Finite Group Databasebeta

← #636798

Subgroups — #636798

Order 512.

5713 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#124C211yesmaximal, characteristic, complemented
2C2×#2853C211yesmaximal, characteristic, complemented
3C2×#2411C212yesmaximal, complemented
4#16505C214yesmaximal, complemented
5#16504C214yesmaximal, complemented
6#14932C214yesmaximal, complemented
7#14928C214yesmaximal, complemented
8#14874C211yesmaximal, characteristic, complemented
9#14873C211yesmaximal, characteristic, complemented
10#14724C211yesmaximal, characteristic, complemented
11#14715C211yesmaximal, characteristic
12#14262C212yesmaximal, complemented
13#14255C212yesmaximal, complemented
14#13868C212yesmaximal, complemented
15#13756C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C24×D4C2211yescharacteristic
18C22×#124—28no—
19C22×#124—216no—
20C22×#124C2214yes—
21C22×#124C2214yes—
22C22×#123C2211yescharacteristic
23C22×#120C2212yes—
24C22×#117C2212yes—
25C23×C22⋊C4C2212yes—
26C23×C22⋊C4C2212yes—
27C23×C42C2211yescharacteristic
28#2853—28no—
29#2411—216no—
30C2×#396C2214yescomplemented
31C2×#395C2214yescomplemented
32C2×#394C2214yes—
33C2×#394C2214yescomplemented
34C2×#393C2214yescomplemented
35C2×#392C2214yes—
36C2×#390C2214yescomplemented
37C2×#388C2212yescomplemented
38C2×#388C2212yescomplemented
39C2×#388C2214yescomplemented
40C2×#387C2212yescomplemented
41C2×#387C2212yescomplemented
42C2×#386C2211yescharacteristic
43C2×#386C2211yescharacteristic
44C2×#386C2212yes—
45C2×#385C2211yescharacteristic
46C2×#385C2211yescharacteristic
47C2×#385C2212yes—
48C2×#381C2211yescharacteristic
49C2×#379C2212yes—
50C2×#379C2212yes—
51C2×#379C2212yes—
52#1446C2214yescomplemented
53#1446C2214yescomplemented
54#1440C2218yescomplemented
55#1439C2218yescomplemented
56#1439C2218yescomplemented
57#1439C2218yescomplemented
58#1438C2214yes—
59#1438C2214yes—
60#1438C2218yescomplemented
61#1437C2218yescomplemented
62C4⋊#95C2214yescomplemented
63C22⋊#95C2212yes—
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2212yes—
66C22⋊#95C2211yescharacteristic
67C22⋊#95C2212yes—
68C4×#95C2214yescomplemented
69C26C2312yes—
70C23×Q8—28no—
71C23×Q8—24no—
72C23×Q8—48no—
73C23×Q8C2314yes—
74C23×D4—28no—
75C23×D4—24no—
76C23×D4—48no—
77C23×D4C2311yescharacteristic
78C23×D4C2314yes—
79C23×D4C2314yes—
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2312yes—
83C24×C4C2311yescharacteristic
84C2×#124—2128no—
85C2×#124—232no—
86C2×#124—232no—
87C2×#124—216no—
88C2×#124—216no—
89C2×#123—24no—
90C2×#123—28no—
91C2×#120—28no—
92C2×#120—28no—
93C2×#120—44no—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—28no—
105C22×C22⋊C4—28no—
106C22×C22⋊C4—28no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4D4116yescomplemented
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2318yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2318yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2318yes—
127C22×C42—28no—
128C22×C42D418yescomplemented
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#396—216no—
133#395—48no—
134#394—216no—
135#394—48no—
136#393—48no—
137#392—48no—
138#390—216no—
139#388—216no—
140#388—28no—
141#388—28no—
142#387—28no—
143#387—28no—
144#386—24no—
145#386—24no—
146#386—28no—
147#385—24no—
148#385—24no—
149#385—28no—
150#381—24no—
151#379—28no—
152#379—28no—
153#379—44no—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2318yes—
165C2×#95C2318yes—
166C2×#95C2318yes—
167C25—24no—
168C25—24no—
169C25—24no—
170C25—24no—
171C25—24no—
172C25—24no—
173C25—48no—
174C25—48no—
175C25C2×D418yes—
176C25C2411yescharacteristic
177C25C2414yes—
178C25C2411yescharacteristic
179C22×Q8—232no—
180C22×Q8—216no—
181C22×Q8—28no—
182C22×Q8—28no—
183C22×Q8—28no—
184C22×Q8—48no—
185C22×Q8—464no—
186C22×Q8—432no—
187C22×D4—232no—
188C22×D4—216no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—28no—
195C22×D4—28no—
196C22×D4—28no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—48no—
200C22×D4—464no—
201C22×D4—432no—
202C22×D4—44no—
203C22×D4—48no—
204C22×D4—48no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—28no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—44no—
214C23×C4—48no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D418yes—
218C23×C4C2×D414yes—
219C23×C4C2×D418yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232#124—2128no—
233#124—2128no—
234#123—416no—
235#120—432no—
236#117—432no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—48no—
250C2×C22⋊C4—264no—
251C2×C22⋊C4—216no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—264no—
255C2×C22⋊C4—216no—
256C2×C22⋊C4—216no—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4—48no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4C2×D4116yes—
295C2×C22⋊C4C2×D4116yes—
296C2×C22⋊C4#4118yes—
297C2×C42—232no—
298C2×C42—28no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—28no—
302C2×C42—44no—
303C2×C42C2×D418yes—
304C2×C42C2×D418yes—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—28no—
314#95—28no—
315#95—48no—
316#95—48no—
317#95—48no—
318C24—24no—
319C24—24no—
320C24—216no—
321C24—24no—
322C24—24no—
323C24—24no—
324C24—24no—
325C24—24no—
326C24—28no—
327C24—28no—
328C24—24no—
329C24—28no—
330C24—28no—
331C24—216no—
332C24—216no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—21no—
338C24—21no—
339C24—24no—
340C24—416no—
341C24—416no—
342C24—416no—
343C24—416no—
344C24—416no—
345C24—416no—
346C24—48no—
347C24—48no—
348C24—48no—
349C24—48no—
350C24—816no—
351C24—816no—
352C24—816no—
353C24—816no—
354C24C22×D418yes—
355C24C22×D412yes—
356C24C22×D414yes—
357C24C2×#4114yes—
358C24H(5,𝔽2)12yes—
359C242−1+412yes—
360C24C2511yescharacteristic
361C2×Q8—232no—
362C2×Q8—432no—
363C2×Q8—432no—
364C2×Q8—4128no—
365C2×Q8—416no—
366C2×Q8—416no—
367C2×D4—232no—
368C2×D4—432no—
369C2×D4—432no—
370C2×D4—4128no—
371C2×D4—416no—
372C2×D4—416no—
373C2×D4—416no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—416no—
380C2×D4—816no—
381C2×D4—88no—
382C2×D4—816no—
383C22×C4—232no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—216no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—28no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—28no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—48no—
437C22×C4—416no—
438C22×C4—416no—
439C22×C4—416no—
440C22×C4—416no—
441C22×C4#120116yes—
442C22×C4C22×D414yes—
443C22×C4C22×D418yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C2×#4118yes—
450C22×C4H(5,𝔽2)18yes—
451C4⋊C4—416no—
452C4⋊C4—48no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—48no—
456C4⋊C4—88no—
457C22⋊C4—264no—
458C22⋊C4—216no—
459C22⋊C4—232no—
460C22⋊C4—232no—
461C22⋊C4—416no—
462C22⋊C4—48no—
463C22⋊C4—416no—
464C22⋊C4—44no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C22⋊C4—48no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—88no—
472C22⋊C4—816no—
473C22⋊C4—88no—
474C22⋊C4—816no—
475C42—216no—
476C42—216no—
477C42—416no—
478C23—24no—
479C23—24no—
480C23—28no—
481C23—28no—
482C23—216no—
483C23—28no—
484C23—28no—
485C23—28no—
486C23—28no—
487C23—28no—
488C23—28no—
489C23—28no—
490C23—28no—
491C23—28no—
492C23—432no—
493C23—432no—
494C23—44no—
495C23—48no—
496C23—44no—
497C23—48no—
498C23—48no—
499C23—432no—
500C23—44no—
501C23—44no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23—48no—
508C23—48no—
509C23—48no—
510C23—48no—
511C23—41no—
512C23—41no—
513C23—42no—
514C23—41no—
515C23—42no—
516C23—44no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—48no—
521C23—44no—
522C23—44no—
523C23—832no—
524C23—832no—
525C23—832no—
526C23—832no—
527C23—832no—
528C23—832no—
529C23C2×#12014yes—
530C23#53214yes—
531C23#53714yes—
532C23#54618yes—
533C23#54718yes—
534C23C23×D411yescharacteristic
535C23C23×D412yes—
536C23C22×#4112yes—
537C23C2×H(5,𝔽2)14yes—
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×2−1+411yescharacteristic
540C23#58514yes—
541Q8—464no—
542D4—464no—
543D4—832no—
544D4—88no—
545C2×C4—232no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—232no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—216no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—216no—
566C2×C4—416no—
567C2×C4—432no—
568C2×C4—48no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—48no—
573C2×C4—48no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—48no—
595C2×C4—48no—
596C2×C4—44no—
597C2×C4—48no—
598C2×C4—44no—
599C2×C4C2×#12014yes—
600C2×C4C2×#12014yes—
601C2×C4#53518yes—
602C22—28no—
603C22—24no—
604C22—216no—
605C22—24no—
606C22—216no—
607C22—28no—
608C22—48no—
609C22—48no—
610C22—48no—
611C22—48no—
612C22—48no—
613C22—48no—
614C22—48no—
615C22—864no—
616C22—88no—
617C22—88no—
618C22—88no—
619C22—88no—
620C22—82no—
621C22—81no—
622C22—84no—
623C22—81no—
624C22—84no—
625C22—81no—
626C22—82no—
627C22#243018yes—
628C22C22×#12011yescharacteristic
629C22C2×#53211yescharacteristic
630C22C2×#53512yes—
631C22C2×#53711yescharacteristic
632C22#344912yes—
633C22#344914yes—
634C22C2×#54612yes—
635C22C2×#54712yes—
636C22#348414yes—
637C22#351418yes—
638C22#355814yes—
639C22#356114yes—
640C4—28no—
641C4—24no—
642C4—24no—
643C4—44no—
644C4—48no—
645C4—48no—
646C4—82no—
647C4—81no—
648C4—84no—
649C4—81no—
650C4—81no—
651C4—82no—
652C2—28no—
653C2—88no—
654C2—161no—
655C2#3463211yescharacteristic
656C2C2×#243011yescharacteristic
657C2#3513112yes—
658C2#3522211yescharacteristic
659C2#3754612yes—
660C2#3802814yes—
661C2#3804414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#14715C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C23×#124C2semidirect
2. C2×#2853C2semidirect
3. C2×#2411C2semidirect
4. #16505C2semidirect
5. #16504C2semidirect
6. #14932C2semidirect
7. #14928C2semidirect
8. #14874C2semidirect
9. #14873C2semidirect
10. #14724C2semidirect
12. #14262C2semidirect
13. #14255C2semidirect
14. #13868C2semidirect
15. #13756C2semidirect
30. C2×#396C22semidirect
31. C2×#395C22semidirect
33. C2×#394C22semidirect
34. C2×#393C22semidirect
36. C2×#390C22semidirect
37. C2×#388C22semidirect
38. C2×#388C22semidirect
39. C2×#388C22semidirect
40. C2×#387C22semidirect
41. C2×#387C22semidirect
52. #1446C22semidirect
53. #1446C22semidirect
54. #1440C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
60. #1438C22semidirect
61. #1437C22semidirect
62. C4⋊#95C22semidirect
68. C4×#95C22semidirect
108. C22×C22⋊C4D4semidirect
128. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C23×#1242
C2×#28532
C2×#24112
#165052
#165042
#149322
#149282
#148742
#148732
#147242
#147152
#142622
#142552
#138682
#137562