Finite Group Databasebeta

← #4867711

Subgroups — #4867711

Order 512.

3502 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2539C212yesmaximal, complemented
2#17168C211yesmaximal, characteristic, complemented
3#17102C211yesmaximal, characteristic, complemented
4#17088C212yesmaximal, complemented
5#16291C214yesmaximal, complemented
6#16277C214yesmaximal, complemented
7#16250C214yesmaximal, complemented
8#16249C214yesmaximal, complemented
9#15428C214yesmaximal, complemented
10#14964C214yesmaximal, complemented
11#14341C211yesmaximal, characteristic, complemented
12C22×#123C2214yes—
13C22×#121C2212yes—
14C22×#120C2214yes—
15C23×C22⋊C4C2211yescharacteristic
16#2539—216no—
17C2×#399C2212yes—
18C2×#399C2212yes—
19C2×#394C2214yes—
20C2×#394C2214yes—
21C2×#394C2214yescomplemented
22C2×#392C2212yes—
23C2×#392C2212yes—
24C2×#392C2214yes—
25C2×#392C2214yes—
26C2×#392C2214yes—
27C2×#392C2214yescomplemented
28C2×#390C2214yes—
29C2×#390C2214yes—
30C2×#388C2214yescomplemented
31C2×#387C2214yescomplemented
32C2×#379C2212yes—
33C2×C4×C22⋊C4C2212yes—
34#1442C2214yescomplemented
35#1441C2214yescomplemented
36#1441C2214yescomplemented
37#1440C2211yescharacteristic
38#1440C2212yes—
39#1440C2214yes—
40#1440C2211yescharacteristic
41#1440C2214yes—
42#1440C2214yes—
43#1440C2214yescomplemented
44#1439C2214yes—
45#1439C2214yescomplemented
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1438C2214yes—
50#1438C2214yes—
51#1438C2214yes—
52#1438C2214yes—
53#1438C2214yescomplemented
54#1438C2214yescomplemented
55#1437C2214yescomplemented
56#1436C2214yescomplemented
57C22⋊#95C2214yes—
58C26C2311yescharacteristic
59C23×D4C2314yes—
60C23×D4C2314yes—
61C23×D4C2314yes—
62C23×D4C2312yes—
63C23×D4C2314yes—
64C24×C4C2312yes—
65C2×#123—216no—
66C2×#123—48no—
67C2×#123—48no—
68C2×#121—28no—
69C2×#121—44no—
70C2×#121—44no—
71C2×#120—216no—
72C2×#120—48no—
73C2×#120—48no—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2314yes—
76C22×C4⋊C4C2314yes—
77C22×C4⋊C4C2312yes—
78C22×C22⋊C4—44no—
79C22×C22⋊C4D418yescomplemented
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2311yescharacteristic
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2311yescharacteristic
99C22×C42C2314yes—
100#399—28no—
101#399—28no—
102#394—48no—
103#394—48no—
104#394—48no—
105#392—28no—
106#392—28no—
107#392—48no—
108#392—48no—
109#392—48no—
110#392—48no—
111#390—216no—
112#390—216no—
113#388—48no—
114#387—48no—
115#379—44no—
116C4×C22⋊C4—28no—
117C2×#95C2312yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C25—24no—
132C25—44no—
133C25—44no—
134C25—44no—
135C25C2×D414yes—
136C25C2414yes—
137C25C2411yescharacteristic
138C25C2411yescharacteristic
139C25C2411yescharacteristic
140C22×D4—28no—
141C22×D4—28no—
142C22×D4—28no—
143C22×D4—28no—
144C22×D4—28no—
145C22×D4—24no—
146C22×D4—28no—
147C22×D4—28no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4—44no—
152C22×D4—42no—
153C22×D4—42no—
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—44no—
164C22×D4—44no—
165C22×D4—44no—
166C22×D4—44no—
167C22×D4—44no—
168C22×D4—44no—
169C22×D4—44no—
170C22×D4—44no—
171C22×D4—44no—
172C22×D4—44no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4C2×D418yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2414yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2414yes—
184C23×C4C2414yes—
185#123—464no—
186#121—432no—
187#120—464no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—28no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—416no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
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—44no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
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⋊C4C2×D414yes—
252C2×C22⋊C4C2×D418yes—
253C2×C22⋊C4C2×D414yes—
254C2×C42—28no—
255C2×C42—44no—
256C2×C42—44no—
257#95—24no—
258#95—24no—
259#95—28no—
260#95—28no—
261#95—44no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271C24—24no—
272C24—24no—
273C24—24no—
274C24—24no—
275C24—24no—
276C24—22no—
277C24—22no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—28no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—48no—
299C24—416no—
300C24—416no—
301C24—84no—
302C24—88no—
303C24—84no—
304C24—88no—
305C24—88no—
306C24—88no—
307C24—88no—
308C24C22×D412yes—
309C24C22×D412yes—
310C24C22×D412yes—
311C24H(5,𝔽2)14yes—
312C24C2511yescharacteristic
313C2×D4—416no—
314C2×D4—416no—
315C2×D4—48no—
316C2×D4—416no—
317C2×D4—416no—
318C2×D4—416no—
319C2×D4—416no—
320C2×D4—416no—
321C2×D4—88no—
322C2×D4—88no—
323C2×D4—88no—
324C2×D4—88no—
325C2×D4—88no—
326C2×D4—88no—
327C2×D4—88no—
328C2×D4—88no—
329C2×D4—88no—
330C2×D4—88no—
331C2×D4—88no—
332C2×D4—88no—
333C2×D4—84no—
334C2×D4—88no—
335C2×D4—84no—
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—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—216no—
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—416no—
389C22×C4—84no—
390C22×C4—88no—
391C22×C4—84no—
392C22×C4C22×D414yes—
393C22×C4C22×D414yes—
394C22×C4H(5,𝔽2)18yes—
395C22×C4H(5,𝔽2)18yes—
396C4⋊C4—416no—
397C4⋊C4—416no—
398C4⋊C4—416no—
399C4⋊C4—48no—
400C22⋊C4—232no—
401C22⋊C4—44no—
402C22⋊C4—48no—
403C22⋊C4—44no—
404C22⋊C4—416no—
405C22⋊C4—416no—
406C22⋊C4—416no—
407C22⋊C4—416no—
408C22⋊C4—416no—
409C22⋊C4—416no—
410C22⋊C4—88no—
411C22⋊C4—88no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C22⋊C4—84no—
415C22⋊C4—88no—
416C22⋊C4—88no—
417C22⋊C4—88no—
418C22⋊C4—84no—
419C42—88no—
420C23—24no—
421C23—28no—
422C23—28no—
423C23—24no—
424C23—28no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23—24no—
429C23—24no—
430C23—28no—
431C23—28no—
432C23—416no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—44no—
445C23—42no—
446C23—42no—
447C23—44no—
448C23—42no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
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—44no—
485C23—44no—
486C23—44no—
487C23—44no—
488C23—816no—
489C23—832no—
490C23—832no—
491C23—832no—
492C23#53414yes—
493C23C23×D411yescharacteristic
494C23C2×H(5,𝔽2)12yes—
495C23C2×H(5,𝔽2)14yes—
496C23C2×H(5,𝔽2)14yes—
497C23#58514yes—
498D4—832no—
499D4—832no—
500D4—816no—
501D4—832no—
502D4—832no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—28no—
506C2×C4—28no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
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—816no—
549C22—24no—
550C22—28no—
551C22—48no—
552C22—48no—
553C22—48no—
554C22—42no—
555C22—42no—
556C22—44no—
557C22—44no—
558C22—48no—
559C22—48no—
560C22—832no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22—84no—
572C22—84no—
573C22—84no—
574C22—84no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—84no—
579C22—84no—
580C22C2×#53411yescharacteristic
581C22#344912yes—
582C22#348414yes—
583C22#353314yes—
584C22#353414yes—
585C22#354214yes—
586C22#354312yes—
587C22#354314yes—
588C22#354312yes—
589C22#354614yes—
590C22#355814yes—
591C4—44no—
592C4—44no—
593C4—48no—
594C4—48no—
595C4—84no—
596C4—84no—
597C4—84no—
598C2—44no—
599C2—82no—
600C2—82no—
601C2—164no—
602C2#3723212yes—
603C2#3745411yescharacteristic
604C2#4059714yes—
605C2#5348514yes—
606C2#5484514yes—
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. C2×#2539C2semidirect
2. #17168C2semidirect
3. #17102C2semidirect
4. #17088C2semidirect
5. #16291C2semidirect
6. #16277C2semidirect
7. #16250C2semidirect
8. #16249C2semidirect
9. #15428C2semidirect
10. #14964C2semidirect
11. #14341C2semidirect
21. C2×#394C22semidirect
27. C2×#392C22semidirect
30. C2×#388C22semidirect
31. C2×#387C22semidirect
34. #1442C22semidirect
35. #1441C22semidirect
36. #1441C22semidirect
43. #1440C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
53. #1438C22semidirect
54. #1438C22semidirect
55. #1437C22semidirect
56. #1436C22semidirect
79. C22×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25392
#171682
#171022
#170882
#162912
#162772
#162502
#162492
#154282
#149642
#143412