Finite Group Databasebeta

← #6660403

Subgroups — #6660403

Order 512.

2696 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35250C212yesmaximal, complemented
2#32712C214yesmaximal, complemented
3#29899C214yesmaximal, complemented
4#29742C214yesmaximal, complemented
5#29602C214yesmaximal, complemented
6#27609C211yesmaximal, characteristic, complemented
7#24560C214yesmaximal, complemented
8#24384C214yesmaximal, complemented
9#24015C212yesmaximal, complemented
10#23925C211yesmaximal, characteristic, complemented
11#20580C211yesmaximal, characteristic, complemented
12C23×#41C2211yescharacteristic
13C2×#535C2212yes—
14C2×#535C2212yescomplemented
15C22×#121C2212yescomplemented
16C2×#482C2214yescomplemented
17C2×#469C2214yes—
18C2×#447C2214yescomplemented
19C2×#407C2212yescomplemented
20C2×#407C2214yescomplemented
21#2847C2214yes—
22#2847C2214yescomplemented
23#2615C2214yes—
24#2615C2214yescomplemented
25#2436—22no—
26#2433—22no—
27#2431—22no—
28#2431—22no—
29#2404—22no—
30#2404—22no—
31#2403—22no—
32#2403—22no—
33#2311C2212yes—
34#2311C2212yescomplemented
35#2280C2212yescomplemented
36#2077C2214yescomplemented
37#2077C2214yescomplemented
38#2077C2214yescomplemented
39#2075C2214yescomplemented
40#2034C2214yescomplemented
41#2034C2214yescomplemented
42#2034C2214yescomplemented
43#2034C2214yescomplemented
44#2034C2214yescomplemented
45#2033C2214yescomplemented
46#2033C2214yescomplemented
47#2033C2214yescomplemented
48#1999C2214yescomplemented
49#1999C2214yescomplemented
50#1999C2214yescomplemented
51#1999C2214yescomplemented
52#1895C2214yescomplemented
53#1895C2214yescomplemented
54#1875C2214yescomplemented
55#1874C2214yescomplemented
56#1854C2214yescomplemented
57#1853C2214yescomplemented
58#1825C2214yescomplemented
59#1809C2214yescomplemented
60#1786C2214yescomplemented
61#1780C2211yescharacteristic
62#1780C2211yescharacteristic
63#1780C2214yescomplemented
64#1780C2212yescomplemented
65C22×#41—44no—
66C22×#41—48no—
67C22×#41D411yescharacteristic
68C22×#41D412yes—
69C22×#41D411yescharacteristic
70C22×#41C2311yescharacteristic
71C22×#41C2312yes—
72C22×#41C2311yescharacteristic
73C23×Q8D411yescharacteristic
74C23×D4—22no—
75C23×D4D411yescharacteristic
76C23×D4D412yes—
77C23×D4C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C22×D8C2314yes—
81C22×M16C2314yes—
82#535—48no—
83#535—48no—
84C2×#127—22no—
85C2×#127—22no—
86C2×#124—22no—
87C2×#124—22no—
88C2×#121—22no—
89C2×#121—22no—
90C2×#121—44no—
91C2×#121—44no—
92C2×#121C2314yes—
93C2×#121C2314yes—
94C2×#121C2314yes—
95C2×#121C2314yes—
96C2×#121C2314yes—
97C2×#121C2314yes—
98C2×#120—22no—
99C2×#120—22no—
100C2×#120—22no—
101C2×#120—22no—
102C2×#120—24no—
103C2×#120—24no—
104C2×#117C2312yes—
105C2×#117C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—22no—
108#482—48no—
109#469—48no—
110#447—48no—
111C2×#107C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120#407—48no—
121#407—44no—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2314yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2312yes—
129#393—24no—
130#393—24no—
131#392—24no—
132#392—24no—
133#392—24no—
134#392—24no—
135#392—24no—
136#392—24no—
137#391—22no—
138#391—22no—
139#390—22no—
140#390—22no—
141#386—22no—
142#386—22no—
143#386—22no—
144#386—22no—
145#386—24no—
146#386—24no—
147#386—24no—
148#386—24no—
149#386—24no—
150#386—24no—
151#381—22no—
152#380C2314yes—
153#380C2314yes—
154#380C2314yes—
155#380C2314yes—
156#380C2314yes—
157#380C2314yes—
158#379—22no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—22no—
161#340C2314yes—
162#340C2314yes—
163#340C2314yes—
164#340C2314yes—
165#340C2314yes—
166#340C2314yes—
167C25—22no—
168C25C2×D412yes—
169C25C2×D411yescharacteristic
170C2×#41—28no—
171C2×#41—28no—
172C2×#41—42no—
173C2×#41—44no—
174C2×#41—42no—
175C2×#41—42no—
176C2×#41—42no—
177C2×#41—42no—
178C2×#41—44no—
179C2×#41—42no—
180C2×#41—44no—
181C2×#41—44no—
182C2×#41—44no—
183C2×#41—44no—
184C2×#41—44no—
185C2×#41—464no—
186C22×Q8—42no—
187C22×Q8—44no—
188C22×Q8C2×D411yescharacteristic
189C22×Q8C2×D412yes—
190C22×Q8C2×D411yescharacteristic
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—28no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—42no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—42no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—48no—
208C22×D4—88no—
209C22×D4C2×D411yescharacteristic
210C22×D4C2×D412yes—
211C22×D4C2×D412yes—
212C22×D4C2×D412yes—
213C22×D4C2×D412yes—
214C22×D4C2×D411yescharacteristic
215C22×D4C2×D412yes—
216C22×D4C2414yes—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—28no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D411yescharacteristic
229C23×C4C2×D411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C2×D8—44no—
236C2×D8—44no—
237C2×D8—44no—
238C2×M16—28no—
239C2×M16—44no—
240C2×M16—44no—
241C22×C8C2414yes—
242C22×C8C2414yes—
243#127—48no—
244#127—48no—
245#124—48no—
246#124—48no—
247#121—28no—
248#121—28no—
249#121—28no—
250#121—28no—
251#121—44no—
252#121—44no—
253#121—44no—
254#121—44no—
255#121—44no—
256#121—44no—
257#121—44no—
258#121—44no—
259#121—48no—
260#121—48no—
261#120—48no—
262#120—48no—
263#120—48no—
264#120—48no—
265#120—88no—
266#120—88no—
267#117—28no—
268#117—44no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2414yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—48no—
290C2×C42—22no—
291C2×C42—22no—
292#107—44no—
293#106—44no—
294#106—44no—
295#102—44no—
296#102—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300#102—44no—
301C22⋊C8—28no—
302C22⋊C8—28no—
303C22⋊C8—44no—
304C22⋊C8—44no—
305C22⋊C8—42no—
306C22⋊C8—44no—
307C22⋊C8—42no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314C24—24no—
315C24—24no—
316C24—22no—
317C24—24no—
318C24—42no—
319C24—42no—
320C24—44no—
321C24—44no—
322C24—44no—
323C24—44no—
324C24—44no—
325C24—44no—
326C24—84no—
327C24—84no—
328C24C22×D412yes—
329C24C22×D411yescharacteristic
330C24C22×D411yescharacteristic
331C24C22×D412yes—
332C24C22×D411yescharacteristic
333#41—416no—
334#41—416no—
335#41—416no—
336#41—416no—
337#41—432no—
338#41—416no—
339#41—416no—
340C2×Q8—24no—
341C2×Q8—24no—
342C2×Q8—42no—
343C2×Q8—41no—
344C2×Q8—42no—
345C2×Q8—41no—
346C2×Q8—42no—
347C2×Q8—416no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—24no—
356C2×D4—24no—
357C2×D4—24no—
358C2×D4—24no—
359C2×D4—24no—
360C2×D4—42no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—42no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—432no—
376C2×D4—416no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—88no—
387C2×D4—88no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—84no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4—84no—
395C2×D4—84no—
396C2×D4—84no—
397C2×D4—84no—
398C2×D4—832no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—42no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—416no—
430C22×C4—416no—
431C22×C4—416no—
432C22×C4—44no—
433C22×C4—84no—
434C22×C4C22×D412yes—
435C22×C4C22×D411yescharacteristic
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D411yescharacteristic
438C22×C4C22×D412yes—
439C22×C4H(5,𝔽2)18yes—
440C22×C4H(5,𝔽2)14yes—
441C22×C4C2511yescharacteristic
442D8—88no—
443M16—88no—
444C2×C8—24no—
445C2×C8—24no—
446C2×C8—24no—
447C2×C8—24no—
448C2×C8—24no—
449C2×C8—24no—
450C4⋊C4—24no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—24no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—48no—
461C22⋊C4—44no—
462C22⋊C4—48no—
463C22⋊C4—44no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C22⋊C4—84no—
469C22⋊C4—84no—
470C22⋊C4—84no—
471C42—44no—
472C42—44no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—22no—
484C23—21no—
485C23—21no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—48no—
491C23—44no—
492C23—48no—
493C23—44no—
494C23—44no—
495C23—44no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—84no—
501C23—84no—
502C23—88no—
503C23—88no—
504C23—88no—
505C23—88no—
506C23—88no—
507C23—88no—
508C23—168no—
509C23—168no—
510C23#53414yes—
511C23#53412yes—
512C23C23×D411yescharacteristic
513Q8—44no—
514Q8—48no—
515Q8—44no—
516D4—48no—
517D4—48no—
518D4—44no—
519D4—48no—
520D4—48no—
521D4—48no—
522D4—44no—
523D4—44no—
524D4—44no—
525D4—44no—
526D4—44no—
527D4—88no—
528D4—816no—
529D4—816no—
530D4—168no—
531D4—168no—
532D4—168no—
533D4—168no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—28no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—48no—
548C2×C4—48no—
549C2×C4—48no—
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—88no—
567C2×C4#53414yes—
568C2×C4#53412yes—
569C2×C4C2×H(5,𝔽2)12yes—
570C2×C4C2×H(5,𝔽2)14yes—
571C8—44no—
572C8—44no—
573C22—22no—
574C22—24no—
575C22—24no—
576C22—28no—
577C22—42no—
578C22—42no—
579C22—44no—
580C22—44no—
581C22—42no—
582C22—41no—
583C22—41no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—44no—
589C22—44no—
590C22—44no—
591C22—44no—
592C22—816no—
593C22—88no—
594C22—84no—
595C22—84no—
596C22—84no—
597C22—84no—
598C22—84no—
599C22—84no—
600C22—84no—
601C22—1616no—
602C22C2×#53411yescharacteristic
603C22C2×#53412yes—
604C22#358514yes—
605C4—22no—
606C4—24no—
607C4—42no—
608C4—41no—
609C4—41no—
610C4—84no—
611C4—84no—
612C4—84no—
613C4#353214yes—
614C2—42no—
615C2—44no—
616C2—82no—
617C2—81no—
618C2—81no—
619C2—164no—
620C2#3743611yescharacteristic
621C2#6075312yes—
622C2#6077114yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35250C2semidirect
2. #32712C2semidirect
3. #29899C2semidirect
4. #29742C2semidirect
5. #29602C2semidirect
6. #27609C2semidirect
7. #24560C2semidirect
8. #24384C2semidirect
9. #24015C2semidirect
10. #23925C2semidirect
11. #20580C2semidirect
14. C2×#535C22semidirect
15. C22×#121C22semidirect
16. C2×#482C22semidirect
18. C2×#447C22semidirect
19. C2×#407C22semidirect
20. C2×#407C22semidirect
22. #2847C22semidirect
24. #2615C22semidirect
34. #2311C22semidirect
35. #2280C22semidirect
36. #2077C22semidirect
37. #2077C22semidirect
38. #2077C22semidirect
39. #2075C22semidirect
40. #2034C22semidirect
41. #2034C22semidirect
42. #2034C22semidirect
43. #2034C22semidirect
44. #2034C22semidirect
45. #2033C22semidirect
46. #2033C22semidirect
47. #2033C22semidirect
48. #1999C22semidirect
49. #1999C22semidirect
50. #1999C22semidirect
51. #1999C22semidirect
52. #1895C22semidirect
53. #1895C22semidirect
54. #1875C22semidirect
55. #1874C22semidirect
56. #1854C22semidirect
57. #1853C22semidirect
58. #1825C22semidirect
59. #1809C22semidirect
60. #1786C22semidirect
63. #1780C22semidirect
64. #1780C22semidirect

Maximal subgroups

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

subgroupindex
#352502
#327122
#298992
#297422
#296022
#276092
#245602
#243842
#240152
#239252
#205802