Finite Group Databasebeta

← #5486624

Subgroups — #5486624

Order 512.

3052 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C211yesmaximal, characteristic, complemented
2#17255C214yesmaximal, complemented
3#16797C214yesmaximal, complemented
4#16283C214yesmaximal, complemented
5#16005C214yesmaximal, complemented
6#15948C211yesmaximal, characteristic, complemented
7#15376C214yesmaximal, complemented
8#15261C211yesmaximal, characteristic, complemented
9#15258C211yesmaximal, characteristic
10#14718C214yesmaximal, complemented
11#14661C211yesmaximal, characteristic, complemented
12#14658C211yesmaximal, characteristic, complemented
13#14134C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#123C2211yescharacteristic
16C22×#120C2211yescharacteristic
17C22×#120C2212yes—
18C23×C22⋊C4C2212yes—
19C2×#394C2214yes—
20C2×#394C2214yes—
21C2×#394C2212yescomplemented
22C2×#392C2214yes—
23C2×#392C2212yes—
24C2×#386C2212yes—
25C2×#386C2214yes—
26C2×#386C2214yes—
27C2×#386C2214yes—
28C2×#386C2212yescomplemented
29C2×#385C2214yes—
30C2×#383C2212yescomplemented
31C2×#380C2212yescomplemented
32C2×#379C2212yes—
33#1445C2214yescomplemented
34#1443C2214yescomplemented
35#1443C2214yescomplemented
36#1442C2214yescomplemented
37#1442C2214yescomplemented
38#1441C2214yescomplemented
39#1440C2214yes—
40#1440C2212yes—
41#1440C2212yes—
42#1440C2214yes—
43#1440C2212yescomplemented
44#1440C2214yescomplemented
45#1440C2214yescomplemented
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1438C2211yescharacteristic
53#1438C2211yescharacteristic
54#1438C2211yescharacteristic
55#1438C2211yescharacteristic
56#1438C2214yes—
57#1438C2212yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2214yescomplemented
61#1438C2212yescomplemented
62#1438C2212yescomplemented
63#1437C2212yes—
64#1436C2212yescomplemented
65C22⋊#95C2214yes—
66C22⋊#95C2214yes—
67C26C2311yescharacteristic
68C23×D4C2312yes—
69C23×D4C2314yes—
70C23×D4C2312yes—
71C23×D4C2312yes—
72C23×D4C2311yescharacteristic
73C23×D4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2314yes—
76C2×#127—24no—
77C2×#127—44no—
78C2×#123—24no—
79C2×#123—44no—
80C2×#120—24no—
81C2×#120—44no—
82C2×#120—44no—
83C2×#120—48no—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2314yes—
87C22×C22⋊C4—48no—
88C22×C22⋊C4—44no—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2311yescharacteristic
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C42C2311yescharacteristic
111#394—44no—
112#394—48no—
113#394—48no—
114#392—48no—
115#392—44no—
116#386—44no—
117#386—48no—
118#386—44no—
119#386—48no—
120#386—48no—
121#385—48no—
122#383—44no—
123#380—44no—
124#379—44no—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2312yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2312yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C25—22no—
145C25—24no—
146C25—28no—
147C25—44no—
148C25—42no—
149C25—42no—
150C25C2412yes—
151C25C2411yescharacteristic
152C25C2411yescharacteristic
153C25C2412yes—
154C22×D4—24no—
155C22×D4—28no—
156C22×D4—24no—
157C22×D4—28no—
158C22×D4—24no—
159C22×D4—24no—
160C22×D4—28no—
161C22×D4—28no—
162C22×D4—22no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—42no—
166C22×D4—44no—
167C22×D4—42no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—42no—
171C22×D4—42no—
172C22×D4—42no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—44no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—44no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—42no—
182C22×D4—42no—
183C22×D4#4114yes—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—28no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204#127—416no—
205#123—416no—
206#120—416no—
207#120—432no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—42no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—432no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C42—22no—
272C2×C42—24no—
273#95—28no—
274#95—28no—
275#95—28no—
276#95—44no—
277#95—44no—
278#95—42no—
279#95—44no—
280#95—42no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—42no—
289#95—42no—
290#95—44no—
291#95—44no—
292C24—22no—
293C24—24no—
294C24—24no—
295C24—22no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—28no—
300C24—22no—
301C24—22no—
302C24—24no—
303C24—22no—
304C24—24no—
305C24—22no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—22no—
311C24—24no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—24no—
316C24—22no—
317C24—24no—
318C24—24no—
319C24—416no—
320C24—416no—
321C24—48no—
322C24—48no—
323C24—48no—
324C24—48no—
325C24—88no—
326C24—88no—
327C24—88no—
328C24—82no—
329C24—82no—
330C24—84no—
331C24—84no—
332C24C2×#4112yes—
333C24C2×#4112yes—
334C24H(5,𝔽2)14yes—
335C24H(5,𝔽2)18yes—
336C24C2511yescharacteristic
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—48no—
341C2×D4—48no—
342C2×D4—48no—
343C2×D4—44no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—416no—
347C2×D4—416no—
348C2×D4—416no—
349C2×D4—48no—
350C2×D4—416no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—84no—
354C2×D4—84no—
355C2×D4—88no—
356C2×D4—88no—
357C2×D4—84no—
358C2×D4—88no—
359C2×D4—88no—
360C2×D4—88no—
361C2×D4—88no—
362C2×D4—88no—
363C2×D4—88no—
364C2×D4—88no—
365C2×D4—88no—
366C2×D4—88no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—28no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
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—416no—
418C22×C4—88no—
419C22×C4—82no—
420C22×C4—82no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4—88no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4C2×#4112yes—
428C22×C4H(5,𝔽2)14yes—
429C22×C4H(5,𝔽2)14yes—
430C4⋊C4—48no—
431C4⋊C4—88no—
432C4⋊C4—88no—
433C22⋊C4—416no—
434C22⋊C4—48no—
435C22⋊C4—416no—
436C22⋊C4—48no—
437C22⋊C4—44no—
438C22⋊C4—44no—
439C22⋊C4—48no—
440C22⋊C4—48no—
441C22⋊C4—48no—
442C22⋊C4—416no—
443C22⋊C4—88no—
444C22⋊C4—88no—
445C22⋊C4—84no—
446C22⋊C4—84no—
447C22⋊C4—88no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—84no—
451C22⋊C4—84no—
452C22⋊C4—88no—
453C22⋊C4—84no—
454C42—28no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23—24no—
459C23—28no—
460C23—28no—
461C23—28no—
462C23—28no—
463C23—28no—
464C23—28no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—28no—
472C23—28no—
473C23—432no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23—42no—
478C23—42no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—42no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—42no—
498C23—44no—
499C23—44no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—832no—
526C23—88no—
527C23—816no—
528C23—816no—
529C23—832no—
530C23#56018yes—
531C23C22×#4111yescharacteristic
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)12yes—
535C23C2×H(5,𝔽2)14yes—
536C23#58514yes—
537D4—88no—
538D4—816no—
539D4—816no—
540D4—88no—
541D4—816no—
542D4—832no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—42no—
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—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—88no—
596C2×C4—832no—
597C22—216no—
598C22—28no—
599C22—48no—
600C22—48no—
601C22—44no—
602C22—48no—
603C22—44no—
604C22—48no—
605C22—48no—
606C22—44no—
607C22—44no—
608C22—44no—
609C22—44no—
610C22—42no—
611C22—42no—
612C22—44no—
613C22—44no—
614C22—832no—
615C22—84no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22—82no—
621C22—82no—
622C22—82no—
623C22—84no—
624C22—84no—
625C22—84no—
626C22—84no—
627C22—84no—
628C22#349011yescharacteristic
629C22#349011yescharacteristic
630C22#349011yescharacteristic
631C22#351314yes—
632C22C2×#56012yes—
633C22#353212yes—
634C22#353312yes—
635C22#353312yes—
636C22#353414yes—
637C22#353712yes—
638C22#353714yes—
639C22#354214yes—
640C22#355112yes—
641C22#355814yes—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—82no—
646C4—84no—
647C4—84no—
648C4—84no—
649C4—82no—
650C4—84no—
651C2—48no—
652C2—82no—
653C2—82no—
654C2—162no—
655C2#4553712yes—
656C2#4638411yescharacteristic
657C2#4968112yes—
658C2#5199312yes—
659C2#5231314yes—
660C2#5503514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15258C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17273C2semidirect
2. #17255C2semidirect
3. #16797C2semidirect
4. #16283C2semidirect
5. #16005C2semidirect
6. #15948C2semidirect
7. #15376C2semidirect
8. #15261C2semidirect
10. #14718C2semidirect
11. #14661C2semidirect
12. #14658C2semidirect
13. #14134C2semidirect
21. C2×#394C22semidirect
28. C2×#386C22semidirect
30. C2×#383C22semidirect
31. C2×#380C22semidirect
33. #1445C22semidirect
34. #1443C22semidirect
35. #1443C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
43. #1440C22semidirect
44. #1440C22semidirect
45. #1440C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
60. #1438C22semidirect
61. #1438C22semidirect
62. #1438C22semidirect
64. #1436C22semidirect

Maximal subgroups

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

subgroupindex
#172732
#172552
#167972
#162832
#160052
#159482
#153762
#152612
#152582
#147182
#146612
#146582
#141342