Finite Group Databasebeta

← #189448

Subgroups — #189448

Order 512.

836 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60290C211yesmaximal, characteristic
2#21362C211yesmaximal, characteristic, complemented
3#17816C211yesmaximal, characteristic, complemented
4#17581C211yesmaximal, characteristic
5#17537C211yesmaximal, characteristic
6#8697C212yesmaximal, complemented
7#8696C212yesmaximal, complemented
8#8695C212yesmaximal, complemented
9#8694C212yesmaximal, complemented
10#7953C211yesmaximal, characteristic, complemented
11#7948C211yesmaximal, 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
31#2285C2211yescharacteristic
32C4×#117C2211yescharacteristic
33#1948—21no—
34#1947—21no—
35#1854—22no—
36C4×#107C2212yescomplemented
37C4×#106C2212yescomplemented
38#1572—22no—
39#1571—21no—
40#1570—21no—
41#1568C2212yescomplemented
42#1568C2212yescomplemented
43#1520—21no—
44#1519—21no—
45#1512—22no—
46#1509C2212yes—
47#1509C2212yes—
48#1509C2212yes—
49#1509C2212yes—
50#1508—21no—
51#1507—21no—
52#1500—22no—
53#1285C2211yescharacteristic, complemented
54#1285C2211yescharacteristic, complemented
55#1285C2211yescharacteristic, complemented
56#1285C2211yescharacteristic, complemented
57#1285C2212yescomplemented
58#1285C2212yescomplemented
59#529—21no—
60#529—21no—
61#529—21no—
62#529—21no—
63#529C2×C412yes—
64#529D412yes—
65#529Q812yes—
66#529C2312yes—
67C2×#126—21no—
68C2×#125Q811yescharacteristic
69C2×#124Q811yescharacteristic
70C2×#123—21no—
71C2×C22⋊Q8D411yescharacteristic
72C2×#121D411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
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—22no—
110#432—21no—
111#432—21no—
112#432—21no—
113#432—21no—
114#432C2311yescharacteristic
115#432C2311yescharacteristic
116#432C2312yes—
117#432C2311yescharacteristic
118#432C2311yescharacteristic
119#431—21no—
120#431—22no—
121#431—21no—
122#431—21no—
123#431C2×C411yescharacteristic
124#431C2×C411yescharacteristic
125#431C2×C411yescharacteristic
126#431C2×C411yescharacteristic
127#431C2×C412yes—
128#431D411yescharacteristic
129#431D411yescharacteristic
130#431D412yes—
131C2×#107—22no—
132C2×#106—22no—
133#424—22no—
134#424—22no—
135#423—21no—
136#423C2×C411yescharacteristic
137#423C2×C411yescharacteristic
138C2×C4⋊C8—21no—
139C2×C4⋊C8D411yescharacteristic
140C2×C4⋊C8D411yescharacteristic
141#407—21no—
142#407—22no—
143#407—21no—
144C2×C22⋊C8—22no—
145C2×C22⋊C8—21no—
146C2×C22⋊C8D412yes—
147#405—22no—
148#405—22no—
149C4×M16—21no—
150C4×M16—21no—
151C2×#97—21no—
152C2×C4×C8C2312yes—
153#384—21no—
154#382—21no—
155#380—21no—
156#380—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4C2311yescharacteristic
160C4×C4⋊C4C2311yescharacteristic
161C4×C4⋊C4C2312yes—
162C4×C22⋊C4—21no—
163C4×C22⋊C4—21no—
164#376—21no—
165C43—21no—
166#340—22no—
167#340—22no—
168#335—22no—
169#335—22no—
170#334—22no—
171#334—22no—
172#328—22no—
173#328—22no—
174#328—22no—
175#328—22no—
176#327—22no—
177#327—22no—
178#327—22no—
179#327—22no—
180C22×Q8C4⋊C411yescharacteristic
181C22×D4—21no—
182C22×D4C4⋊C411yescharacteristic
183C23×C4—21no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4111yescharacteristic
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—41no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—41no—
195C22×C8C2×D412yes—
196C22×C8C2×D412yes—
197#126—22no—
198#126—22no—
199#126—22no—
200#126—22no—
201#125—21no—
202#125—21no—
203#125C4⋊C412yes—
204#125C2×Q812yes—
205#124—21no—
206#124—21no—
207#124C4⋊C412yes—
208#124C2×Q812yes—
209#123—22no—
210#123—22no—
211#123—22no—
212#123—22no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8C4⋊C412yes—
216C22⋊Q8C2×D412yes—
217#121—21no—
218#121—21no—
219#121C4⋊C412yes—
220#121C2×D412yes—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8C4⋊C412yes—
224C4×Q8C22×C412yes—
225C4×D4—22no—
226C4×D4—22no—
227C4×D4—22no—
228C4×D4—22no—
229C4×D4—21no—
230C4×D4—21no—
231C4×D4C4⋊C412yes—
232C4×D4C22×C412yes—
233#117—22no—
234#117—22no—
235#117—22no—
236#117—22no—
237#117—42no—
238#117—42no—
239#117—82no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4C4⋊C411yescharacteristic
250C2×C4⋊C4C4⋊C411yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—41no—
257C2×C22⋊C4—41no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×Q811yescharacteristic
271C2×C42C2×Q811yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42#4111yescharacteristic
275C2×C42C2411yescharacteristic
276#107—44no—
277#106—44no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—22no—
287C4⋊C8—41no—
288C4⋊C8—41no—
289C4⋊C8C22⋊C412yes—
290C4⋊C8C22⋊C412yes—
291C4⋊C8C22⋊C411yescharacteristic
292C4⋊C8C22⋊C411yescharacteristic
293C4⋊C8C22⋊C411yescharacteristic
294C4⋊C8C22⋊C411yescharacteristic
295C4⋊C8C22×C411yescharacteristic
296C4⋊C8C22×C411yescharacteristic
297C4⋊C8C22×C411yescharacteristic
298C4⋊C8C22×C411yescharacteristic
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—21no—
302C22⋊C8—21no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—21no—
308C22⋊C8—21no—
309C22⋊C8—22no—
310C22⋊C8—41no—
311C22⋊C8—41no—
312C22⋊C8C22⋊C412yes—
313C22⋊C8C22⋊C412yes—
314C22⋊C8C22⋊C412yes—
315C22⋊C8C22⋊C412yes—
316#97—22no—
317#97—22no—
318#97—21no—
319#97—21no—
320#97—21no—
321#97—21no—
322#97—22no—
323#97—41no—
324C4×C8—22no—
325C4×C8—22no—
326C4×C8—21no—
327C4×C8—21no—
328C4×C8—41no—
329C4×C8C22×C412yes—
330C4×C8C22×C412yes—
331C4×C8C2×D412yes—
332C4×C8C2×D412yes—
333#95—21no—
334#95—21no—
335#95—22no—
336#95—41no—
337#95—41no—
338C24—21no—
339C24#12111yescharacteristic
340C2×Q8—21no—
341C2×Q8—42no—
342C2×Q8C2×C4⋊C412yes—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—42no—
347C2×D4—81no—
348C2×D4—81no—
349C2×D4C2×C4⋊C412yes—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C4⋊C411yescharacteristic
368C22×C4C2×C4⋊C411yescharacteristic
369C22×C4#11711yescharacteristic
370C22×C4#11711yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4C22⋊Q811yescharacteristic
377C22×C4C22⋊Q811yescharacteristic
378C22×C4C22⋊Q811yescharacteristic
379C22×C4C22⋊Q811yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D411yescharacteristic
382C22×C4C2×#4111yescharacteristic
383M16—82no—
384M16—82no—
385M16—81no—
386M16—81no—
387M16—82no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—42no—
393C2×C8—42no—
394C2×C8—42no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—82no—
398C2×C8—82no—
399C2×C8—82no—
400C2×C8C2×C22⋊C412yes—
401C2×C8C2×C22⋊C412yes—
402C2×C8C2×C22⋊C412yes—
403C2×C8C2×C22⋊C412yes—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—21no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—82no—
421C4⋊C4—82no—
422C4⋊C4C2×C4⋊C412yes—
423C4⋊C4C2×C4⋊C412yes—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—44no—
435C22⋊C4—82no—
436C42—22no—
437C42—22no—
438C42—44no—
439C42—81no—
440C42—81no—
441C42—82no—
442C42—82no—
443C42C2×C22⋊C411yescharacteristic
444C42C2×C22⋊C411yescharacteristic
445C42C2×C4⋊C411yescharacteristic
446C42C2×C4⋊C411yescharacteristic
447C42#11711yescharacteristic
448C42#11711yescharacteristic
449C42#12111yescharacteristic
450C42#12111yescharacteristic
451C42C22⋊Q811yescharacteristic
452C42C22⋊Q811yescharacteristic
453C42C22⋊Q811yescharacteristic
454C42C22⋊Q811yescharacteristic
455C42C23×C411yescharacteristic
456C42C22×D411yescharacteristic
457C42C22×Q811yescharacteristic
458C42C2×#4111yescharacteristic
459C23—21no—
460C23—21no—
461C23—22no—
462C23—41no—
463C23—41no—
464C23—42no—
465C23—81no—
466C23—81no—
467C23#38011yescharacteristic
468C23#38011yescharacteristic
469C23C2×#12111yescharacteristic
470Q8—42no—
471Q8—42no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—82no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—82no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4#38011yescharacteristic
519C2×C4#38011yescharacteristic
520C2×C4#38011yescharacteristic
521C2×C4#38011yescharacteristic
522C2×C4#38011yescharacteristic
523C2×C4#38011yescharacteristic
524C2×C4C22×C22⋊C411yescharacteristic
525C2×C4C22×C4⋊C411yescharacteristic
526C2×C4C2×#11711yescharacteristic
527C2×C4C2×#12111yescharacteristic
528C2×C4C2×C22⋊Q811yescharacteristic
529C2×C4C2×C22⋊Q811yescharacteristic
530C8—22no—
531C8—22no—
532C8—22no—
533C8—22no—
534C8—41no—
535C8—41no—
536C8—41no—
537C8—41no—
538C8—81no—
539C8—81no—
540C8—82no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—81no—
553C22—81no—
554C22—162no—
555C22C2×#38011yescharacteristic
556C22#306811yescharacteristic
557C22#306911yescharacteristic
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—42no—
569C4—42no—
570C4—42no—
571C4—82no—
572C4—82no—
573C4—162no—
574C4#181212yes—
575C4#290111yescharacteristic
576C4#290111yescharacteristic
577C2—21no—
578C2—21no—
579C2—41no—
580C2—41no—
581C2—82no—
582C2#2065511yescharacteristic
583C2C2×#181211yescharacteristic
584C2#2067911yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21362C2semidirect
3. #17816C2semidirect
6. #8697C2semidirect
7. #8696C2semidirect
8. #8695C2semidirect
9. #8694C2semidirect
10. #7953C2semidirect
11. #7948C2semidirect
15. #2993C4semidirect
16. #2993C4semidirect
17. #2983C4semidirect
18. #2982C4semidirect
19. #2975C4semidirect
21. #2972C4semidirect
36. C4×#107C22semidirect
37. C4×#106C22semidirect
41. #1568C22semidirect
42. #1568C22semidirect
53. #1285C22semidirect
54. #1285C22semidirect
55. #1285C22semidirect
56. #1285C22semidirect
57. #1285C22semidirect
58. #1285C22semidirect

Maximal subgroups

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

subgroupindex
#602902
#213622
#178162
#175812
#175372
#86972
#86962
#86952
#86942
#79532
#79482