Finite Group Databasebeta

← #6500760

Subgroups — #6500760

Order 512.

2410 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35149C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic, complemented
3#27594C212yesmaximal, complemented
4#27592C212yesmaximal, complemented
5#24889C214yesmaximal, complemented
6#24885C214yesmaximal, complemented
7#24883C214yesmaximal, complemented
8#24879C214yesmaximal, complemented
9#24015C212yesmaximal, complemented
10#24014C212yesmaximal, complemented
11#21944C212yesmaximal, complemented
12#21743C211yesmaximal, characteristic, complemented
13#20601C212yesmaximal, complemented
14C23×#41C2211yescharacteristic
15C2×#536C2211yescharacteristic
16C2×#535C2211yescharacteristic
17C2×#525C2211yescharacteristic
18C2×#484C2212yescomplemented
19C2×#482C2214yescomplemented
20C2×#483C2212yescomplemented
21C2×#417C2214yes—
22C2×#417C2214yes—
23C2×#407C2212yes—
24#2849—22no—
25#2613—22no—
26#2432—22no—
27#2431—22no—
28#2407C2214yes—
29#2407C2214yes—
30#2376—22no—
31#2358—22no—
32#2352C2214yescomplemented
33D4×C4⋊C4C2214yescomplemented
34#2329—22no—
35#2325—22no—
36#2314C2212yes—
37#2311C2212yes—
38#2310C2211yescharacteristic
39#2309C2211yescharacteristic
40#2280C2211yescharacteristic
41C2×#380C2212yes—
42#2097C2214yescomplemented
43#2094C2212yescomplemented
44#2093C2212yescomplemented
45#2073C2212yescomplemented
46#2073C2212yescomplemented
47#2072C2214yescomplemented
48#2036C2214yescomplemented
49#2035C2214yescomplemented
50#2034C2214yescomplemented
51#2033C2214yescomplemented
52#1928C2214yescomplemented
53#1927C2214yescomplemented
54#1926C2214yescomplemented
55#1925C2214yescomplemented
56#1874C2214yescomplemented
57#1873C2212yescomplemented
58#1873C2212yescomplemented
59#1870C2214yescomplemented
60#1870C2214yescomplemented
61#1854C2214yes—
62#1854C2214yes—
63#1853C2212yescomplemented
64#1853C2212yescomplemented
65#1852C2212yescomplemented
66#1852C2212yescomplemented
67#1818C2212yes—
68#1809C2214yescomplemented
69#1809C2214yescomplemented
70#1788C2214yes—
71#1788C2214yes—
72#1787C2214yes—
73#1786C2214yes—
74#1780C2212yes—
75C22×#41—22no—
76C22×#41—28no—
77C22×#41—44no—
78C22×#41—44no—
79C22×#41C2314yes—
80C22×#41C2311yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4—21no—
83C23×D4C2311yescharacteristic
84C24×C4—21no—
85C24×C4C2311yescharacteristic
86C22×M16C2314yes—
87#536—44no—
88#535—44no—
89C2×C4⋊Q8—21no—
90C2×#127—21no—
91C2×#124—21no—
92C2×#124—21no—
93#525—28no—
94C2×#123—22no—
95C2×#123D412yes—
96C2×C22⋊Q8D412yes—
97C2×C22⋊Q8C2312yes—
98C2×C22⋊Q8C2312yes—
99C2×#121D412yes—
100C2×#121C2314yes—
101C2×#120—22no—
102C2×#120D412yes—
103C2×C4×D4—24no—
104C2×C4×D4C2314yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—22no—
113#484—44no—
114#483—44no—
115#482—48no—
116C2×#107C2314yes—
117C2×#107C2314yes—
118C2×#106C2314yes—
119C2×#106C2314yes—
120#417—48no—
121#417—48no—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130#407—28no—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2312yes—
137#394—24no—
138#394—24no—
139#392—24no—
140#392—24no—
141#391—22no—
142#391—22no—
143#391C2314yes—
144#390—22no—
145#390—22no—
146#390C2312yes—
147#390C2312yes—
148#389—24no—
149#388—24no—
150#388—24no—
151#387—24no—
152#387—24no—
153#386—22no—
154#386—22no—
155#386—24no—
156#386—24no—
157#386—24no—
158#385—24no—
159#385—24no—
160#385—24no—
161#385—24no—
162#384C2312yes—
163#382C2312yes—
164#381—21no—
165#381—22no—
166#380—22no—
167#380—22no—
168#380C2314yes—
169#380C2314yes—
170#380C2314yes—
171#380C2312yes—
172#380C2312yes—
173#379—21no—
174#379—22no—
175C4×C4⋊C4C2314yes—
176C4×C22⋊C4—22no—
177C4×C22⋊C4—24no—
178C2×#95—22no—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182#340C2314yes—
183#336C2314yes—
184#336C2312yes—
185#336C2312yes—
186C25—21no—
187C25C2×D411yescharacteristic
188C2×#41—24no—
189C2×#41—28no—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—28no—
193C2×#41—42no—
194C2×#41—42no—
195C2×#41—44no—
196C2×#41—44no—
197C2×#41—48no—
198C2×#41—48no—
199C2×#41—464no—
200C2×#41—48no—
201C22×Q8—24no—
202C22×Q8—21no—
203C22×Q8—24no—
204C22×Q8—42no—
205C22×Q8C2×D411yescharacteristic
206C22×Q8C2412yes—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—21no—
211C22×D4—24no—
212C22×D4—42no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4C2×D414yes—
217C22×D4C2×D412yes—
218C22×D4C2×D411yescharacteristic
219C22×D4C2412yes—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—24no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2×D412yes—
231C23×C4C2×D411yescharacteristic
232C23×C4#4118yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C2×M16—28no—
237C2×M16—44no—
238C2×M16—44no—
239C22×C8C2414yes—
240C22×C8C2414yes—
241C4⋊Q8—44no—
242#127—44no—
243#124—44no—
244#124—44no—
245#123—28no—
246#123—48no—
247C22⋊Q8—28no—
248C22⋊Q8—24no—
249C22⋊Q8—24no—
250C22⋊Q8—42no—
251C22⋊Q8—42no—
252#121—28no—
253#121—28no—
254#121—44no—
255#120—28no—
256#120—48no—
257C4×D4—28no—
258C4×D4—28no—
259C4×D4—88no—
260#117—24no—
261#117—42no—
262#117—42no—
263#117—44no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2414yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2411yescharacteristic
280C2×C4⋊C4C2411yescharacteristic
281C2×C22⋊C4—24no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—84no—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D414yes—
297C2×C22⋊C4C2×D414yes—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42C2412yes—
303#107—44no—
304#107—44no—
305#106—44no—
306#106—44no—
307#103—28no—
308#103—28no—
309#103—44no—
310#103—44no—
311#102—28no—
312#102—28no—
313#102—44no—
314#102—44no—
315C22⋊C8—28no—
316C22⋊C8—24no—
317C22⋊C8—24no—
318C22⋊C8—44no—
319C22⋊C8—42no—
320C22⋊C8—42no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330C24—24no—
331C24—22no—
332C24—42no—
333C24—41no—
334C24—41no—
335C24—42no—
336C24—42no—
337C24—42no—
338C24—82no—
339C24—82no—
340C24#12314yes—
341C24C22×D412yes—
342C24C22×D411yescharacteristic
343C24C22×D412yes—
344#41—232no—
345#41—416no—
346#41—416no—
347#41—416no—
348#41—464no—
349#41—88no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—416no—
358C2×Q8—42no—
359C2×Q8—42no—
360C2×Q8H(5,𝔽2)14yes—
361C2×D4—24no—
362C2×D4—28no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—22no—
368C2×D4—416no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—48no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—48no—
378C2×D4—432no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—84no—
383C2×D4—84no—
384C2×D4H(5,𝔽2)14yes—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—416no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—48no—
406C22×C4—48no—
407C22×C4—42no—
408C22×C4—432no—
409C22×C4—42no—
410C22×C4—48no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—82no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4—82no—
417C22×C4—82no—
418C22×C4—84no—
419C22×C4#12318yes—
420C22×C4#12314yes—
421C22×C4C22×D414yes—
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C2×#4114yes—
426C22×C4C2×#4114yes—
427C22×C4C2×#4114yes—
428C22×C4C2511yescharacteristic
429M16—88no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—24no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—84no—
451C22⋊C4—24no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—48no—
455C22⋊C4—44no—
456C22⋊C4—44no—
457C22⋊C4—48no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—84no—
461C22⋊C4—84no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—84no—
465C22⋊C4—82no—
466C22⋊C4—84no—
467C42—24no—
468C42—42no—
469C42—42no—
470C42—82no—
471C42—84no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—24no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—22no—
488C23—44no—
489C23—48no—
490C23—48no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—42no—
495C23—88no—
496C23—88no—
497C23—84no—
498C23—82no—
499C23—82no—
500C23—84no—
501C23—84no—
502C23C2×#12312yes—
503C23C2×#12314yes—
504C23C23×D411yescharacteristic
505Q8—28no—
506Q8—48no—
507Q8—44no—
508Q8—82no—
509D4—28no—
510D4—48no—
511D4—416no—
512D4—432no—
513D4—44no—
514D4—82no—
515D4—84no—
516D4—168no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—44no—
532C2×C4—48no—
533C2×C4—432no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—44no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—164no—
553C2×C4—164no—
554C2×C4—164no—
555C2×C4C2×#12314yes—
556C2×C4C2×#12312yes—
557C2×C4C22×#4112yes—
558C2×C4C2×H(5,𝔽2)12yes—
559C2×C4C2×H(5,𝔽2)12yes—
560C8—44no—
561C8—44no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—42no—
573C22—41no—
574C22—44no—
575C22—42no—
576C22—44no—
577C22—44no—
578C22—48no—
579C22—42no—
580C22—42no—
581C22—88no—
582C22—816no—
583C22—84no—
584C22—82no—
585C22—82no—
586C22C22×#12311yescharacteristic
587C22#348411yescharacteristic
588C22#348411yescharacteristic
589C22#358512yes—
590C22#358612yes—
591C4—24no—
592C4—22no—
593C4—42no—
594C4—44no—
595C4—44no—
596C4—84no—
597C4—82no—
598C4—81no—
599C4—84no—
600C4—81no—
601C4#349014yes—
602C2—44no—
603C2—42no—
604C2—82no—
605C2—84no—
606C2#3535011yescharacteristic
607C2#6051712yes—
608C2#6164712yes—
609C2#6165111yescharacteristic
610C2#6165211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35149C2
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
2. #34360C2semidirect
3. #27594C2semidirect
4. #27592C2semidirect
5. #24889C2semidirect
6. #24885C2semidirect
7. #24883C2semidirect
8. #24879C2semidirect
9. #24015C2semidirect
10. #24014C2semidirect
11. #21944C2semidirect
12. #21743C2semidirect
13. #20601C2semidirect
18. C2×#484C22semidirect
19. C2×#482C22semidirect
20. C2×#483C22semidirect
32. #2352C22semidirect
33. D4×C4⋊C4C22semidirect
42. #2097C22semidirect
43. #2094C22semidirect
44. #2093C22semidirect
45. #2073C22semidirect
46. #2073C22semidirect
47. #2072C22semidirect
48. #2036C22semidirect
49. #2035C22semidirect
50. #2034C22semidirect
51. #2033C22semidirect
52. #1928C22semidirect
53. #1927C22semidirect
54. #1926C22semidirect
55. #1925C22semidirect
56. #1874C22semidirect
57. #1873C22semidirect
58. #1873C22semidirect
59. #1870C22semidirect
60. #1870C22semidirect
63. #1853C22semidirect
64. #1853C22semidirect
65. #1852C22semidirect
66. #1852C22semidirect
68. #1809C22semidirect
69. #1809C22semidirect

Maximal subgroups

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

subgroupindex
#351492
#343602
#275942
#275922
#248892
#248852
#248832
#248792
#240152
#240142
#219442
#217432
#206012