Finite Group Databasebeta

← #6459005

Subgroups — #6459005

Order 512.

1825 conjugacy classes of subgroups.

The subgroup lattice has 615 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34269C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#27677C212yesmaximal, complemented
4#27650C212yesmaximal, complemented
5#27640C214yesmaximal, complemented
6#27615C212yesmaximal, complemented
7#27608C212yesmaximal, complemented
8#27582C212yesmaximal, complemented
9#24015C212yesmaximal
10#23761C212yesmaximal, complemented
11#23754C212yesmaximal, complemented
12#21590C214yesmaximal, complemented
13#21368C212yesmaximal, complemented
14#20744C212yesmaximal, complemented
15#20720C211yesmaximal, characteristic, complemented
16C2×#535C2211yescharacteristic
17C2×#518C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#482C2214yes—
20C2×#433C2212yes—
21#2367—22no—
22#2366—22no—
23#2358—22no—
24#2358—22no—
25#2355—22no—
26#2354—22no—
27#2352—22no—
28#2352—22no—
29#2324C2212yes—
30#2324C2212yes—
31#2314C2212yes—
32#2314C2212yescomplemented
33#2311C2212yes—
34#2311C2211yescharacteristic
35C4×#121C2212yes—
36C4×#121C2212yes—
37#2296C2211yescharacteristic
38C2×#390C2211yescharacteristic
39#2285C2212yes—
40#2285C2212yescomplemented
41#2285C2211yescharacteristic, complemented
42C2×#384C2211yescharacteristic, complemented
43#2274C2211yescharacteristic
44#2274C2212yescomplemented
45#2099C2212yescomplemented
46#2099C2212yescomplemented
47#2093C2212yes—
48#2093C2212yes—
49#2084C2212yescomplemented
50#2083C2212yescomplemented
51#2076C2214yescomplemented
52#2075C2214yescomplemented
53#2071C2212yes—
54#2071C2212yes—
55#2034C2214yes—
56#2033C2214yes—
57#1949C2214yescomplemented
58#1947C2214yescomplemented
59#1925C2214yes—
60#1925C2214yes—
61#1921C2214yescomplemented
62#1919C2214yescomplemented
63#1874C2214yes—
64#1854C2214yes—
65#1852C2214yes—
66#1852C2214yescomplemented
67#1825C2214yes—
68#1825C2214yescomplemented
69#1817C2212yes—
70#1804C2214yes—
71#1802C2214yes—
72#1798C2212yescomplemented
73#1797C2212yescomplemented
74#1786C2214yes—
75#1780C2212yes—
76#1776C2214yescomplemented
77C4×#107C2214yescomplemented
78#1762C2214yes—
79C4×#102C2214yes—
80#1750C2212yescomplemented
81#1749C2211yescharacteristic, complemented
82C4×C22⋊C8C2211yescharacteristic, complemented
83C22×#41C2311yescharacteristic
84C23×D4C2311yescharacteristic
85#535—28no—
86C2×#127—21no—
87C2×#124—21no—
88C2×#121C2312yes—
89C2×#121C2312yes—
90C2×#120—21no—
91C2×#120—21no—
92#518—28no—
93C4×#41—28no—
94C2×C4×Q8D412yes—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4—22no—
98C2×C4×D4D412yes—
99C2×C4×D4D412yes—
100C2×C4×D4D412yes—
101C2×C4×D4C2312yes—
102C2×C4×D4C2312yes—
103C2×#117—22no—
104C2×#117—22no—
105C2×#117C2312yes—
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—22no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—22no—
115C22×C22⋊C4—22no—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#482—216no—
120#433—28no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×#97C2312yes—
135C2×C4×C8C2312yes—
136#391—22no—
137#391—21no—
138#390—22no—
139#390—21no—
140#390—22no—
141#390—22no—
142#390—22no—
143#390C2312yes—
144#390C2312yes—
145#389C2312yes—
146#389C2312yes—
147#388—22no—
148#388—22no—
149#388—24no—
150#388—24no—
151#388—22no—
152#388—22no—
153#387—24no—
154#387—24no—
155#386—22no—
156#386—22no—
157#386—21no—
158#386—21no—
159#386—22no—
160#386—22no—
161#386—22no—
162#386—22no—
163#385—22no—
164#385—22no—
165#385—22no—
166#384—22no—
167#384C2312yes—
168#384C2312yes—
169#384C2312yes—
170#382—22no—
171#382—22no—
172#382C2312yes—
173#380—22no—
174#380—22no—
175#380—22no—
176#380C2314yes—
177#380C2314yes—
178#380C2312yes—
179#380C2312yes—
180C4×C4⋊C4—24no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4—22no—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2314yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—24no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4—22no—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2314yes—
205C25—21no—
206C2×#41—24no—
207C2×#41—24no—
208C2×#41—42no—
209C22×Q8C2×D411yescharacteristic
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—24no—
213C22×D4—21no—
214C22×D4—24no—
215C22×D4—42no—
216C22×D4C2×D411yescharacteristic
217C22×D4C2×D411yescharacteristic
218C22×D4C2×D411yescharacteristic
219C22×D4C2412yes—
220C23×C4—21no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4C2×D412yes—
225C23×C4C2×D412yes—
226C23×C4C2×D412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C22×C8C2414yes—
231C22×C8C2414yes—
232#127—28no—
233#124—28no—
234#121—24no—
235#121—24no—
236#121—24no—
237#121#4118yes—
238#120—28no—
239#120—28no—
240C4×Q8—28no—
241C4×D4—28no—
242C4×D4—28no—
243C4×D4—28no—
244C4×D4—28no—
245C4×D4—28no—
246C4×D4—48no—
247C4×D4—84no—
248C4×D4—84no—
249#117—24no—
250#117—24no—
251#117—24no—
252#117—24no—
253#117—28no—
254#117—84no—
255#117—84no—
256#117#4118yes—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—84no—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2414yes—
277C2×C4⋊C4C2412yes—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—84no—
295C2×C22⋊C4C2×D412yes—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C42—21no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2411yescharacteristic
315#107—28no—
316#107—44no—
317#107—44no—
318#106—28no—
319C4⋊C8—28no—
320#102—28no—
321#102—28no—
322#102—44no—
323#102—44no—
324C22⋊C8—28no—
325C22⋊C8—28no—
326C22⋊C8—42no—
327C22⋊C8—42no—
328#97—24no—
329C4×C8—24no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—22no—
341C24—24no—
342C24—22no—
343C24—42no—
344C24—42no—
345C24—82no—
346C24—82no—
347C24C22×D411yescharacteristic
348C24C22×D411yescharacteristic
349C24C22×D411yescharacteristic
350#41—416no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—41no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—416no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—82no—
373C2×D4—82no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4C2×#4114yes—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—44no—
399C22×C4—42no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—82no—
407C22×C4—84no—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4114yes—
417C22×C4C2511yescharacteristic
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—42no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—82no—
441C4⋊C4—84no—
442C4⋊C4#12318yes—
443C4⋊C4C2×#4114yes—
444C22⋊C4—24no—
445C22⋊C4—28no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—28no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—82no—
456C22⋊C4—82no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4#12318yes—
462C42—24no—
463C42—24no—
464C42—24no—
465C42—24no—
466C42—22no—
467C42—22no—
468C42—42no—
469C42—44no—
470C42—84no—
471C42—84no—
472C42—82no—
473C42—82no—
474C42C2×#4114yes—
475C42C2×#4114yes—
476C23—22no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—21no—
486C23—24no—
487C23—21no—
488C23—21no—
489C23—22no—
490C23—48no—
491C23—42no—
492C23—42no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—88no—
498C23—164no—
499C23—162no—
500C23—162no—
501C23C2×#12312yes—
502C23#52512yes—
503C23C23×D411yescharacteristic
504Q8—44no—
505D4—28no—
506D4—44no—
507D4—44no—
508D4—48no—
509D4—44no—
510D4—44no—
511D4—88no—
512D4—88no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—24no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—42no—
550C2×C4—164no—
551C2×C4—164no—
552C2×C4—164no—
553C2×C4—162no—
554C2×C4C2×#12312yes—
555C2×C4C2×#12314yes—
556C2×C4C2×#12314yes—
557C2×C4#52512yes—
558C2×C4#57614yes—
559C2×C4C22×#4112yes—
560C2×C4C22×#4112yes—
561C2×C4#58512yes—
562C8—44no—
563C8—44no—
564C22—24no—
565C22—24no—
566C22—22no—
567C22—22no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—41no—
581C22—44no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—168no—
587C22C22×#12311yescharacteristic
588C22C2×#52511yescharacteristic
589C22#344911yescharacteristic
590C22C2×#57612yes—
591C22#358612yes—
592C4—24no—
593C4—24no—
594C4—24no—
595C4—22no—
596C4—22no—
597C4—42no—
598C4—44no—
599C4—41no—
600C4—82no—
601C4—84no—
602C4—82no—
603C4—84no—
604C4#347514yes—
605C2—42no—
606C2—42no—
607C2—81no—
608C2—84no—
609C2#3480911yescharacteristic
610C2#6052112yes—
611C2#6059612yes—
612C2#6101111yescharacteristic
613C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34188C2semidirect
3. #27677C2semidirect
4. #27650C2semidirect
5. #27640C2semidirect
6. #27615C2semidirect
7. #27608C2semidirect
8. #27582C2semidirect
10. #23761C2semidirect
11. #23754C2semidirect
12. #21590C2semidirect
13. #21368C2semidirect
14. #20744C2semidirect
15. #20720C2semidirect
32. #2314C22semidirect
40. #2285C22semidirect
41. #2285C22semidirect
42. C2×#384C22semidirect
44. #2274C22semidirect
45. #2099C22semidirect
46. #2099C22semidirect
49. #2084C22semidirect
50. #2083C22semidirect
51. #2076C22semidirect
52. #2075C22semidirect
57. #1949C22semidirect
58. #1947C22semidirect
61. #1921C22semidirect
62. #1919C22semidirect
66. #1852C22semidirect
68. #1825C22semidirect
72. #1798C22semidirect
73. #1797C22semidirect
76. #1776C22semidirect
77. C4×#107C22semidirect
80. #1750C22semidirect
81. #1749C22semidirect
82. C4×C22⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#342692
#341882
#276772
#276502
#276402
#276152
#276082
#275822
#240152
#237612
#237542
#215902
#213682
#207442
#207202