Finite Group Databasebeta

← #8196701

Subgroups — #8196701

Order 512.

5068 conjugacy classes of subgroups.

The subgroup lattice has 653 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59628C214yesmaximal, complemented
2#59248C212yesmaximal, complemented
3#59143C212yesmaximal, complemented
4#56508C218yesmaximal, complemented
5#52990C214yesmaximal, complemented
6#52054C214yesmaximal, complemented
7#46998C214yesmaximal, complemented
8#45683C214yesmaximal, complemented
9#45380C214yesmaximal, complemented
10#42523C218yesmaximal, complemented
11#39020C218yesmaximal, complemented
12#38941C214yesmaximal, complemented
13#36580C214yesmaximal, complemented
14#35272C211yesmaximal, characteristic
15#34971C212yesmaximal, complemented
16C2×#559C2212yes—
17C2×#559C2214yescomplemented
18C2×#535C2212yes—
19C22×#127C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C22×C4×Q8C2212yescomplemented
22#2856C2214yes—
23#2850C2218yescomplemented
24#2840C2214yes—
25#2832C22116yescomplemented
26#2824C2218yescomplemented
27#2789C2218yescomplemented
28#2778C22116yescomplemented
29#2760C2218yescomplemented
30#2746C22116yescomplemented
31#2741C22116yescomplemented
32#2737C22116yescomplemented
33#2737C22116yescomplemented
34#2736C2218yes—
35#2736C2218yes—
36#2717C2218yescomplemented
37#2712C2218yescomplemented
38#2709C2218yes—
39#2709C2218yescomplemented
40#2686C22116yescomplemented
41#2684C22116yes—
42#2653C22116yescomplemented
43#2640C2218yescomplemented
44#2632C2218yescomplemented
45#2619C2214yes—
46#2600C2218yescomplemented
47#2599C2218yes—
48#2599C2214yes—
49#2599C2214yescomplemented
50#2592C2218yescomplemented
51#2582C2212yescomplemented
52#2581C2214yescomplemented
53#2580C2218yescomplemented
54#2563C2218yescomplemented
55#2551C2212yescomplemented
56#2548C2214yes—
57#2548C2214yes—
58#2544C2218yes—
59#2541C2214yes—
60#2536C2214yes—
61#2514C2218yescomplemented
62#2513C2218yescomplemented
63#2494C2218yes—
64#2492C2218yes—
65#2491C2218yescomplemented
66#2491C2218yescomplemented
67#2486C2212yes—
68#2486C2212yescomplemented
69#2484C2214yescomplemented
70#2483C2214yes—
71#2459C22116yescomplemented
72#2458C22116yescomplemented
73#2458C22116yescomplemented
74#2448C2218yescomplemented
75#2445C2218yes—
76#2443C22116yescomplemented
77#2436C2218yescomplemented
78#2435C2218yes—
79#2427C2218yescomplemented
80#2420C2218yescomplemented
81#2418C22116yes—
82#2409C2212yes—
83#2409C2212yescomplemented
84#2408C2214yes—
85#2408C2214yes—
86#2406C2214yes—
87#2405C2218yes—
88#2399C2214yescomplemented
89#2398C2214yes—
90#2380C2214yescomplemented
91#2367C2218yescomplemented
92#2366C2212yes—
93#2365C2218yescomplemented
94#2361C2218yescomplemented
95#2355C2214yescomplemented
96D4×C22⋊C4C2218yes—
97#2319C2218yescomplemented
98#2312C2218yes—
99#2309C2212yes—
100C4×C22⋊Q8C2214yes—
101C4×C22⋊Q8C2214yescomplemented
102C4×#123C2214yes—
103C4×#123C2214yescomplemented
104C4×#121C2218yescomplemented
105#2285C2211yescharacteristic
106#2283C2212yescomplemented
107C22×#41C2311yescharacteristic
108C23×Q8C2312yes—
109C23×D4C2312yes—
110C23×D4C2314yes—
111#559—48no—
112#559—416no—
113#535—48no—
114C2×C4⋊Q8C2318yes—
115C2×#127—44no—
116C2×#127C2312yes—
117C2×#127C2314yes—
118C2×#127C2312yes—
119C2×#127C2314yes—
120C2×#124C2318yes—
121C2×#124C2318yes—
122C2×#124C2314yes—
123C2×#124C23116yes—
124C2×#123C2314yes—
125C2×#123C2318yes—
126C2×#123C2318yes—
127C2×#123C2314yes—
128C2×C22⋊Q8C2314yes—
129C2×C22⋊Q8C2314yes—
130C2×#121C2318yes—
131C2×#121C2318yes—
132C2×#121C2318yes—
133C2×#120C2318yes—
134C4×#41—44no—
135C2×C4×Q8—48no—
136C2×C4×Q8C2314yes—
137C2×C4×Q8C2314yes—
138C2×C4×Q8C2312yes—
139C2×C4×Q8C2314yes—
140C2×C4×Q8C2314yes—
141C2×C4×Q8C2318yes—
142C2×C4×D4C2318yes—
143C2×C4×D4C2318yes—
144C2×C4×D4C23116yes—
145C2×C4×D4C2312yes—
146C2×C4×D4C2312yes—
147C2×C4×D4C2312yes—
148C2×#117C2311yescharacteristic
149C2×#117C2312yes—
150C2×#117C2312yes—
151C22×C4⋊C4C2312yes—
152C22×C4⋊C4C2314yes—
153C22×C22⋊C4C2318yes—
154C22×C42C2312yes—
155C22×C42C2311yescharacteristic
156C22×C42C2314yes—
157#401C23116yes—
158#400C23116yes—
159#400C2318yes—
160#399C23116yes—
161#398C2318yes—
162#398C2314yes—
163#398C2314yes—
164#398C2318yes—
165#397C23116yes—
166#397C23116yes—
167#397C23116yes—
168#397C2314yes—
169#397C2318yes—
170#397C2314yes—
171#397C23116yes—
172#395C2314yes—
173#395C2318yes—
174#395C2314yes—
175#395C2318yes—
176#394C23116yes—
177#394C23116yes—
178#394C2318yes—
179#394C23116yes—
180#394C23116yes—
181#394C2318yes—
182#394C23116yes—
183#394C2318yes—
184#394C23116yes—
185#393C23116yes—
186#393C23116yes—
187#393C2314yes—
188#393C23116yes—
189#393C23116yes—
190#393C23116yes—
191#393C2314yes—
192#393C2318yes—
193#392C2318yes—
194#392C2318yes—
195#391C23116yes—
196#391C2314yes—
197#391C2314yes—
198#391C2318yes—
199#391C2318yes—
200#390C2318yes—
201#390C2318yes—
202#390C2318yes—
203#390C2318yes—
204#390C2318yes—
205#390C2318yes—
206#390C2318yes—
207#390C2318yes—
208#389C2314yes—
209#389C2318yes—
210#389C2318yes—
211#389C2314yes—
212#388C2318yes—
213#388C23116yes—
214#388C23116yes—
215#388C23116yes—
216#388C23116yes—
217#388C23116yes—
218#388C23116yes—
219#388C23116yes—
220#388C23116yes—
221#388C2318yes—
222#388C23116yes—
223#387C2318yes—
224#387C2318yes—
225#387C23116yes—
226#387C23116yes—
227#387C2318yes—
228#387C2318yes—
229#387C23116yes—
230#386C23116yes—
231#386C23116yes—
232#386C23116yes—
233#386C23116yes—
234#386C2318yes—
235#386C2314yes—
236#386C2318yes—
237#386C23116yes—
238#386C2314yes—
239#386C23116yes—
240#385C2314yes—
241#385C23116yes—
242#385C23116yes—
243#385C2318yes—
244#385C2318yes—
245#385C2314yes—
246#384C2312yes—
247#383C2314yes—
248#383C2318yes—
249#382C2314yes—
250#382C2318yes—
251#382C2312yes—
252#381C23116yes—
253#381C2314yes—
254#381C2314yes—
255#381C2314yes—
256#380C23116yes—
257#380C23116yes—
258#380C2314yes—
259#380C2314yes—
260#380C2312yes—
261#380C2312yes—
262#380C2314yes—
263#379C2314yes—
264#379C2314yes—
265#379C2318yes—
266#379C23116yes—
267C4×C4⋊C4C2318yes—
268C4×C4⋊C4C2318yes—
269C4×C4⋊C4C2318yes—
270C4×C4⋊C4C2314yes—
271C4×C4⋊C4C2314yes—
272C4×C4⋊C4C2318yes—
273C4×C22⋊C4C23116yes—
274C4×C22⋊C4C2318yes—
275C4×C22⋊C4C2318yes—
276C4×C22⋊C4C2314yes—
277#376C2314yes—
278C25C2414yes—
279C2×#41—24no—
280C2×#41—42no—
281C2×#41—42no—
282C22×Q8—44no—
283C22×Q8—48no—
284C22×Q8C2414yes—
285C22×Q8C2411yescharacteristic
286C22×Q8C2414yes—
287C22×Q8C2418yes—
288C22×D4—44no—
289C22×D4—48no—
290C22×D4—44no—
291C22×D4—416no—
292C22×D4—44no—
293C22×D4C2418yes—
294C22×D4C2412yes—
295C22×D4C24116yes—
296C22×D4C2414yes—
297C22×D4C2414yes—
298C22×D4C2418yes—
299C22×D4C2411yescharacteristic
300C23×C4C2412yes—
301C23×C4C2411yescharacteristic
302C23×C4C2412yes—
303C23×C4C2412yes—
304C23×C4C2412yes—
305C23×C4C2414yes—
306C23×C4C2418yes—
307C23×C4C2418yes—
308C23×C4C2414yes—
309C4⋊Q8—416no—
310#127—216no—
311#127—28no—
312#127—44no—
313#127—48no—
314#124—416no—
315#124—48no—
316#124—416no—
317#124—432no—
318#123—216no—
319#123—48no—
320#123—416no—
321#123—416no—
322C22⋊Q8—48no—
323C22⋊Q8—48no—
324#121—416no—
325#121—416no—
326#121—416no—
327#120—416no—
328C4×Q8—216no—
329C4×Q8—48no—
330C4×Q8—44no—
331C4×Q8—48no—
332C4×Q8—416no—
333C4×Q8—48no—
334C4×D4—416no—
335C4×D4—416no—
336C4×D4—44no—
337C4×D4—44no—
338C4×D4—44no—
339C4×D4—432no—
340#117—42no—
341#117—44no—
342#117—44no—
343C2×C4⋊C4—44no—
344C2×C4⋊C4—48no—
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2418yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2418yes—
349C2×C4⋊C4C2418yes—
350C2×C4⋊C4C2414yes—
351C2×C4⋊C4C2414yes—
352C2×C4⋊C4C2418yes—
353C2×C4⋊C4C2412yes—
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2414yes—
357C2×C4⋊C4C2418yes—
358C2×C4⋊C4C2418yes—
359C2×C4⋊C4C2418yes—
360C2×C4⋊C4C2418yes—
361C2×C4⋊C4C24116yes—
362C2×C22⋊C4—416no—
363C2×C22⋊C4C2418yes—
364C2×C22⋊C4C2418yes—
365C2×C22⋊C4C2414yes—
366C2×C22⋊C4C24116yes—
367C2×C22⋊C4C2411yescharacteristic
368C2×C22⋊C4C2411yescharacteristic
369C2×C22⋊C4C2412yes—
370C2×C22⋊C4C2412yes—
371C2×C22⋊C4C2414yes—
372C2×C22⋊C4C24116yes—
373C2×C22⋊C4C24116yes—
374C2×C22⋊C4C24116yes—
375C2×C22⋊C4C24116yes—
376C2×C22⋊C4C24116yes—
377C2×C22⋊C4C24116yes—
378C2×C22⋊C4C2418yes—
379C2×C22⋊C4C2418yes—
380C2×C22⋊C4C2418yes—
381C2×C22⋊C4C2418yes—
382C2×C22⋊C4C2414yes—
383C2×C22⋊C4C24116yes—
384C2×C22⋊C4C24116yes—
385C2×C42—28no—
386C2×C42—24no—
387C2×C42—48no—
388C2×C42C2412yes—
389C2×C42C2411yescharacteristic
390C2×C42C2414yes—
391C2×C42C2412yes—
392C2×C42C2412yes—
393C2×C42C2418yes—
394C2×C42C2412yes—
395C2×C42C2418yes—
396C2×C42C24116yes—
397C2×C42C2418yes—
398C2×C42C2411yescharacteristic
399#95C2418yes—
400#95C2414yes—
401#95C2414yes—
402#95C2418yes—
403#95C2418yes—
404#95C2418yes—
405#95C2418yes—
406#95C2418yes—
407#95C24116yes—
408#95C2418yes—
409#95C24116yes—
410#95C2418yes—
411#95C2418yes—
412#95C2418yes—
413#95C2418yes—
414#95C2418yes—
415#95C24116yes—
416#95C24116yes—
417#95C2418yes—
418#95C2418yes—
419C24—28no—
420C24—44no—
421C24—416no—
422C24—44no—
423C24C2518yes—
424C24C2512yes—
425C24C2511yescharacteristic
426#41—416no—
427C2×Q8—28no—
428C2×Q8—22no—
429C2×Q8—28no—
430C2×Q8—28no—
431C2×Q8—44no—
432C2×Q8—41no—
433C2×Q8—48no—
434C2×Q8—41no—
435C2×Q8—48no—
436C2×Q8—416no—
437C2×Q8—432no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—232no—
441C2×D4—216no—
442C2×D4—28no—
443C2×D4—28no—
444C2×D4—216no—
445C2×D4—216no—
446C2×D4—41no—
447C2×D4—42no—
448C2×D4—41no—
449C2×D4—44no—
450C2×D4—416no—
451C2×D4—416no—
452C2×D4—44no—
453C2×D4—48no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—416no—
457C2×D4—816no—
458C2×D4—832no—
459C2×D4H(5,𝔽2)14yes—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—28no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—28no—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—416no—
482C22×C4C2512yes—
483C22×C4C2518yes—
484C22×C4C2518yes—
485C22×C4C2512yes—
486C22×C4C2518yes—
487C22×C4C2518yes—
488C22×C4C2512yes—
489C22×C4C2518yes—
490C22×C4C2512yes—
491C22×C4C2511yescharacteristic
492C22×C4C2511yescharacteristic
493C22×C4C2512yes—
494C4⋊C4—28no—
495C4⋊C4—216no—
496C4⋊C4—216no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—28no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C22⋊C4—216no—
512C22⋊C4—28no—
513C22⋊C4—216no—
514C22⋊C4—416no—
515C22⋊C4—41no—
516C22⋊C4—42no—
517C22⋊C4—41no—
518C22⋊C4—42no—
519C22⋊C4—44no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—416no—
523C22⋊C4—416no—
524C22⋊C4—416no—
525C22⋊C4—416no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—416no—
532C22⋊C4—416no—
533C42—24no—
534C42—22no—
535C42—24no—
536C42—24no—
537C42—24no—
538C42—216no—
539C42—28no—
540C42—216no—
541C42—48no—
542C42—416no—
543C42H(5,𝔽2)14yes—
544C23—21no—
545C23—22no—
546C23—21no—
547C23—24no—
548C23—24no—
549C23—22no—
550C23—24no—
551C23—28no—
552C23—28no—
553C23—216no—
554C23—28no—
555C23—216no—
556C23—88no—
557C23—816no—
558C23—816no—
559C23—168no—
560C23C2×H(5,𝔽2)14yes—
561C23#58512yes—
562C23C2611yescharacteristic
563Q8—44no—
564Q8—416no—
565Q8—416no—
566Q8—816no—
567D4—44no—
568D4—48no—
569D4—432no—
570D4—88no—
571D4—88no—
572D4—832no—
573D4—816no—
574C2×C4—28no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—22no—
589C2×C4—28no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—216no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—48no—
603C2×C4—416no—
604C2×C4—48no—
605C2×C4—48no—
606C2×C4—48no—
607C2×C4—88no—
608C2×C4—168no—
609C2×C4—168no—
610C2×C4C2×H(5,𝔽2)12yes—
611C2×C4C2×H(5,𝔽2)12yes—
612C2×C4C2×H(5,𝔽2)14yes—
613C2×C4#58518yes—
614C22—28no—
615C22—24no—
616C22—22no—
617C22—24no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—416no—
624C22—416no—
625C22—48no—
626C22—48no—
627C22—48no—
628C22—1616no—
629C22C22×H(5,𝔽2)11yescharacteristic
630C22C22×H(5,𝔽2)11yescharacteristic
631C22C2×#58514yes—
632C22C2×#58511yescharacteristic
633C4—22no—
634C4—22no—
635C4—42no—
636C4—42no—
637C4—42no—
638C4—48no—
639C4—48no—
640C4—48no—
641C4—48no—
642C4—42no—
643C4—48no—
644C4#354018yes—
645C2—44no—
646C2—42no—
647C2—88no—
648C2C2×#354012yes—
649C2#6281811yescharacteristic
650C2#6302112yes—
651C2#6303012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC23C26
prime radical#35272C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ G
LECSG ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59628C2semidirect
2. #59248C2semidirect
3. #59143C2semidirect
4. #56508C2semidirect
5. #52990C2semidirect
6. #52054C2semidirect
7. #46998C2semidirect
8. #45683C2semidirect
9. #45380C2semidirect
10. #42523C2semidirect
11. #39020C2semidirect
12. #38941C2semidirect
13. #36580C2semidirect
15. #34971C2semidirect
17. C2×#559C22semidirect
21. C22×C4×Q8C22semidirect
23. #2850C22semidirect
25. #2832C22semidirect
26. #2824C22semidirect
27. #2789C22semidirect
28. #2778C22semidirect
29. #2760C22semidirect
30. #2746C22semidirect
31. #2741C22semidirect
32. #2737C22semidirect
33. #2737C22semidirect
36. #2717C22semidirect
37. #2712C22semidirect
39. #2709C22semidirect
40. #2686C22semidirect
42. #2653C22semidirect
43. #2640C22semidirect
44. #2632C22semidirect
46. #2600C22semidirect
49. #2599C22semidirect
50. #2592C22semidirect
51. #2582C22semidirect
52. #2581C22semidirect
53. #2580C22semidirect
54. #2563C22semidirect
55. #2551C22semidirect
61. #2514C22semidirect
62. #2513C22semidirect
65. #2491C22semidirect
66. #2491C22semidirect
68. #2486C22semidirect
69. #2484C22semidirect
71. #2459C22semidirect
72. #2458C22semidirect
73. #2458C22semidirect
74. #2448C22semidirect
76. #2443C22semidirect
77. #2436C22semidirect
79. #2427C22semidirect
80. #2420C22semidirect
83. #2409C22semidirect
88. #2399C22semidirect
90. #2380C22semidirect
91. #2367C22semidirect
93. #2365C22semidirect
94. #2361C22semidirect
95. #2355C22semidirect
97. #2319C22semidirect
101. C4×C22⋊Q8C22semidirect
103. C4×#123C22semidirect
104. C4×#121C22semidirect
106. #2283C22semidirect

Maximal subgroups

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

subgroupindex
#596282
#592482
#591432
#565082
#529902
#520542
#469982
#456832
#453802
#425232
#390202
#389412
#365802
#352722
#349712