Finite Group Databasebeta

← #4474585

Subgroups — #4474585

Order 512.

3506 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17233C214yesmaximal, complemented
2#16904C211yesmaximal, characteristic, complemented
3#16893C214yesmaximal, complemented
4#16284C214yesmaximal, complemented
5#16252C211yesmaximal, characteristic, complemented
6#15376C214yesmaximal, complemented
7#15263C214yesmaximal, complemented
8#15257C214yesmaximal, complemented
9#14716C212yesmaximal, complemented
10#14417C212yesmaximal, complemented
11#14338C211yesmaximal, characteristic, complemented
12C22×#121C2212yes—
13C22×#121C2214yes—
14C22×#120C2214yes—
15C23×C22⋊C4C2211yescharacteristic
16C2×#394C2214yescomplemented
17C2×#392C2214yes—
18C2×#392C2212yes—
19C2×#392C2212yes—
20C2×#392C2214yes—
21C2×#392C2214yescomplemented
22C2×#386C2214yes—
23C2×#386C2214yes—
24C2×#386C2214yes—
25C2×#386C2214yescomplemented
26C2×#386C2214yescomplemented
27C2×#385C2214yescomplemented
28C2×#380C2212yes—
29C2×#379C2212yes—
30#1445C2214yescomplemented
31#1443C2214yescomplemented
32#1443C2214yescomplemented
33#1442C2214yescomplemented
34#1441C2214yescomplemented
35#1440C2214yes—
36#1440C2214yescomplemented
37#1440C2214yescomplemented
38#1439C2214yes—
39#1439C2214yes—
40#1439C2214yes—
41#1439C2212yes—
42#1439C2212yes—
43#1439C2214yes—
44#1439C2214yescomplemented
45#1439C2214yescomplemented
46#1439C2214yescomplemented
47#1438C2214yes—
48#1438C2214yes—
49#1438C2214yes—
50#1438C2214yes—
51#1438C2214yes—
52#1438C2214yes—
53C22⋊#95C2211yescharacteristic
54C22⋊#95C2211yescharacteristic
55C22⋊#95C2212yes—
56C22⋊#95C2214yescomplemented
57C26C2311yescharacteristic
58C23×D4C2314yes—
59C23×D4C2312yes—
60C23×D4C2314yes—
61C23×D4C2314yes—
62C23×D4C2314yes—
63C24×C4C2311yescharacteristic
64C24×C4C2314yes—
65C24×C4C2311yescharacteristic
66C2×#121—28no—
67C2×#121—48no—
68C2×#121—48no—
69C2×#121—48no—
70C2×#121—48no—
71C2×#120—48no—
72C2×#120—48no—
73C2×#120—48no—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—44no—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2312yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96#394—48no—
97#392—44no—
98#392—44no—
99#392—48no—
100#392—48no—
101#392—48no—
102#386—48no—
103#386—48no—
104#386—48no—
105#386—48no—
106#386—48no—
107#385—48no—
108#380—44no—
109#379—44no—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2312yes—
114C2×#95C2312yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C25—22no—
127C25—22no—
128C25—22no—
129C25—24no—
130C25—42no—
131C25—44no—
132C25—44no—
133C25C2411yescharacteristic
134C25C2414yes—
135C25C2411yescharacteristic
136C25C2411yescharacteristic
137C22×D4—28no—
138C22×D4—28no—
139C22×D4—28no—
140C22×D4—28no—
141C22×D4—28no—
142C22×D4—28no—
143C22×D4—28no—
144C22×D4—28no—
145C22×D4—28no—
146C22×D4—28no—
147C22×D4—28no—
148C22×D4—24no—
149C22×D4—28no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—44no—
153C22×D4—44no—
154C22×D4—44no—
155C22×D4—44no—
156C22×D4—44no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—44no—
160C22×D4—44no—
161C22×D4—44no—
162C22×D4—44no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C23×C4—22no—
171C23×C4—24no—
172C23×C4—28no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4#4114yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2414yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2414yes—
190#121—432no—
191#121—464no—
192#120—464no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—416no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
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—44no—
253#95—28no—
254#95—28no—
255#95—28no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—42no—
265#95—42no—
266#95—44no—
267#95—44no—
268#95—44no—
269C24—24no—
270C24—21no—
271C24—22no—
272C24—24no—
273C24—21no—
274C24—28no—
275C24—21no—
276C24—21no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—21no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—22no—
294C24—22no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—21no—
301C24—48no—
302C24—48no—
303C24—44no—
304C24—416no—
305C24—44no—
306C24—84no—
307C24—84no—
308C24—88no—
309C24—88no—
310C24—88no—
311C24—88no—
312C24—88no—
313C24C2×#4112yes—
314C24H(5,𝔽2)12yes—
315C24H(5,𝔽2)18yes—
316C24H(5,𝔽2)14yes—
317C242−1+412yes—
318C24C2511yescharacteristic
319C2×D4—216no—
320C2×D4—416no—
321C2×D4—416no—
322C2×D4—416no—
323C2×D4—416no—
324C2×D4—416no—
325C2×D4—416no—
326C2×D4—416no—
327C2×D4—416no—
328C2×D4—416no—
329C2×D4—416no—
330C2×D4—416no—
331C2×D4—84no—
332C2×D4—84no—
333C2×D4—88no—
334C2×D4—88no—
335C2×D4—88no—
336C2×D4—88no—
337C2×D4—88no—
338C2×D4—88no—
339C2×D4—88no—
340C2×D4—88no—
341C2×D4—88no—
342C2×D4—88no—
343C2×D4—88no—
344C2×D4—88no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—88no—
348C2×D4—88no—
349C2×D4—88no—
350C2×D4—88no—
351C2×D4—88no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—28no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
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—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—44no—
396C22×C4—48no—
397C22×C4—416no—
398C22×C4—88no—
399C22×C4—88no—
400C22×C4—88no—
401C22×C4—88no—
402C22×C4—84no—
403C22×C4—88no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4C2×#4114yes—
408C22×C4H(5,𝔽2)14yes—
409C22×C4H(5,𝔽2)18yes—
410C4⋊C4—48no—
411C4⋊C4—416no—
412C4⋊C4—88no—
413C22⋊C4—416no—
414C22⋊C4—48no—
415C22⋊C4—48no—
416C22⋊C4—416no—
417C22⋊C4—416no—
418C22⋊C4—416no—
419C22⋊C4—416no—
420C22⋊C4—416no—
421C22⋊C4—416no—
422C22⋊C4—88no—
423C22⋊C4—84no—
424C22⋊C4—88no—
425C22⋊C4—84no—
426C22⋊C4—88no—
427C22⋊C4—88no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C23—22no—
431C23—24no—
432C23—22no—
433C23—24no—
434C23—28no—
435C23—24no—
436C23—24no—
437C23—28no—
438C23—28no—
439C23—22no—
440C23—22no—
441C23—28no—
442C23—28no—
443C23—28no—
444C23—28no—
445C23—28no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—24no—
451C23—416no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—44no—
462C23—44no—
463C23—42no—
464C23—42no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—42no—
504C23—816no—
505C23—816no—
506C23—832no—
507C23—832no—
508C23—88no—
509C23—88no—
510C23#54014yes—
511C23C22×#4111yescharacteristic
512C23C2×H(5,𝔽2)12yes—
513C23C2×H(5,𝔽2)14yes—
514C23C2×H(5,𝔽2)12yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×H(5,𝔽2)14yes—
517C23C2×2−1+411yescharacteristic
518D4—816no—
519D4—832no—
520D4—832no—
521D4—832no—
522D4—832no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—88no—
569C2×C4—88no—
570C2×C4—832no—
571C22—28no—
572C22—24no—
573C22—44no—
574C22—48no—
575C22—42no—
576C22—48no—
577C22—42no—
578C22—48no—
579C22—44no—
580C22—44no—
581C22—48no—
582C22—48no—
583C22—48no—
584C22—48no—
585C22—48no—
586C22—48no—
587C22—48no—
588C22—48no—
589C22—42no—
590C22—42no—
591C22—832no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—81no—
597C22—81no—
598C22—84no—
599C22—82no—
600C22—82no—
601C22—84no—
602C22—84no—
603C22—84no—
604C22—84no—
605C22—81no—
606C22—81no—
607C22—84no—
608C22C2×#54011yescharacteristic
609C22#349012yes—
610C22#349011yescharacteristic
611C22#349211yescharacteristic
612C22#350812yes—
613C22#353214yes—
614C22#353214yes—
615C22#353614yes—
616C22#353612yes—
617C22#353712yes—
618C22#353714yes—
619C22#353714yes—
620C22#353714yes—
621C4—48no—
622C4—44no—
623C4—44no—
624C4—84no—
625C4—84no—
626C4—84no—
627C4—84no—
628C2—44no—
629C2—88no—
630C2—82no—
631C2—82no—
632C2#3672711yescharacteristic
633C2#4054012yes—
634C2#4638411yescharacteristic
635C2#4645411yescharacteristic
636C2#4788112yes—
637C2#5361314yes—
638C2#5634614yes—
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. #17233C2semidirect
2. #16904C2semidirect
3. #16893C2semidirect
4. #16284C2semidirect
5. #16252C2semidirect
6. #15376C2semidirect
7. #15263C2semidirect
8. #15257C2semidirect
9. #14716C2semidirect
10. #14417C2semidirect
11. #14338C2semidirect
16. C2×#394C22semidirect
21. C2×#392C22semidirect
25. C2×#386C22semidirect
26. C2×#386C22semidirect
27. C2×#385C22semidirect
30. #1445C22semidirect
31. #1443C22semidirect
32. #1443C22semidirect
33. #1442C22semidirect
34. #1441C22semidirect
36. #1440C22semidirect
37. #1440C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
56. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
#172332
#169042
#168932
#162842
#162522
#153762
#152632
#152572
#147162
#144172
#143382