yes—
350C2×C22⋊C4C24124yes—
351C2×C22⋊C4C24124yes—
352C2×C22⋊C4C24112yes—
353C2×C22⋊C4C24112yes—
354C2×C22⋊C4C2414yes—
355C2×C22⋊C4C2414yes—
356C2×C22⋊C4C24112yes—
357C2×C22⋊C4C24112yes—
358C2×C22⋊C4C24112yes—
359C2×C22⋊C4C24112yes—
360C2×C22⋊C4C2416yes—
361C2×C22⋊C4C24124yes—
362C2×C22⋊C4C2416yes—
363C2×C22⋊C4C2416yes—
364C2×C22⋊C4C24112yes—
365C2×C22⋊C4C24112yes—
366C2×C22⋊C4C24112yes—
367C2×C22⋊C4C2412yes—
368C2×C22⋊C4C24112yes—
369C2×C42C24112yes—
370C2×C42C2413yes—
371C2×C42C2413yes—
372C2×C42C2416yes—
373C2×C42C2411yescharacteristic
374C2×C42C2416yes—
375#95C2414yes—
376#95C2412yes—
377#95C2416yes—
378#95C24112yes—
379#95C24112yes—
380#95C2416yes—
381#95C24112yes—
382#95C24112yes—
383C24—224no—
384C24—212no—
385C24—248no—
386C24—46no—
387C24—412no—
388C24—412no—
389C24—46no—
390C24—412no—
391C24—424no—
392C24—412no—
393C24C2516yes—
394C24C25112yes—
395C24C2512yes—
396#41—2128no—
397#41—496no—
398#41—4192no—
399#41—432no—
400C2×Q8—28no—
401C2×Q8—224no—
402C2×Q8—212no—
403C2×Q8—42no—
404C2×Q8—432no—
405C2×Q8C22×D418yes—
406C2×Q8H(5,𝔽2)14yes—
407C2×D4—224no—
408C2×D4—248no—
409C2×D4—224no—
410C2×D4—224no—
411C2×D4—224no—
412C2×D4—224no—
413C2×D4—224no—
414C2×D4—248no—
415C2×D4—224no—
416C2×D4—224no—
417C2×D4—224no—
418C2×D4—224no—
419C2×D4—224no—
420C2×D4—212no—
421C2×D4—26no—
422C2×D4—4192no—
423C2×D4—41no—
424C2×D4—412no—
425C2×D4—46no—
426C2×D4—412no—
427C2×D4—46no—
428C2×D4—412no—
429C2×D4—412no—
430C2×D4—412no—
431C2×D4—412no—
432C2×D4—46no—
433C2×D4—41no—
434C2×D4—43no—
435C2×D4—424no—
436C2×D4—412no—
437C2×D4—424no—
438C2×D4—43no—
439C2×D4—412no—
440C2×D4—412no—
441C2×D4—46no—
442C2×D4—496no—
443C2×D4—496no—
444C2×D4—8192no—
445C2×D4C22×D4148yes—
446C2×D4C22×D4124yes—
447C2×D4C22×D4148yes—
448C2×D4H(5,𝔽2)14yescomplemented
449C22×C4—224no—
450C22×C4—224no—
451C22×C4—224no—
452C22×C4—224no—
453C22×C4—224no—
454C22×C4—224no—
455C22×C4—28no—
456C22×C4—212no—
457C22×C4—212no—
458C22×C4—248no—
459C22×C4—42no—
460C22×C4—412no—
461C22×C4—496no—
462C22×C4—424no—
463C22×C4—424no—
464C22×C4C22×D4148yes—
465C22×C4C2514yes—
466C22×C4C2513yes—
467C22×C4C25112yes—
468C22×C4C2516yes—
469C22×C4C2512yes—
470C22×C4C2516yes—
471C22×C4C2514yes—
472C22×C4C2512yes—
473C22×C4C2511yescharacteristic
474C22×C4C2513yes—
475C4⋊C4—224no—
476C4⋊C4—224no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—224no—
480C4⋊C4—224no—
481C4⋊C4—26no—
482C4⋊C4—42no—
483C4⋊C4—41no—
484C4⋊C4—412no—
485C4⋊C4—43no—
486C4⋊C4—46no—
487C4⋊C4—46no—
488C4⋊C4C22×D4124yes—
489C4⋊C4C22×D418yes—
490C22⋊C4—248no—
491C22⋊C4—224no—
492C22⋊C4—224no—
493C22⋊C4—212no—
494C22⋊C4—224no—
495C22⋊C4—248no—
496C22⋊C4—224no—
497C22⋊C4—224no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—224no—
501C22⋊C4—224no—
502C22⋊C4—42no—
503C22⋊C4—46no—
504C22⋊C4—46no—
505C22⋊C4—424no—
506C22⋊C4—412no—
507C22⋊C4—412no—
508C22⋊C4—412no—
509C22⋊C4C22×D4148yes—
510C42—224no—
511C42—26no—
512C42—26no—
513C42—41no—
514C42—46no—
515C42C22×D4124yes—
516C23—24no—
517C23—212no—
518C23—212no—
519C23—22no—
520C23—212no—
521C23—212no—
522C23—212no—
523C23—24no—
524C23—22no—
525C23—212no—
526C23—212no—
527C23—26no—
528C23—26no—
529C23—212no—
530C23—26no—
531C23—424no—
532C23—448no—
533C23—448no—
534C23—824no—
535C23—812no—
536C23—848no—
537C23—848no—
538C23—812no—
539C23—1612no—
540C23—1612no—
541C23C23×D4124yes—
542C23C2×H(5,𝔽2)14yes—
543C23C2611yescharacteristic
544Q8—48no—
545Q8—424no—
546Q8D42116yescomplemented
547D4—296no—
548D4—448no—
549D4—424no—
550D4—44no—
551D4—448no—
552D4—448no—
553D4—424no—
554D4—448no—
555D4—448no—
556D4—424no—
557D4—86no—
558D4—824no—
559D4—848no—
560D4—166no—
561C2×C4—24no—
562C2×C4—21no—
563C2×C4—26no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—26no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—296no—
570C2×C4—24no—
571C2×C4—212no—
572C2×C4—212no—
573C2×C4—24no—
574C2×C4—23no—
575C2×C4—212no—
576C2×C4—22no—
577C2×C4—26no—
578C2×C4—26no—
579C2×C4—24no—
580C2×C4—212no—
581C2×C4—24no—
582C2×C4—48no—
583C2×C4—448no—
584C2×C4—424no—
585C2×C4—424no—
586C2×C4—848no—
587C2×C4C23×D414yes—
588C2×C4C23×D4112yes—
589C2×C4C23×D418yes—
590C2×C4C23×D4112yes—
591C2×C4C2×H(5,𝔽2)12yes—
592C2×C4C2×H(5,𝔽2)16yes—
593C22—28no—
594C22—224no—
595C22—224no—
596C22—24no—
597C22—224no—
598C22—412no—
599C22—42no—
600C22—412no—
601C22—44no—
602C22—412no—
603C22—412no—
604C22—42no—
605C22—412no—
606C22—46no—
607C22—412no—
608C22—46no—
609C22—46no—
610C22—848no—
611C22—1648no—
612C22C24×D412yes—
613C22C22×H(5,𝔽2)11yescharacteristic
614C22C22×H(5,𝔽2)11yescharacteristic
615C22H(7,𝔽2)12yes—
616C222−1+611yescharacteristic
617C4—212no—
618C4—24no—
619C4—22no—
620C4—28no—
621C4—412no—
622C4—43no—
623C4—44no—
624C4—42no—
625C4—46no—
626C4C2×D42112yes—
627C2—44no—
628C2—424no—
629C2—86no—
630C2C22×D4211yescharacteristic
631C2#6285312yes—
632C2#6289511yescharacteristic
633C2D4×H(5,𝔽2)12yes—
634C2#6306611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#3488C2semidirect
2. #59816C2semidirect
3. #58997C2semidirect
4. #52103C2semidirect
5. #45103C2semidirect
6. #44877C2semidirect
7. #39355C2semidirect
8. #37648C2semidirect
9. #36465C2semidirect
10. #36428C2semidirect
11. #36091C2semidirect
12. #35831C2semidirect
13. #34313C2semidirect
14. C22×H(5,𝔽2)C22semidirect
15. C23×#41C22semidirect
18. C2×#550C22semidirect
19. C2×#550C22semidirect
20. C2×D4×Q8C22semidirect
21. C2×#548C22semidirect
22. C2×#548C22semidirect
23. C2×#546C22semidirect
24. C2×#546C22semidirect
25. C2×D42C22semidirect
26. C2×D42C22semidirect
27. C2×D42C22semidirect
28. C2×#537C22semidirect
29. C2×#535C22semidirect
30. C2×#525C22semidirect
31. C22×#121C22semidirect
32. C2×#520C22semidirect
33. #2858C22semidirect
34. #2848C22semidirect
35. #2683C22semidirect
36. #2682C22semidirect
37. #2682C22semidirect
38. #2681C22semidirect
39. #2680C22semidirect
40. #2679C22semidirect
41. #2679C22semidirect
42. #2679C22semidirect
43. #2678C22semidirect
44. #2675C22semidirect
45. #2671C22semidirect
46. #2669C22semidirect
47. #2668C22semidirect
48. #2620C22semidirect
49. #2615C22semidirect
50. #2544C22semidirect
51. #2542C22semidirect
52. #2541C22semidirect
53. #2540C22semidirect
54. #2537C22semidirect
55. #2537C22semidirect
56. #2437C22semidirect
57. #2435C22semidirect
58. #2433C22semidirect
59. #2433C22semidirect
60. #2432C22semidirect
61. #2428C22semidirect
62. #2427C22semidirect
63. #2424C22semidirect
64. #2424C22semidirect
65. #2417C22semidirect
66. #2416C22semidirect
67. #2410C22semidirect
68. #2408C22semidirect
69. #2408C22semidirect
70. #2408C22semidirect
71. #2408C22semidirect
72. #2407C22semidirect
73. #2403C22semidirect
74. #2392C22semidirect
75. #2388C22semidirect
76. #2366C22semidirect
77. #2361C22semidirect
78. Q8×C4⋊C4C22semidirect
79. D4×C4⋊C4C22semidirect
80. #2341C22semidirect
81. #2341C22semidirect
82. Q8×C22⋊C4C22semidirect
83. #2324C22semidirect
84. C4×#127C22semidirect
85. C4×#127C22semidirect
86. C4×#120C22semidirect
87. #2293C22semidirect
88. C42×Q8C22semidirect
105. #550D4semidirect
247. C2×#41C2×D4semidirect
289. #127C2×D4semidirect
310. #121C2×D4semidirect
320. C4×Q8C2×D4semidirect
330. C4×D4C2×D4semidirect
448. C2×D4H(5,𝔽2)semidirect
546. Q8D42semidirect

Maximal subgroups

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

subgroupindex
C2×#34882
#598162
#589972
#521032
#451032
#448772
#393552
#376482
#364652
#364282
#360912
#358312
#343132