Finite Group Databasebeta

← #4868892

Subgroups — #4868892

Order 512.

4184 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2404C212yesmaximal, complemented
2#17224C214yesmaximal, complemented
3#16512C214yesmaximal, complemented
4#16397C214yesmaximal, complemented
5#16268C214yesmaximal, complemented
6#16249C214yesmaximal, complemented
7#16014C211yesmaximal, characteristic, complemented
8#15947C211yesmaximal, characteristic, complemented
9#15266C212yesmaximal, complemented
10#14725C214yesmaximal, complemented
11#14337C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C22×#120C2212yes—
14C22×#120C2214yes—
15C22×#120C2212yescomplemented
16C23×C22⋊C4C2212yescomplemented
17#2404—216no—
18C2×#394C2214yes—
19C2×#394C2214yes—
20C2×#394C2214yescomplemented
21C2×#393C2214yescomplemented
22C2×#392C2214yes—
23C2×#392C2214yes—
24C2×#386C2214yes—
25C2×#386C2212yes—
26C2×#386C2214yes—
27C2×#386C2214yescomplemented
28C2×#386C2212yescomplemented
29C2×#381C2212yescomplemented
30C2×#380C2214yescomplemented
31C2×#380C2212yescomplemented
32C2×#379C2214yescomplemented
33#1446C2212yescomplemented
34#1445C2214yescomplemented
35#1443C2214yescomplemented
36#1443C2214yescomplemented
37#1441C2214yescomplemented
38#1440C2214yescomplemented
39#1440C2214yescomplemented
40#1440C2214yescomplemented
41#1440C2214yescomplemented
42#1440C2214yescomplemented
43#1440C2214yescomplemented
44#1439C2214yescomplemented
45#1439C2214yescomplemented
46#1439C2214yescomplemented
47#1439C2214yescomplemented
48#1439C2214yescomplemented
49#1438C2211yescharacteristic
50#1438C2211yescharacteristic
51#1438C2214yes—
52#1438C2214yes—
53#1438C2214yes—
54#1438C2214yes—
55#1438C2214yescomplemented
56#1438C2214yescomplemented
57C22⋊#95C2214yescomplemented
58C26C2312yes—
59C22×#41—44no—
60C22×#41—48no—
61C22×#41D418yescomplemented
62C23×Q8C2311yescharacteristic
63C23×D4C2314yes—
64C23×D4C2311yescharacteristic
65C23×D4C2314yes—
66C23×D4C2314yes—
67C23×D4C2312yes—
68C24×C4C2312yes—
69C24×C4C2311yescharacteristic
70C24×C4C2312yes—
71C2×#120—28no—
72C2×#120—28no—
73C2×#120—48no—
74C2×#120—48no—
75C2×#120—48no—
76C2×#120—48no—
77C2×#120—48no—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C22⋊C4—28no—
81C22×C22⋊C4—44no—
82C22×C22⋊C4—44no—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100#394—48no—
101#394—48no—
102#394—48no—
103#393—216no—
104#392—216no—
105#392—48no—
106#386—28no—
107#386—28no—
108#386—216no—
109#386—216no—
110#386—48no—
111#381—28no—
112#380—48no—
113#380—44no—
114#379—48no—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2312yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C25—24no—
132C25—28no—
133C25—28no—
134C25—42no—
135C25—42no—
136C25—44no—
137C25—44no—
138C25—44no—
139C25—44no—
140C25C2411yescharacteristic
141C25C2414yes—
142C25C2412yes—
143C2×#41—416no—
144C2×#41—432no—
145C2×#41—464no—
146C22×Q8—42no—
147C22×Q8—44no—
148C22×Q8C2×D414yes—
149C22×D4—28no—
150C22×D4—28no—
151C22×D4—28no—
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—28no—
158C22×D4—28no—
159C22×D4—42no—
160C22×D4—44no—
161C22×D4—42no—
162C22×D4—42no—
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—
173C22×D4—44no—
174C22×D4—44no—
175C22×D4—44no—
176C22×D4C2×D414yes—
177C22×D4C2×D418yes—
178C23×C4—28no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—42no—
186C23×C4—42no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4C2×D418yes—
190C23×C4C2×D414yes—
191C23×C4C2414yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2411yescharacteristic
198C23×C4C2411yescharacteristic
199#120—432no—
200#120—432no—
201#120—464no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
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—28no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
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—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—432no—
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—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—44no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—44no—
263#95—44no—
264#95—44no—
265#95—42no—
266#95—44no—
267#95—42no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274C24—24no—
275C24—24no—
276C24—24no—
277C24—22no—
278C24—22no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—416no—
300C24—48no—
301C24—416no—
302C24—416no—
303C24—48no—
304C24—416no—
305C24—48no—
306C24—48no—
307C24—48no—
308C24—88no—
309C24—88no—
310C24—88no—
311C24—88no—
312C24—88no—
313C24—88no—
314C24—88no—
315C24—88no—
316C24—84no—
317C24—84no—
318C24—88no—
319C24—88no—
320C24C22×D412yes—
321C24C22×D414yes—
322C24H(5,𝔽2)18yes—
323C24H(5,𝔽2)18yes—
324C24C2511yescharacteristic
325#41—4128no—
326C2×Q8—44no—
327C2×Q8—48no—
328C2×Q8—416no—
329C2×D4—216no—
330C2×D4—216no—
331C2×D4—48no—
332C2×D4—44no—
333C2×D4—48no—
334C2×D4—416no—
335C2×D4—416no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—416no—
339C2×D4—416no—
340C2×D4—416no—
341C2×D4—416no—
342C2×D4—416no—
343C2×D4—416no—
344C2×D4—416no—
345C2×D4—88no—
346C2×D4—88no—
347C2×D4—88no—
348C2×D4—88no—
349C2×D4—88no—
350C2×D4—88no—
351C2×D4—88no—
352C2×D4—88no—
353C2×D4—88no—
354C2×D4—88no—
355C2×D4—88no—
356C2×D4—88no—
357C2×D4—88no—
358C22×C4—24no—
359C22×C4—216no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—216no—
367C22×C4—28no—
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—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—48no—
404C22×C4—48no—
405C22×C4—416no—
406C22×C4—416no—
407C22×C4—416no—
408C22×C4—48no—
409C22×C4—88no—
410C22×C4—84no—
411C22×C4—84no—
412C22×C4—88no—
413C22×C4C22×D414yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4H(5,𝔽2)14yes—
417C4⋊C4—88no—
418C4⋊C4—88no—
419C22⋊C4—416no—
420C22⋊C4—416no—
421C22⋊C4—416no—
422C22⋊C4—48no—
423C22⋊C4—416no—
424C22⋊C4—416no—
425C22⋊C4—416no—
426C22⋊C4—48no—
427C22⋊C4—416no—
428C22⋊C4—88no—
429C22⋊C4—88no—
430C22⋊C4—88no—
431C22⋊C4—88no—
432C22⋊C4—88no—
433C22⋊C4—88no—
434C22⋊C4—88no—
435C22⋊C4—88no—
436C23—24no—
437C23—24no—
438C23—28no—
439C23—28no—
440C23—28no—
441C23—28no—
442C23—28no—
443C23—28no—
444C23—28no—
445C23—28no—
446C23—22no—
447C23—24no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—28no—
455C23—28no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—42no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—432no—
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—44no—
489C23—42no—
490C23—44no—
491C23—42no—
492C23—44no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—832no—
504C23—816no—
505C23—832no—
506C23—832no—
507C23—832no—
508C23—832no—
509C23—816no—
510C23—832no—
511C23#53418yes—
512C23C23×D411yescharacteristic
513C23C2×H(5,𝔽2)14yes—
514C23C2×H(5,𝔽2)12yes—
515C23C2×H(5,𝔽2)14yes—
516C23#58514yes—
517Q8—416no—
518D4—416no—
519D4—432no—
520D4—832no—
521D4—832no—
522D4—832no—
523C2×C4—24no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—432no—
531C2×C4—416no—
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—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—44no—
569C2×C4—44no—
570C2×C4—816no—
571C2×C4#53414yes—
572C22—28no—
573C22—216no—
574C22—48no—
575C22—48no—
576C22—44no—
577C22—44no—
578C22—48no—
579C22—48no—
580C22—48no—
581C22—42no—
582C22—48no—
583C22—48no—
584C22—48no—
585C22—44no—
586C22—48no—
587C22—48no—
588C22—48no—
589C22—48no—
590C22—48no—
591C22—864no—
592C22—84no—
593C22—84no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—84no—
602C22C2×#53411yescharacteristic
603C22C2×#53412yes—
604C22#348814yes—
605C22#353214yes—
606C22#353214yes—
607C22#353414yes—
608C22#353414yes—
609C22#353712yes—
610C22#354012yes—
611C22#354214yes—
612C22#354514yes—
613C4—24no—
614C4—44no—
615C4—42no—
616C4—84no—
617C4—84no—
618C4—84no—
619C4—84no—
620C4—84no—
621C2—48no—
622C2—82no—
623C2—88no—
624C2#3726912yes—
625C2#3743611yescharacteristic
626C2#4198114yes—
627C2#5394614yes—
628C2#5795814yes—
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×#2404C2semidirect
2. #17224C2semidirect
3. #16512C2semidirect
4. #16397C2semidirect
5. #16268C2semidirect
6. #16249C2semidirect
7. #16014C2semidirect
8. #15947C2semidirect
9. #15266C2semidirect
10. #14725C2semidirect
11. #14337C2semidirect
15. C22×#120C22semidirect
16. C23×C22⋊C4C22semidirect
20. C2×#394C22semidirect
21. C2×#393C22semidirect
27. C2×#386C22semidirect
28. C2×#386C22semidirect
29. C2×#381C22semidirect
30. C2×#380C22semidirect
31. C2×#380C22semidirect
32. C2×#379C22semidirect
33. #1446C22semidirect
34. #1445C22semidirect
35. #1443C22semidirect
36. #1443C22semidirect
37. #1441C22semidirect
38. #1440C22semidirect
39. #1440C22semidirect
40. #1440C22semidirect
41. #1440C22semidirect
42. #1440C22semidirect
43. #1440C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
55. #1438C22semidirect
56. #1438C22semidirect
57. C22⋊#95C22semidirect
61. C22×#41D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24042
#172242
#165122
#163972
#162682
#162492
#160142
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