Finite Group Databasebeta

← #4453269

Subgroups — #4453269

Order 512.

5690 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic, complemented
2C2×#2408C212yesmaximal, complemented
3C22×#394C211yesmaximal, characteristic, complemented
4#16655C214yesmaximal, complemented
5#16633C214yesmaximal, complemented
6#16253C214yesmaximal, complemented
7#16250C214yesmaximal, complemented
8#14911C214yesmaximal, complemented
9#14717C212yesmaximal, complemented
10#14716C212yesmaximal, complemented
11#14339C211yesmaximal, characteristic, complemented
12#13960C212yesmaximal, complemented
13C24×D4C2211yescharacteristic
14C2×#539—216no—
15C2×#539—28no—
16C22×#126C2212yescomplemented
17C22×#125C2212yescomplemented
18C22×#123C2212yes—
19C22×#123C2212yescomplemented
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yescomplemented
22C22×#121C2211yescharacteristic
23C22×#121C2214yes—
24C22×#120C2214yes—
25C22×C4×D4C2212yes—
26C23×C4⋊C4C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28C23×C22⋊C4C2212yes—
29#2408—216no—
30C2×#394—28no—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#394C2214yes—
34C2×#393C2214yescomplemented
35C2×#392C2214yes—
36C2×#392C2214yescomplemented
37C2×#390C2214yes—
38C2×#386C2214yes—
39C2×#381C2212yes—
40C2×#380C2212yes—
41C2×#379C2214yes—
42C22×#95C2211yescharacteristic
43#1441C2214yescomplemented
44#1441C2214yescomplemented
45#1441C2214yescomplemented
46#1439C2214yescomplemented
47#1439C2218yescomplemented
48#1439C2214yescomplemented
49#1439C2218yescomplemented
50#1439C2218yescomplemented
51#1439C2214yescomplemented
52#1438C2218yescomplemented
53#1438C2218yescomplemented
54#1438C2218yescomplemented
55#1434C2214yescomplemented
56C22⋊#95C2212yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yescomplemented
59C26C2312yes—
60C23×Q8C2311yescharacteristic
61C23×D4—24no—
62C23×D4—28no—
63C23×D4—48no—
64C23×D4C2312yes—
65C23×D4C2314yes—
66C23×D4C2311yescharacteristic
67C23×D4C2318yes—
68C24×C4C2311yescharacteristic
69C24×C4C2312yes—
70C24×C4C2311yescharacteristic
71C24×C4C2312yes—
72C24×C4C2311yescharacteristic
73#539—2128no—
74C2×#126—28no—
75C2×#126—28no—
76C2×#126—28no—
77C2×#125—28no—
78C2×#125—28no—
79C2×#125—28no—
80C2×#123—28no—
81C2×#123—28no—
82C2×#123—28no—
83C2×#123—28no—
84C2×#123—28no—
85C2×#123—28no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—28no—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×#121—28no—
92C2×#121—24no—
93C2×#121—216no—
94C2×#121—416no—
95C2×#120—216no—
96C2×#120—48no—
97C2×#120—48no—
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4D4116yescomplemented
101C22×C4⋊C4—28no—
102C22×C4⋊C4D418yescomplemented
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4—28no—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—48no—
115C22×C22⋊C4—48no—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2318yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2318yes—
126C22×C22⋊C4C2318yes—
127C22×C22⋊C4C2318yes—
128C22×C22⋊C4C2318yes—
129C22×C42C2312yes—
130#394—216no—
131#394—216no—
132#394—216no—
133#393—48no—
134#392—216no—
135#392—48no—
136#390—216no—
137#386—48no—
138#381—44no—
139#380—28no—
140#379—216no—
141C2×#95—24no—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2318yes—
146C2×#95C2314yes—
147C2×#95C2318yes—
148C2×#95C2318yes—
149C2×#95C2318yes—
150C25—24no—
151C25—28no—
152C25—28no—
153C25—28no—
154C25—24no—
155C25—24no—
156C25—44no—
157C25—44no—
158C25—44no—
159C25C2412yes—
160C25C2414yes—
161C25C2411yescharacteristic
162C22×Q8—22no—
163C22×Q8—24no—
164C22×Q8—44no—
165C22×D4—24no—
166C22×D4—24no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—216no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—24no—
173C22×D4—216no—
174C22×D4—216no—
175C22×D4—28no—
176C22×D4—22no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—48no—
180C22×D4—48no—
181C22×D4—48no—
182C22×D4—48no—
183C22×D4—48no—
184C22×D4—48no—
185C22×D4—44no—
186C22×D4—464no—
187C22×D4—432no—
188C22×D4—416no—
189C22×D4C2×D418yes—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—48no—
203C23×C4—44no—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4C2×D418yes—
207C23×C4C2×D418yes—
208C23×C4C2×D414yes—
209C23×C4C2414yes—
210C23×C4C2418yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218#126—264no—
219#125—264no—
220#123—264no—
221#123—264no—
222C22⋊Q8—264no—
223C22⋊Q8—232no—
224#121—232no—
225#121—464no—
226#120—464no—
227C4×D4—264no—
228C2×C4⋊C4—232no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—48no—
250C2×C4⋊C4C2×D418yes—
251C2×C4⋊C4C2×D418yes—
252C2×C4⋊C4C2×D418yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—216no—
275C2×C22⋊C4—216no—
276C2×C22⋊C4—216no—
277C2×C22⋊C4—216no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4—48no—
283C2×C22⋊C4—48no—
284C2×C22⋊C4—48no—
285C2×C22⋊C4—432no—
286C2×C22⋊C4—432no—
287C2×C22⋊C4—48no—
288C2×C22⋊C4—48no—
289C2×C22⋊C4—48no—
290C2×C22⋊C4—48no—
291C2×C22⋊C4—48no—
292C2×C22⋊C4C2×D418yes—
293C2×C22⋊C4C2×D418yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42C2×D418yes—
297#95—28no—
298#95—28no—
299#95—48no—
300#95—48no—
301#95—48no—
302#95—44no—
303#95—48no—
304#95—44no—
305C24—24no—
306C24—24no—
307C24—216no—
308C24—22no—
309C24—22no—
310C24—24no—
311C24—28no—
312C24—22no—
313C24—28no—
314C24—24no—
315C24—24no—
316C24—28no—
317C24—28no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—28no—
324C24—28no—
325C24—24no—
326C24—432no—
327C24—432no—
328C24—416no—
329C24—416no—
330C24—48no—
331C24—48no—
332C24—48no—
333C24—48no—
334C24—48no—
335C24—48no—
336C24—416no—
337C24—416no—
338C24—816no—
339C24—816no—
340C24—88no—
341C24—88no—
342C24C22×D414yes—
343C24C22×D414yes—
344C24C22×D412yes—
345C24H(5,𝔽2)18yes—
346C24H(5,𝔽2)12yes—
347C24H(5,𝔽2)14yes—
348C242−1+412yes—
349C24C2511yescharacteristic
350C2×Q8—28no—
351C2×Q8—88no—
352C2×Q8—84no—
353C2×D4—232no—
354C2×D4—216no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—28no—
358C2×D4—432no—
359C2×D4—432no—
360C2×D4—416no—
361C2×D4—4128no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—432no—
367C2×D4—432no—
368C2×D4—416no—
369C2×D4—84no—
370C2×D4—88no—
371C2×D4—816no—
372C2×D4—816no—
373C2×D4—816no—
374C2×D4—816no—
375C2×D4—816no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—28no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—28no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—216no—
404C22×C4—24no—
405C22×C4—28no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—416no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—416no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—48no—
425C22×C4—88no—
426C22×C4—88no—
427C22×C4—88no—
428C22×C4—816no—
429C22×C4#120116yes—
430C22×C4C22×D414yes—
431C22×C4C22×D414yes—
432C22×C4C22×D414yes—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C4⋊C4—28no—
440C4⋊C4—216no—
441C4⋊C4—216no—
442C4⋊C4—28no—
443C4⋊C4—216no—
444C4⋊C4—216no—
445C4⋊C4—216no—
446C4⋊C4—216no—
447C4⋊C4—416no—
448C22⋊C4—216no—
449C22⋊C4—216no—
450C22⋊C4—216no—
451C22⋊C4—216no—
452C22⋊C4—216no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—432no—
456C22⋊C4—432no—
457C22⋊C4—432no—
458C22⋊C4—416no—
459C22⋊C4—816no—
460C22⋊C4—816no—
461C42—216no—
462C23—24no—
463C23—28no—
464C23—24no—
465C23—28no—
466C23—216no—
467C23—28no—
468C23—28no—
469C23—24no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—216no—
474C23—28no—
475C23—28no—
476C23—28no—
477C23—216no—
478C23—28no—
479C23—432no—
480C23—432no—
481C23—432no—
482C23—44no—
483C23—44no—
484C23—48no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—48no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—48no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—48no—
499C23—48no—
500C23—44no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—48no—
505C23—48no—
506C23—48no—
507C23—44no—
508C23—48no—
509C23—48no—
510C23—44no—
511C23—48no—
512C23—864no—
513C23—864no—
514C23—816no—
515C23—816no—
516C23—816no—
517C23—816no—
518C23C2×#12014yes—
519C23#53518yes—
520C23#53514yes—
521C23#53714yes—
522C23#53718yes—
523C23C23×D411yescharacteristic
524C23C23×D412yes—
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×H(5,𝔽2)12yes—
527C23C2×H(5,𝔽2)14yes—
528C23C2×2−1+411yescharacteristic
529C23#58514yes—
530Q8—88no—
531D4—464no—
532D4—432no—
533D4—88no—
534D4—864no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—28no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—416no—
553C2×C4—416no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—48no—
570C2×C4—44no—
571C2×C4—48no—
572C2×C4—44no—
573C2×C4—48no—
574C2×C4—48no—
575C2×C4—48no—
576C2×C4—48no—
577C2×C4—48no—
578C2×C4—48no—
579C2×C4—48no—
580C2×C4—816no—
581C2×C4—88no—
582C2×C4—816no—
583C2×C4C2×#12018yes—
584C22—24no—
585C22—216no—
586C22—216no—
587C22—28no—
588C22—24no—
589C22—28no—
590C22—44no—
591C22—48no—
592C22—48no—
593C22—416no—
594C22—44no—
595C22—48no—
596C22—416no—
597C22—48no—
598C22—44no—
599C22—48no—
600C22—864no—
601C22—84no—
602C22—88no—
603C22—84no—
604C22—84no—
605C22—88no—
606C22—84no—
607C22—84no—
608C22—84no—
609C22—84no—
610C22—84no—
611C22#261518yes—
612C22C22×#12011yescharacteristic
613C22C2×#53512yes—
614C22C2×#53511yescharacteristic
615C22C2×#53712yes—
616C22C2×#53711yescharacteristic
617C22#348414yes—
618C22#348412yes—
619C22#348712yes—
620C22#350014yes—
621C22#353418yes—
622C22#353414yes—
623C22#353514yes—
624C4—28no—
625C4—42no—
626C4—44no—
627C4—42no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—88no—
632C4—84no—
633C2—28no—
634C2—84no—
635C2—88no—
636C2C2×#261511yescharacteristic
637C2#3534611yescharacteristic
638C2#3567611yescharacteristic
639C2#3672212yes—
640C2#3718512yes—
641C2#4049214yes—
642C2#4049414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#539C2semidirect
2. C2×#2408C2semidirect
3. C22×#394C2semidirect
4. #16655C2semidirect
5. #16633C2semidirect
6. #16253C2semidirect
7. #16250C2semidirect
8. #14911C2semidirect
9. #14717C2semidirect
10. #14716C2semidirect
11. #14339C2semidirect
12. #13960C2semidirect
16. C22×#126C22semidirect
17. C22×#125C22semidirect
19. C22×#123C22semidirect
21. C22×C22⋊Q8C22semidirect
34. C2×#393C22semidirect
36. C2×#392C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1439C22semidirect
52. #1438C22semidirect
53. #1438C22semidirect
54. #1438C22semidirect
55. #1434C22semidirect
58. C22⋊#95C22semidirect
100. C2×C4×D4D4semidirect
102. C22×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#24082
C22×#3942
#166552
#166332
#162532
#162502
#149112
#147172
#147162
#143392
#139602