Finite Group Databasebeta

← #182939

Subgroups — #182939

Order 512.

920 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60290C211yesmaximal, characteristic
2#20470C211yesmaximal, characteristic, complemented
3#17650C211yesmaximal, characteristic
4#17537C211yesmaximal, characteristic
5#17327C211yesmaximal, characteristic, complemented
6#8250C212yesmaximal, complemented
7#8249C212yesmaximal, complemented
8#8248C212yesmaximal, complemented
9#8247C212yesmaximal, complemented
10#8086C211yesmaximal, characteristic, complemented
11#8068C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2994—21no—
14#2994—21no—
15#2993C412yescomplemented
16#2993C412yescomplemented
17#2983C411yescharacteristic, complemented
18#2982C411yescharacteristic, complemented
19#2975C411yescharacteristic, complemented
20#2974—22no—
21#2972C411yescharacteristic, complemented
22#2969—21no—
23#2969—21no—
24#2966C2212yes—
25#2966C2212yes—
26#2924—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1767—21no—
34C4×C4⋊C8—21no—
35#1750—22no—
36#1746C2212yescomplemented
37#1726C2212yescomplemented
38#1532C414yescomplemented
39#1528C412yescomplemented
40#1527C412yescomplemented
41#1524C2212yes—
42#1524C2212yes—
43#1509C414yes—
44#1508C412yes—
45#1507C412yes—
46#1500C2212yes—
47#1500C2212yes—
48#1453—22no—
49#1451—21no—
50#1451C2212yescomplemented
51C8×M16—21no—
52C8×M16C2212yescomplemented
53#1302C2212yescomplemented
54#1302C2212yescomplemented
55#1300C2211yescharacteristic, complemented
56#1300C2211yescharacteristic, complemented
57#1300C2211yescharacteristic, complemented
58#1300C2211yescharacteristic, complemented
59C22×M16—21no—
60#529—21no—
61#529—21no—
62#529—21no—
63#529—21no—
64#529C2×C412yes—
65#529C2×C412yes—
66#529C2×C412yes—
67#529C2312yes—
68C2×#126—21no—
69C2×#125C2×C411yescharacteristic
70C2×#124C2×C411yescharacteristic
71C2×#123—21no—
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×#117—21no—
78C2×#117—21no—
79C22×C42C2311yescharacteristic
80#446—21no—
81#446—21no—
82#446C2×C412yes—
83#446C2×C412yes—
84C8×Q8—21no—
85C8×Q8—21no—
86C8×Q8C2×C412yes—
87C8×Q8C2×C412yes—
88#436—21no—
89#436—21no—
90#436—22no—
91#436—22no—
92#436C2×C412yes—
93#436C2×C412yes—
94#435—22no—
95#435—22no—
96#435—22no—
97#435—22no—
98C8×D4—22no—
99C8×D4—22no—
100C8×D4—21no—
101C8×D4—21no—
102C8×D4C2×C412yes—
103C8×D4C2×C412yes—
104#433—22no—
105#433—22no—
106#433—22no—
107#433—22no—
108#432—21no—
109#432—21no—
110#432—21no—
111#432—21no—
112#432—22no—
113#432C2×C412yes—
114#432C2×C412yes—
115#432C2×C412yes—
116#432C2311yescharacteristic
117#432C2311yescharacteristic
118#431—21no—
119#431—21no—
120#431—22no—
121#431C2×C411yescharacteristic
122#431C2×C411yescharacteristic
123#431C2×C411yescharacteristic
124#431C2×C411yescharacteristic
125#431C2×C412yes—
126#431D412yes—
127#431D411yescharacteristic
128#431D411yescharacteristic
129#431C2312yes—
130#424—22no—
131#424—22no—
132#423—21no—
133#423C2×C411yescharacteristic
134#423C2×C411yescharacteristic
135C2×C4⋊C8—21no—
136C2×C4⋊C8D411yescharacteristic
137C2×C4⋊C8D411yescharacteristic
138#407—21no—
139#407—22no—
140#407C2×C412yes—
141C2×C22⋊C8—21no—
142C2×C22⋊C8—22no—
143C2×C22⋊C8D412yes—
144#405—22no—
145#405—22no—
146C4×M16—21no—
147C4×M16—21no—
148C4×M16—21no—
149C4×M16—21no—
150C4×M16C2×C412yes—
151C4×M16C2311yescharacteristic
152C4×M16C2311yescharacteristic
153C2×#97—22no—
154C2×#97C2×C412yes—
155C2×#97C2312yes—
156C2×C4×C8—21no—
157C2×C4×C8—22no—
158C2×C4×C8C2312yes—
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2×C412yes—
161C4×C22⋊C4—21no—
162C4×C22⋊C4C2×C412yes—
163#376C2311yescharacteristic
164#376C2311yescharacteristic
165C43—21no—
166#336—22no—
167#336—22no—
168#330C2×C414yes—
169#330C2×C414yes—
170#329C2×C414yes—
171#329C2×C414yes—
172#328C2×C414yes—
173#328C2×C414yes—
174#327C2×C414yes—
175#327C2×C414yes—
176#322—22no—
177#322—22no—
178#322—22no—
179C82—22no—
180C22×Q8C4211yescharacteristic
181C22×D4—21no—
182C22×D4C4211yescharacteristic
183C23×C4—21no—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4111yescharacteristic
187C2×M16—21no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—44no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8C2×D412yes—
201C22×C8C2×D412yes—
202#126—22no—
203#126—22no—
204#126—22no—
205#126—22no—
206#125—21no—
207#125—21no—
208#125C4212yes—
209#125C22×C412yes—
210#124—21no—
211#124—21no—
212#124C4212yes—
213#124C22×C412yes—
214#123—22no—
215#123—22no—
216#123—22no—
217#123—22no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220C22⋊Q8C4212yes—
221C22⋊Q8C22×C412yes—
222#121—21no—
223#121—21no—
224#121C4212yes—
225#121C22×C412yes—
226C4×Q8—21no—
227C4×Q8—21no—
228C4×Q8C4212yes—
229C4×Q8C22×C412yes—
230C4×D4—22no—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—21no—
235C4×D4—21no—
236C4×D4C4212yes—
237C4×D4C22×C412yes—
238#117—22no—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—42no—
243#117—42no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4C4211yescharacteristic
250C2×C4⋊C4C4211yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—22no—
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42C2411yescharacteristic
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8C22⋊C412yes—
285C4⋊C8C22⋊C412yes—
286C4⋊C8C22⋊C411yescharacteristic
287C4⋊C8C22⋊C411yescharacteristic
288C4⋊C8C22⋊C411yescharacteristic
289C4⋊C8C22⋊C411yescharacteristic
290C4⋊C8C22×C411yescharacteristic
291C4⋊C8C22×C411yescharacteristic
292C4⋊C8C22×C411yescharacteristic
293C4⋊C8C22×C411yescharacteristic
294C22⋊C8—21no—
295C22⋊C8—21no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8C22⋊C412yes—
306C22⋊C8C22⋊C412yes—
307C22⋊C8C22⋊C412yes—
308C22⋊C8C22⋊C412yes—
309C22⋊C8C22×C414yes—
310#97—22no—
311#97—22no—
312#97—22no—
313#97—22no—
314#97—21no—
315#97—21no—
316#97—22no—
317#97—44no—
318#97C22×C412yes—
319#97C22×C412yes—
320#97C22×C414yes—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—22no—
326C4×C8—21no—
327C4×C8—22no—
328C4×C8—21no—
329C4×C8—44no—
330C4×C8C22×C412yes—
331C4×C8C22×C412yes—
332C4×C8C2×D412yes—
333C4×C8C2×D412yes—
334#95—21no—
335#95—21no—
336#95C22×C412yes—
337#95C22×C412yes—
338C24—21no—
339C24C4×D411yescharacteristic
340C2×Q8—21no—
341C2×Q8—42no—
342C2×Q8C2×C4212yes—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—42no—
347C2×D4—82no—
348C2×D4C2×C4212yes—
349C22×C4—22no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—81no—
361C22×C4—81no—
362C22×C4C2×C4211yescharacteristic
363C22×C4C2×C4211yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4#11711yescharacteristic
367C22×C4#11711yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C2×#4111yescharacteristic
380M16—24no—
381M16—44no—
382M16—44no—
383M16—42no—
384M16—82no—
385M16—82no—
386M16—81no—
387M16—81no—
388C2×C8—22no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—22no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—44no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—82no—
402C2×C8—82no—
403C2×C8C2×C22⋊C412yes—
404C2×C8C2×C22⋊C412yes—
405C2×C8C2×C22⋊C412yes—
406C2×C8C2×C22⋊C412yes—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—44no—
418C4⋊C4C2×C4212yes—
419C4⋊C4C2×C4212yes—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—44no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—44no—
436C42—41no—
437C42—41no—
438C42—82no—
439C42C2×C4211yescharacteristic
440C42C2×C4211yescharacteristic
441C42C2×C22⋊C411yescharacteristic
442C42C2×C22⋊C411yescharacteristic
443C42#11711yescharacteristic
444C42#11711yescharacteristic
445C42C4×D411yescharacteristic
446C42C4×D411yescharacteristic
447C42C4×D411yescharacteristic
448C42C4×D411yescharacteristic
449C42C4×D411yescharacteristic
450C42C4×D411yescharacteristic
451C42C23×C411yescharacteristic
452C42C23×C411yescharacteristic
453C42C22×D411yescharacteristic
454C42C2×#4111yescharacteristic
455C23—22no—
456C23—22no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—81no—
461C23—81no—
462C23C4×C22⋊C411yescharacteristic
463C23C4×C22⋊C411yescharacteristic
464C23C2×C4×D411yescharacteristic
465Q8—42no—
466Q8—42no—
467D4—42no—
468D4—44no—
469D4—42no—
470D4—82no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—82no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4C4×C22⋊C411yescharacteristic
508C2×C4C4×C22⋊C411yescharacteristic
509C2×C4C4×C22⋊C411yescharacteristic
510C2×C4C4×C22⋊C411yescharacteristic
511C2×C4C4×C22⋊C411yescharacteristic
512C2×C4C4×C22⋊C411yescharacteristic
513C2×C4C22×C4211yescharacteristic
514C2×C4C22×C22⋊C411yescharacteristic
515C2×C4C2×#11711yescharacteristic
516C2×C4C2×C4×D411yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×C4×D411yescharacteristic
519C8—24no—
520C8—22no—
521C8—22no—
522C8—22no—
523C8—22no—
524C8—44no—
525C8—42no—
526C8—44no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—41no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—82no—
543C22—162no—
544C22C2×C4×C22⋊C411yescharacteristic
545C22#295612yes—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—44no—
557C4—42no—
558C4—42no—
559C4—42no—
560C4—82no—
561C4#175712yes—
562C4#290111yescharacteristic
563C4#290111yescharacteristic
564C2—22no—
565C2—41no—
566C2—41no—
567C2—82no—
568C2C2×#175711yescharacteristic
569C2#2040511yescharacteristic
570C2#2042111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC22C2×C4×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20470C2semidirect
5. #17327C2semidirect
6. #8250C2semidirect
7. #8249C2semidirect
8. #8248C2semidirect
9. #8247C2semidirect
10. #8086C2semidirect
11. #8068C2semidirect
15. #2993C4semidirect
16. #2993C4semidirect
17. #2983C4semidirect
18. #2982C4semidirect
19. #2975C4semidirect
21. #2972C4semidirect
36. #1746C22semidirect
37. #1726C22semidirect
38. #1532C4semidirect
39. #1528C4semidirect
40. #1527C4semidirect
50. #1451C22semidirect
52. C8×M16C22semidirect
53. #1302C22semidirect
54. #1302C22semidirect
55. #1300C22semidirect
56. #1300C22semidirect
57. #1300C22semidirect
58. #1300C22semidirect

Maximal subgroups

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

subgroupindex
#602902
#204702
#176502
#175372
#173272
#82502
#82492
#82482
#82472
#80862
#80682