Finite Group Databasebeta

← #949250

Subgroups — #949250

Order 512.

3874 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
1#17273C211yesmaximal, characteristic, complemented
2#17272C212yesmaximal, complemented
3#17137C211yesmaximal, characteristic, complemented
4#16893C214yesmaximal, complemented
5#16320C214yesmaximal, complemented
6#15961C212yesmaximal, complemented
7#15506C212yesmaximal, complemented
8#15015C212yesmaximal, complemented
9#14717C214yesmaximal, complemented
10#14281C212yesmaximal, complemented
11#13877C214yesmaximal, complemented
12#13867C212yesmaximal
13C2×#1438C211yesmaximal, characteristic
14C24×D4C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C23×C22⋊C4C2214yes—
17C23×C22⋊C4C2212yes—
18C2×#393C2214yescomplemented
19C2×#393C2214yescomplemented
20C2×#386C2214yes—
21C2×#386C2214yes—
22C2×#385C2214yescomplemented
23C2×#385C2214yescomplemented
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×#379C2214yes—
27C2×#379C2212yes—
28C22×#95C2211yescharacteristic
29C22×#95C2211yescharacteristic
30#1446C2212yescomplemented
31#1446C2212yescomplemented
32#1445C2212yescomplemented
33#1445C2212yescomplemented
34#1445C2212yescomplemented
35#1445C2212yescomplemented
36#1443C2214yescomplemented
37#1443C2214yescomplemented
38#1442C2212yescomplemented
39#1442C2212yescomplemented
40#1441C2214yescomplemented
41#1441C2214yescomplemented
42#1439C2214yescomplemented
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1439C2214yescomplemented
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1439C2214yescomplemented
52#1438C2211yescharacteristic
53#1438C2211yescharacteristic
54#1438C2211yescharacteristic
55#1438C2211yescharacteristic
56#1438C2214yes—
57#1438C2212yes—
58#1438C2212yes—
59#1438C2212yes—
60#1438C2212yes—
61C22⋊#95C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2214yes—
64C22⋊#95C2212yes—
65C22⋊#95C2214yescomplemented
66C22⋊#95C2214yescomplemented
67C26C2312yes—
68C23×Q8C2312yes—
69C23×D4—42no—
70C23×D4—42no—
71C23×D4—48no—
72C23×D4Q818yescomplemented
73C23×D4C2311yescharacteristic
74C23×D4C2312yes—
75C23×D4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C24×C4C2314yes—
80C2×C22⋊Q8—28no—
81C2×C22⋊Q8—44no—
82C2×C22⋊Q8—44no—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2314yes—
87C22×C4⋊C4C2312yes—
88C22×C22⋊C4—216no—
89C22×C22⋊C4—48no—
90C22×C22⋊C4—48no—
91C22×C22⋊C4—48no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108#393—48no—
109#393—48no—
110#386—48no—
111#386—48no—
112#385—48no—
113#385—48no—
114#380—44no—
115#380—44no—
116#379—48no—
117#379—44no—
118C2×#95—42no—
119C2×#95—42no—
120C2×#95C2312yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2311yescharacteristic
139C2×#95C2312yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143C2×#95C2311yescharacteristic
144C25—28no—
145C25—28no—
146C25—24no—
147C25—28no—
148C25—24no—
149C25—44no—
150C25—44no—
151C25—44no—
152C25C2×Q818yes—
153C25C2412yes—
154C25C2412yes—
155C25C2411yescharacteristic
156C22×Q8—24no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—44no—
160C22×Q8—44no—
161C22×D4—28no—
162C22×D4—28no—
163C22×D4—416no—
164C22×D4—41no—
165C22×D4—41no—
166C22×D4—42no—
167C22×D4—42no—
168C22×D4—41no—
169C22×D4—41no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—432no—
173C22×D4—432no—
174C22×D4—432no—
175C22×D4C2×Q814yes—
176C22×D4C2×Q818yes—
177C22×D4C2×Q814yes—
178C23×C4—22no—
179C23×C4—22no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—24no—
184C23×C4—28no—
185C23×C4—24no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—44no—
191C23×C4—44no—
192C23×C4—44no—
193C23×C4—42no—
194C23×C4—44no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4C2×Q814yes—
198C23×C4#4118yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C22⋊Q8—432no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—44no—
223C2×C4⋊C4—44no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—464no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—432no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267#95—28no—
268#95—24no—
269#95—41no—
270#95—41no—
271#95—42no—
272#95—42no—
273#95—41no—
274#95—41no—
275#95—42no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—44no—
288#95—44no—
289#95—44no—
290#95—44no—
291C24—22no—
292C24—24no—
293C24—22no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—22no—
298C24—24no—
299C24—22no—
300C24—24no—
301C24—216no—
302C24—24no—
303C24—24no—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—22no—
308C24—24no—
309C24—24no—
310C24—216no—
311C24—232no—
312C24—48no—
313C24—416no—
314C24—416no—
315C24—416no—
316C24—416no—
317C24—416no—
318C24—416no—
319C24—416no—
320C24—416no—
321C24—48no—
322C24—48no—
323C24—88no—
324C24—88no—
325C24—88no—
326C24—88no—
327C24—88no—
328C24—84no—
329C24—84no—
330C24C22×Q814yes—
331C24C22×Q814yes—
332C24C22×Q812yes—
333C24C2×#4114yes—
334C24H(5,𝔽2)18yes—
335C24H(5,𝔽2)12yes—
336C24H(5,𝔽2)14yes—
337C24H(5,𝔽2)12yes—
338C24C2511yescharacteristic
339C2×Q8—48no—
340C2×Q8—88no—
341C2×Q8—84no—
342C2×Q8—88no—
343C2×Q8—84no—
344C2×D4—216no—
345C2×D4—216no—
346C2×D4—44no—
347C2×D4—48no—
348C2×D4—44no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4—48no—
353C2×D4—416no—
354C2×D4—416no—
355C2×D4—4128no—
356C2×D4—48no—
357C2×D4—48no—
358C22×C4—28no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—216no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
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—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—48no—
412C22×C4—416no—
413C22×C4—416no—
414C22×C4—48no—
415C22×C4—48no—
416C22×C4—48no—
417C22×C4—416no—
418C22×C4—48no—
419C22×C4—88no—
420C22×C4—84no—
421C22×C4—84no—
422C22×C4—88no—
423C22×C4—84no—
424C22×C4—88no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—88no—
429C22×C4—88no—
430C22×C4C22×Q812yes—
431C22×C4C22×Q812yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4H(5,𝔽2)14yes—
435C4⋊C4—416no—
436C4⋊C4—48no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—88no—
440C22⋊C4—232no—
441C22⋊C4—416no—
442C22⋊C4—416no—
443C22⋊C4—416no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—416no—
447C22⋊C4—416no—
448C22⋊C4—88no—
449C22⋊C4—88no—
450C22⋊C4—88no—
451C22⋊C4—88no—
452C22⋊C4—88no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C23—24no—
456C23—28no—
457C23—28no—
458C23—24no—
459C23—24no—
460C23—24no—
461C23—24no—
462C23—28no—
463C23—28no—
464C23—22no—
465C23—22no—
466C23—24no—
467C23—24no—
468C23—28no—
469C23—28no—
470C23—28no—
471C23—28no—
472C23—28no—
473C23—28no—
474C23—28no—
475C23—28no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—28no—
480C23—28no—
481C23—432no—
482C23—464no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—42no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—42no—
492C23—42no—
493C23—44no—
494C23—42no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—42no—
508C23—42no—
509C23—432no—
510C23—816no—
511C23—832no—
512C23—832no—
513C23—832no—
514C23—832no—
515C23—832no—
516C23#53818yes—
517C23#54318yes—
518C23#54314yes—
519C23#55418yes—
520C23#55914yes—
521C23H(3,𝔽4)14yes—
522C23C23×Q811yescharacteristic
523C23C22×#4112yes—
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×H(5,𝔽2)12yes—
526C23C2×H(5,𝔽2)11yescharacteristic
527C23C2×H(5,𝔽2)14yes—
528C23C2×H(5,𝔽2)12yes—
529C23#58512yes—
530Q8—816no—
531D4—416no—
532D4—432no—
533D4—416no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—416no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
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—44no—
591C2×C4—816no—
592C2×C4—816no—
593C2×C4—832no—
594C22—24no—
595C22—28no—
596C22—24no—
597C22—28no—
598C22—28no—
599C22—28no—
600C22—216no—
601C22—28no—
602C22—216no—
603C22—44no—
604C22—48no—
605C22—44no—
606C22—48no—
607C22—48no—
608C22—48no—
609C22—48no—
610C22—44no—
611C22—44no—
612C22—44no—
613C22—48no—
614C22—48no—
615C22—864no—
616C22—84no—
617C22—84no—
618C22—84no—
619C22—84no—
620C22#280418yes—
621C22C2×#53812yes—
622C22C2×#54311yescharacteristic
623C22C2×#54312yes—
624C22C2×#55412yes—
625C22C2×#55911yescharacteristic
626C22#349014yes—
627C22#349414yes—
628C22#350112yes—
629C22C2×H(3,𝔽4)11yescharacteristic
630C22#353212yes—
631C22#353212yes—
632C22#353412yes—
633C22#353414yes—
634C22#353714yes—
635C22#354311yescharacteristic
636C22#354511yescharacteristic
637C4—42no—
638C4—42no—
639C4—44no—
640C4—44no—
641C4—44no—
642C4—84no—
643C4—82no—
644C4—84no—
645C4—84no—
646C4—84no—
647C2—28no—
648C2—48no—
649C2—84no—
650C2#3713411yescharacteristic
651C2#3747511yescharacteristic
652C2#3787214yes—
653C2#4064012yes—
654C2#4201112yes—
655C2#4317512yes—
656C2C2×#280411yescharacteristic
657C2#5241012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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. #17272C2semidirect
3. #17137C2semidirect
4. #16893C2semidirect
5. #16320C2semidirect
6. #15961C2semidirect
7. #15506C2semidirect
8. #15015C2semidirect
9. #14717C2semidirect
10. #14281C2semidirect
11. #13877C2semidirect
18. C2×#393C22semidirect
19. C2×#393C22semidirect
22. C2×#385C22semidirect
23. C2×#385C22semidirect
30. #1446C22semidirect
31. #1446C22semidirect
32. #1445C22semidirect
33. #1445C22semidirect
34. #1445C22semidirect
35. #1445C22semidirect
36. #1443C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1441C22semidirect
41. #1441C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
65. C22⋊#95C22semidirect
66. C22⋊#95C22semidirect
72. C23×D4Q8semidirect

Maximal subgroups

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

subgroupindex
#172732
#172722
#171372
#168932
#163202
#159612
#155062
#150152
#147172
#142812
#138772
#138672
C2×#14382