Finite Group Databasebeta

← #205522

Subgroups — #205522

Order 512.

882 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60846C211yesmaximal, characteristic
2#20548C211yesmaximal, characteristic, complemented
3#17806C211yesmaximal, characteristic, complemented
4#17576C211yesmaximal, characteristic
5#17562C211yesmaximal, characteristic, complemented
6#17498C211yesmaximal, characteristic
7#17486C211yesmaximal, characteristic, complemented
8#9754C212yesmaximal, complemented
9#9752C212yesmaximal, complemented
10#9751C212yesmaximal, complemented
11#9749C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#533C2211yescharacteristic
14#3233—22no—
15#3232—22no—
16#3231C411yescharacteristic, complemented
17#3231C411yescharacteristic, complemented
18#3230C411yescharacteristic, complemented
19#3229C411yescharacteristic, complemented
20#3228C411yescharacteristic, complemented
21#3228C411yescharacteristic, complemented
22#3227C411yescharacteristic, complemented
23#3226C411yescharacteristic, complemented
24#3160C2211yescharacteristic
25#3160C2211yescharacteristic
26#3131C2211yescharacteristic
27#3131C2211yescharacteristic
28#3091—22no—
29#3076—22no—
30#2920C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#423C2211yescharacteristic
34#2283C2211yescharacteristic
35#1770C2212yes—
36C8×C4⋊C4C2212yes—
37#1768—22no—
38#1753—22no—
39#1574—22no—
40#1573—22no—
41#1564C2212yes—
42#1564C2212yes—
43#1521—22no—
44#1516—22no—
45#1515—22no—
46#1511—22no—
47#1503C2212yes—
48#1503C2212yes—
49#1503C2212yes—
50#1502C2212yes—
51#1495—22no—
52#1493—22no—
53#1489—22no—
54#1487—22no—
55#1481C2212yes—
56#1481C2212yes—
57#1481C2212yes—
58#1480C2212yes—
59C22×M16—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66#533C2×C411yescharacteristic
67#533C2×C411yescharacteristic
68#533C2×C411yescharacteristic
69#533C2×C411yescharacteristic
70#533C2311yescharacteristic
71#533C2311yescharacteristic
72#533C2311yescharacteristic
73#532C2×C411yescharacteristic
74#532C2×C411yescharacteristic
75#532C2311yescharacteristic
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×C4⋊Q8C2×C411yescharacteristic
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529—21no—
83#529—21no—
84C2×#125—21no—
85#525—21no—
86#525—21no—
87#525—21no—
88C2×C22⋊Q8—21no—
89C2×C22⋊Q8—21no—
90C4×#41—21no—
91C4×#41—21no—
92C4×#41—21no—
93C2×C4×Q8—21no—
94C2×C4×Q8—21no—
95C2×#117—21no—
96C22×C42C2311yescharacteristic
97#500C2×C411yescharacteristic
98#500C2×C411yescharacteristic
99#500C2×C411yescharacteristic
100#500C2×C411yescharacteristic
101#499—22no—
102#499—22no—
103#498C2×C411yescharacteristic
104#498C2×C411yescharacteristic
105#498C2×C411yescharacteristic
106#498C2×C411yescharacteristic
107#488—22no—
108#488—22no—
109#487C2×C411yescharacteristic
110#487C2×C411yescharacteristic
111#487C2×C411yescharacteristic
112#487C2×C411yescharacteristic
113#486C2×C411yescharacteristic
114#486C2×C411yescharacteristic
115#486C2×C411yescharacteristic
116#486C2×C411yescharacteristic
117#485—22no—
118#485—22no—
119#484—22no—
120#483—22no—
121#482—22no—
122#482—22no—
123#481—22no—
124#480—22no—
125#479—22no—
126#479—22no—
127#478—22no—
128#478—22no—
129#477—22no—
130#476—22no—
131#475—22no—
132#474—22no—
133#432—21no—
134#432—21no—
135#431—21no—
136#431—21no—
137#431—21no—
138#431—21no—
139#431C2×C411yescharacteristic
140#431C2×C411yescharacteristic
141#431C2×C411yescharacteristic
142#431C2×C411yescharacteristic
143#431D412yes—
144#431D412yes—
145#431C2312yes—
146#431C2312yes—
147#427—22no—
148#427—22no—
149#423—21no—
150#423—21no—
151#423—21no—
152#423D412yes—
153#423D412yes—
154C2×C4⋊C8—21no—
155C2×C4⋊C8—21no—
156C2×C4⋊C8D412yes—
157C2×C4⋊C8C2312yes—
158#416—22no—
159#416—22no—
160#407—21no—
161C2×C22⋊C8—21no—
162C2×C22⋊C8D412yes—
163C4×M16—21no—
164C2×#97—22no—
165C2×C4×C8—21no—
166C2×C4×C8—22no—
167C2×C4×C8C2311yescharacteristic
168C2×C4×C8C2311yescharacteristic
169#383D412yes—
170#382—21no—
171#381D412yes—
172#380—21no—
173C4×C4⋊C4C2312yes—
174C4×C22⋊C4—21no—
175#376—21no—
176#336—22no—
177#336—22no—
178#335—22no—
179#335—22no—
180#334—22no—
181#334—22no—
182#328—24no—
183#328—24no—
184#328—24no—
185#327—24no—
186Q8⋊C8—24no—
187Q8⋊C8—24no—
188Q8⋊C8—24no—
189D4⋊C8—24no—
190C2×#41—21no—
191C2×#41—21no—
192C2×#41—21no—
193C22×Q8—21no—
194C22×Q8—21no—
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C23×C4#4111yescharacteristic
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—41no—
201C2×M16—82no—
202C22×C8—22no—
203C22×C8—41no—
204C22×C8C2×D412yes—
205C22×C8C2×D412yes—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8C22×C411yescharacteristic
213C4⋊Q8C22×C411yescharacteristic
214C4⋊Q8C22×C411yescharacteristic
215C4⋊Q8C22×C411yescharacteristic
216C4⋊Q8C22×C411yescharacteristic
217C4⋊Q8C22×C411yescharacteristic
218C4⋊Q8C22×C411yescharacteristic
219#127C22×C411yescharacteristic
220#126—41no—
221#126—41no—
222#125—21no—
223#125—21no—
224#125—21no—
225#125—21no—
226#125—21no—
227#125—21no—
228#125—21no—
229#125—41no—
230#125—41no—
231#124—21no—
232#124—21no—
233#124—21no—
234#123—41no—
235#123—41no—
236#123—41no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241C22⋊Q8—21no—
242C22⋊Q8—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C22⋊Q8—41no—
252C22⋊Q8—41no—
253C22⋊Q8—41no—
254C22⋊Q8—41no—
255#121—21no—
256#121—21no—
257#121—21no—
258#121—21no—
259#121—21no—
260#121—21no—
261#120—41no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—41no—
277C4×Q8—41no—
278C4×Q8—41no—
279C4×Q8—41no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—21no—
283C4×D4—21no—
284C4×D4—21no—
285C4×D4—21no—
286C4×D4—41no—
287C4×D4—41no—
288C4×D4—41no—
289C4×D4—41no—
290C4×D4—41no—
291C4×D4—41no—
292#117—21no—
293#117—21no—
294#117—21no—
295#117—21no—
296#117—21no—
297#117—21no—
298#117—21no—
299#117—41no—
300#117—41no—
301#117—41no—
302#117—41no—
303#117—41no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—41no—
310C2×C4⋊C4—41no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—41no—
313C2×C42—22no—
314C2×C42—41no—
315C2×C42—41no—
316C2×C42C22×C411yescharacteristic
317C2×C42C22×C411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D411yescharacteristic
323C2×C42C2×D411yescharacteristic
324C2×C42#4111yescharacteristic
325C2×C42#4111yescharacteristic
326C2×C42#4111yescharacteristic
327C2×C42C2411yescharacteristic
328#107—22no—
329#107—22no—
330#107—22no—
331#107—22no—
332#106—22no—
333#106—22no—
334#106—22no—
335#106—22no—
336C4⋊C8—22no—
337C4⋊C8—22no—
338C4⋊C8—22no—
339C4⋊C8—22no—
340C4⋊C8—22no—
341C4⋊C8—41no—
342C4⋊C8—41no—
343C4⋊C8C22⋊C412yes—
344C4⋊C8C22⋊C412yes—
345C4⋊C8C22⋊C412yes—
346C4⋊C8C22⋊C412yes—
347C4⋊C8C2×D414yes—
348#103—22no—
349#103—22no—
350#103—22no—
351#103—22no—
352#102—22no—
353#102—22no—
354#102—22no—
355#102—22no—
356C22⋊C8—22no—
357C22⋊C8—22no—
358C22⋊C8—41no—
359C22⋊C8—41no—
360C22⋊C8C22⋊C412yes—
361C22⋊C8C22⋊C412yes—
362C22⋊C8C22⋊C412yes—
363C22⋊C8C22⋊C412yes—
364#97—41no—
365#97—82no—
366C4×C8—22no—
367C4×C8—22no—
368C4×C8—41no—
369C4×C8—82no—
370C4×C8C22×C411yescharacteristic
371C4×C8C22×C411yescharacteristic
372C4×C8C22×C411yescharacteristic
373C4×C8C22×C411yescharacteristic
374C4×C8C2×D412yes—
375C4×C8C2×D412yes—
376#95—22no—
377#95—22no—
378#95—22no—
379#95C2×D414yes—
380C24#12111yescharacteristic
381#41—42no—
382#41—42no—
383#41—82no—
384#41—82no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—41no—
393C2×Q8—41no—
394C2×Q8—82no—
395C2×Q8—82no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4—41no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—42no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—81no—
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C4×D411yescharacteristic
415C22×C4C4×D411yescharacteristic
416C22×C4C4×D411yescharacteristic
417C22×C4C4×D411yescharacteristic
418C22×C4#12111yescharacteristic
419C22×C4#12111yescharacteristic
420C22×C4#12111yescharacteristic
421C22×C4#12111yescharacteristic
422C22×C4#12111yescharacteristic
423C22×C4#12411yescharacteristic
424C22×C4#12411yescharacteristic
425C22×C4#12711yescharacteristic
426C22×C4#12711yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C2×#4111yescharacteristic
430M16—82no—
431M16—82no—
432M16—82no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8—42no—
439C2×C8—42no—
440C2×C8—42no—
441C2×C8—42no—
442C2×C8—82no—
443C2×C8C2×C22⋊C412yes—
444C2×C8C2×C22⋊C412yes—
445C2×C8C2×C22⋊C412yes—
446C2×C8C2×C22⋊C412yes—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—44no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—41no—
479C4⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—82no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—42no—
498C42—42no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42C2×C22⋊C411yescharacteristic
504C42C2×C22⋊C411yescharacteristic
505C42C4×D411yescharacteristic
506C42C4×D411yescharacteristic
507C42C4×D411yescharacteristic
508C42C4×D411yescharacteristic
509C42#12111yescharacteristic
510C42#12111yescharacteristic
511C42#12411yescharacteristic
512C42#12411yescharacteristic
513C42#12711yescharacteristic
514C42#12711yescharacteristic
515C42C23×C411yescharacteristic
516C42C22×D411yescharacteristic
517C42C22×D411yescharacteristic
518C42C2×#4111yescharacteristic
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23#39011yescharacteristic
524C23#39011yescharacteristic
525C23C2×#12111yescharacteristic
526Q8—42no—
527Q8—42no—
528Q8—42no—
529Q8—42no—
530Q8—81no—
531Q8—81no—
532Q8—81no—
533Q8—82no—
534Q8—82no—
535D4—42no—
536D4—42no—
537D4—81no—
538D4—82no—
539D4—82no—
540D4—82no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—82no—
571C2×C4—82no—
572C2×C4—82no—
573C2×C4—82no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4#39011yescharacteristic
577C2×C4#39011yescharacteristic
578C2×C4#39011yescharacteristic
579C2×C4#39011yescharacteristic
580C2×C4#39011yescharacteristic
581C2×C4#39011yescharacteristic
582C2×C4C22×C22⋊C411yescharacteristic
583C2×C4C2×C4×D411yescharacteristic
584C2×C4C2×C4×D411yescharacteristic
585C2×C4C2×#12111yescharacteristic
586C2×C4C2×#12411yescharacteristic
587C2×C4C2×#12711yescharacteristic
588C2×C4#57712yes—
589C2×C4#57812yes—
590C8—22no—
591C8—22no—
592C8—22no—
593C8—22no—
594C8—44no—
595C8—82no—
596C8—82no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—82no—
606C22—81no—
607C22—81no—
608C22—81no—
609C22C2×#39011yescharacteristic
610C22#304612yes—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—84no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4#289812yes—
629C4#295712yes—
630C4#304211yescharacteristic
631C4#304411yescharacteristic
632C2—22no—
633C2—41no—
634C2—41no—
635C2—161no—
636C2#2124011yescharacteristic
637C2#2124411yescharacteristic
638C2#2126211yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20548C2semidirect
3. #17806C2semidirect
5. #17562C2semidirect
7. #17486C2semidirect
8. #9754C2semidirect
9. #9752C2semidirect
10. #9751C2semidirect
11. #9749C2semidirect
16. #3231C4semidirect
17. #3231C4semidirect
18. #3230C4semidirect
19. #3229C4semidirect
20. #3228C4semidirect
21. #3228C4semidirect
22. #3227C4semidirect
23. #3226C4semidirect

Maximal subgroups

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

subgroupindex
#608462
#205482
#178062
#175762
#175622
#174982
#174862
#97542
#97522
#97512
#97492