Finite Group Databasebeta

← #6500762

Subgroups — #6500762

Order 512.

2290 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35242C211yesmaximal, characteristic
2#34361C211yesmaximal, characteristic
3#27596C212yesmaximal, complemented
4#27592C212yesmaximal, complemented
5#24892C214yesmaximal, complemented
6#24888C214yesmaximal, complemented
7#24884C214yesmaximal, complemented
8#24880C214yesmaximal, complemented
9#24016C212yesmaximal, complemented
10#24014C212yesmaximal, complemented
11#21943C212yesmaximal, complemented
12#21743C211yesmaximal, characteristic
13#20599C212yesmaximal, complemented
14C24×Q8C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#536C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#484C2212yescomplemented
19C2×#485C2214yescomplemented
20C2×#483C2212yescomplemented
21C2×#419C2212yes—
22C2×#419C2212yes—
23C2×#419C2212yes—
24C2×#419C2212yes—
25C2×#407C2212yes—
26#2849—22no—
27#2614—22no—
28#2436—22no—
29#2434—22no—
30#2409C2214yes—
31#2409C2214yes—
32#2375—22no—
33#2374—22no—
34#2354C2214yes—
35#2352C2214yes—
36#2333—22no—
37#2332—22no—
38#2314C2212yes—
39#2311C2212yes—
40#2310C2211yescharacteristic
41#2310C2211yescharacteristic
42#2280C2212yes—
43#2280C2211yescharacteristic
44#2097C2214yescomplemented
45#2096C2212yescomplemented
46#2095C2212yescomplemented
47#2074C2212yescomplemented
48#2074C2212yescomplemented
49#2072C2214yescomplemented
50#2036C2214yescomplemented
51#2036C2214yescomplemented
52#2035C2214yescomplemented
53#2035C2214yescomplemented
54#1928C2214yescomplemented
55#1928C2214yescomplemented
56#1927C2214yescomplemented
57#1927C2214yescomplemented
58#1873C2212yescomplemented
59#1873C2212yescomplemented
60#1872C2214yescomplemented
61#1870C2214yes—
62#1870C2214yes—
63#1854C2214yes—
64#1854C2214yes—
65#1854C2214yes—
66#1854C2214yes—
67#1818C2212yes—
68#1809C2214yes—
69#1809C2214yes—
70#1787C2214yes—
71#1787C2214yes—
72#1787C2214yes—
73#1787C2214yes—
74#1780C2212yes—
75C22×#41C2311yescharacteristic
76C23×Q8—21no—
77C23×Q8—22no—
78C23×Q8—28no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8—44no—
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
84C23×Q8C2312yes—
85C23×Q8C2312yes—
86C24×C4—21no—
87C24×C4C2311yescharacteristic
88C22×M16C2312yes—
89C22×M16C2312yes—
90#536—42no—
91#536—42no—
92#536—44no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—21no—
95C2×#124—21no—
96C2×#124—21no—
97#525—28no—
98C2×#123—22no—
99C2×#123—22no—
100C2×#123D412yes—
101C2×C22⋊Q8D412yes—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2314yes—
104C2×C22⋊Q8C2312yes—
105C2×#121D412yes—
106C2×#120D412yes—
107C2×C4×D4C2314yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2312yes—
111C2×#117C2312yes—
112C2×#117C2312yes—
113C2×#117C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115#485—48no—
116#484—44no—
117#483—44no—
118C2×#107C2314yes—
119C2×#107C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122#419—44no—
123#419—44no—
124#419—44no—
125#419—44no—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134#407—28no—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2314yes—
137C2×C22⋊C8C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2312yes—
140#395—24no—
141#395—24no—
142#393—24no—
143#393—24no—
144#391—22no—
145#391—22no—
146#391—22no—
147#391C2314yes—
148#391C2312yes—
149#391C2312yes—
150#390—22no—
151#390—24no—
152#388—24no—
153#388—24no—
154#388—24no—
155#388—24no—
156#387—24no—
157#387—24no—
158#386—22no—
159#386—22no—
160#386—24no—
161#386—24no—
162#386—24no—
163#385—24no—
164#385—24no—
165#384C2312yes—
166#382C2312yes—
167#381—22no—
168#381—21no—
169#381—22no—
170#381—22no—
171#380C2314yes—
172#380C2314yes—
173#380C2314yes—
174#380C2314yes—
175#379—21no—
176#379—22no—
177C4×C4⋊C4—24no—
178C4×C4⋊C4C2314yes—
179C4×C22⋊C4—22no—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—24no—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2312yes—
189#336C2312yes—
190C25C2×D411yescharacteristic
191C2×#41—24no—
192C2×#41—41no—
193C2×#41—41no—
194C2×#41—42no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—21no—
199C22×Q8—21no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—24no—
211C22×Q8—24no—
212C22×Q8—42no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8—42no—
216C22×Q8—41no—
217C22×Q8—41no—
218C22×Q8—42no—
219C22×Q8—44no—
220C22×Q8—44no—
221C22×Q8—432no—
222C22×Q8—432no—
223C22×Q8—48no—
224C22×Q8—48no—
225C22×Q8—48no—
226C22×Q8—44no—
227C22×Q8C2×D411yescharacteristic
228C22×Q8C2411yescharacteristic
229C22×Q8C2412yes—
230C22×Q8C2411yescharacteristic
231C22×D4C2×D414yes—
232C22×D4C2×D412yes—
233C22×D4C2×D411yescharacteristic
234C23×C4—21no—
235C23×C4—21no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—42no—
240C23×C4—41no—
241C23×C4—41no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—44no—
245C23×C4C2×D412yes—
246C23×C4C2×D411yescharacteristic
247C23×C4#4118yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2412yes—
251C2×M16—24no—
252C2×M16—24no—
253C2×M16—42no—
254C2×M16—42no—
255C2×M16—42no—
256C2×M16—42no—
257C22×C8C2414yes—
258C22×C8C2414yes—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261#124—44no—
262#124—44no—
263#123—28no—
264#123—48no—
265#123—48no—
266C22⋊Q8—28no—
267C22⋊Q8—24no—
268C22⋊Q8—24no—
269C22⋊Q8—28no—
270C22⋊Q8—42no—
271C22⋊Q8—44no—
272C22⋊Q8—42no—
273#121—28no—
274#120—28no—
275C4×D4—28no—
276C4×D4—28no—
277#117—24no—
278#117—41no—
279#117—42no—
280#117—41no—
281#117—42no—
282#117—42no—
283#117—42no—
284#117—42no—
285#117—44no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—42no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C2×D412yes—
294C2×C4⋊C4C2×D414yes—
295C2×C4⋊C4C2414yes—
296C2×C4⋊C4C2414yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D414yes—
313C2×C22⋊C4C2×D414yes—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42C2412yes—
321#107—44no—
322#107—44no—
323#106—44no—
324#106—44no—
325#103—28no—
326#103—28no—
327#103—28no—
328#103—28no—
329#103—44no—
330#103—44no—
331#103—44no—
332#103—44no—
333C22⋊C8—28no—
334C22⋊C8—28no—
335C22⋊C8—42no—
336C22⋊C8—42no—
337C22⋊C8—44no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348C24—41no—
349C24—41no—
350C24—44no—
351C24#12314yes—
352C24C22×D412yes—
353C24C22×D412yes—
354C24C22×D411yescharacteristic
355#41—84no—
356#41—84no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—216no—
364C2×Q8—216no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—44no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—48no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—48no—
387C2×Q8—42no—
388C2×Q8—42no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—432no—
392C2×Q8—464no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8H(5,𝔽2)12yes—
398C2×Q8H(5,𝔽2)12yes—
399C2×Q8H(5,𝔽2)14yes—
400C2×D4—28no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—41no—
406C2×D4—41no—
407C2×D4—44no—
408C2×D4—44no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—42no—
432C22×C4—42no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—432no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—82no—
447C22×C4—84no—
448C22×C4#12318yes—
449C22×C4#12314yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2511yescharacteristic
458M16—84no—
459M16—84no—
460C2×C8—24no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C4⋊C4—24no—
467C4⋊C4—28no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—22no—
473C4⋊C4—24no—
474C4⋊C4—22no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—84no—
479C4⋊C4—82no—
480C22⋊C4—24no—
481C22⋊C4—28no—
482C22⋊C4—28no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—84no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—82no—
493C22⋊C4—82no—
494C22⋊C4—82no—
495C42—24no—
496C42—42no—
497C42—42no—
498C42—84no—
499C42—82no—
500C42—82no—
501C42—82no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—21no—
506C23—21no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—44no—
515C23—48no—
516C23—48no—
517C23—48no—
518C23C2×#12312yes—
519C23C2×#12314yes—
520C23C23×D411yescharacteristic
521Q8—24no—
522Q8—24no—
523Q8—24no—
524Q8—24no—
525Q8—48no—
526Q8—48no—
527Q8—416no—
528Q8—416no—
529Q8—42no—
530Q8—42no—
531Q8—42no—
532Q8—42no—
533Q8—82no—
534D4—82no—
535D4—88no—
536D4—84no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—48no—
557C2×C4—44no—
558C2×C4—48no—
559C2×C4—48no—
560C2×C4—416no—
561C2×C4—416no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—164no—
581C2×C4C2×#12314yes—
582C2×C4C2×#12312yes—
583C2×C4C22×#4112yes—
584C2×C4C2×H(5,𝔽2)12yes—
585C2×C4C2×H(5,𝔽2)11yescharacteristic
586C2×C4C2×H(5,𝔽2)11yescharacteristic
587C8—44no—
588C8—44no—
589C22—22no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—416no—
595C22—44no—
596C22—42no—
597C22—44no—
598C22—42no—
599C22—42no—
600C22—44no—
601C22—44no—
602C22—42no—
603C22—42no—
604C22C22×#12311yescharacteristic
605C22#348411yescharacteristic
606C22#348411yescharacteristic
607C22#358614yes—
608C4—22no—
609C4—24no—
610C4—42no—
611C4—44no—
612C4—42no—
613C4—42no—
614C4—44no—
615C4—84no—
616C4—82no—
617C4—82no—
618C4—84no—
619C4#349012yes—
620C4#349012yes—
621C2—42no—
622C2—44no—
623C2—82no—
624C2#3535011yescharacteristic
625C2#6051912yes—
626C2#6164512yes—
627C2#6164612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27596C2semidirect
4. #27592C2semidirect
5. #24892C2semidirect
6. #24888C2semidirect
7. #24884C2semidirect
8. #24880C2semidirect
9. #24016C2semidirect
10. #24014C2semidirect
11. #21943C2semidirect
13. #20599C2semidirect
18. C2×#484C22semidirect
19. C2×#485C22semidirect
20. C2×#483C22semidirect
44. #2097C22semidirect
45. #2096C22semidirect
46. #2095C22semidirect
47. #2074C22semidirect
48. #2074C22semidirect
49. #2072C22semidirect
50. #2036C22semidirect
51. #2036C22semidirect
52. #2035C22semidirect
53. #2035C22semidirect
54. #1928C22semidirect
55. #1928C22semidirect
56. #1927C22semidirect
57. #1927C22semidirect
58. #1873C22semidirect
59. #1873C22semidirect
60. #1872C22semidirect

Maximal subgroups

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

subgroupindex
#352422
#343612
#275962
#275922
#248922
#248882
#248842
#248802
#240162
#240142
#219432
#217432
#205992