Finite Group Databasebeta

← #216026

Subgroups — #216026

Order 512.

732 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35132C211yesmaximal, characteristic
2#21435C211yesmaximal, characteristic
3#20465C211yesmaximal, characteristic
4#9894C211yesmaximal, characteristic, complemented
5#9893C211yesmaximal, characteristic, complemented
6#9892C211yesmaximal, characteristic, complemented
7#9892C211yesmaximal, characteristic, complemented
8#9892C211yesmaximal, characteristic, complemented
9#9891C211yesmaximal, characteristic, complemented
10#9890C211yesmaximal, characteristic, complemented
11#9890C211yesmaximal, characteristic, complemented
12#8720C211yesmaximal, characteristic
13#8720C211yesmaximal, characteristic
14#8719C211yesmaximal, characteristic
15#8718C211yesmaximal, characteristic
16C2×#433—21no—
17#2864C2211yescharacteristic
18#2864C2211yescharacteristic
19#2862C2211yescharacteristic
20#2859C2211yescharacteristic
21#2666—21no—
22#2666—21no—
23#2612—21no—
24#2612—21no—
25#2570C411yescharacteristic
26#2570C411yescharacteristic
27#2550C411yescharacteristic
28#2550C411yescharacteristic
29#2442C411yescharacteristic
30#2442C411yescharacteristic
31#2440C411yescharacteristic
32#2440C411yescharacteristic
33#2438—21no—
34#2438—21no—
35#2329—21no—
36#2329—21no—
37#2283C2211yescharacteristic
38C2×#382C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1924—21no—
41#1923—21no—
42#1917—21no—
43#1916—21no—
44#1768—21no—
45#1767C2211yescharacteristic
46#1767C2211yescharacteristic
47#1767C2211yescharacteristic
48#1767C2211yescharacteristic
49#1750—21no—
50#1749C2211yescharacteristic
51#1749C2211yescharacteristic
52#1749C2211yescharacteristic
53#1749C2211yescharacteristic
54#1746—21no—
55#1745C2211yescharacteristic
56C4×#97C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C4×#97C2211yescharacteristic
59#1295C2211yescharacteristic
60#1295C2211yescharacteristic
61#1295C2211yescharacteristic
62#1295C2211yescharacteristic
63#1295C2211yescharacteristic
64#1295C2211yescharacteristic
65#1295C2211yescharacteristic
66#1295C2211yescharacteristic
67#1295C2211yescharacteristic
68#1295C2211yescharacteristic
69#1295C2211yescharacteristic
70#1295C2211yescharacteristic
71#1295C2211yescharacteristic
72#1295C2211yescharacteristic
73#1295C2211yescharacteristic
74#1295C2211yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×#125C2×C411yescharacteristic
78C2×#125C2×C411yescharacteristic
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×#117D411yescharacteristic
82C2×#117D411yescharacteristic
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C4⋊C4C2×C411yescharacteristic
85C22×C42C2311yescharacteristic
86#433—28no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×#97C2311yescharacteristic
99C2×#97C2311yescharacteristic
100C2×#97C2311yescharacteristic
101C2×#97C2311yescharacteristic
102C2×#97C2311yescharacteristic
103C2×#97C2311yescharacteristic
104C2×#97C2311yescharacteristic
105C2×#97C2311yescharacteristic
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109#400—21no—
110#400—21no—
111#400—21no—
112#400—21no—
113#398—21no—
114#398—21no—
115#398—21no—
116#398—21no—
117#398—21no—
118#398—21no—
119#398—21no—
120#398—21no—
121#395—21no—
122#395—21no—
123#395—21no—
124#395—21no—
125#391—21no—
126#391—21no—
127#391—21no—
128#391—21no—
129#391—21no—
130#391—21no—
131#389—21no—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389C2×C411yescharacteristic
138#389C2×C411yescharacteristic
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#389C2×C411yescharacteristic
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#385—21no—
158#385—21no—
159#385—21no—
160#385—21no—
161#384C2×C411yescharacteristic
162#384C2×C411yescharacteristic
163#383—21no—
164#382—21no—
165#382—21no—
166#382C2×C411yescharacteristic
167#382C2×C411yescharacteristic
168#382C2×C411yescharacteristic
169#382C2×C411yescharacteristic
170#382C2311yescharacteristic
171#382C2311yescharacteristic
172#382C2311yescharacteristic
173#382C2311yescharacteristic
174#381—21no—
175#380—21no—
176#380—21no—
177#380C2×C411yescharacteristic
178#380C2×C411yescharacteristic
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4C2×C411yescharacteristic
181C4×C4⋊C4C2×C411yescharacteristic
182C4×C4⋊C4C2×C411yescharacteristic
183C4×C4⋊C4D411yescharacteristic
184C4×C4⋊C4D411yescharacteristic
185C4×C22⋊C4—21no—
186C4×C22⋊C4C2×C411yescharacteristic
187C4×C22⋊C4C2×C411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190#376—21no—
191#376D411yescharacteristic
192#376D411yescharacteristic
193C2×#95—21no—
194C2×#95—21no—
195C43C2311yescharacteristic
196C43C2311yescharacteristic
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336—21no—
202#336—21no—
203#336—21no—
204#336—21no—
205#336—21no—
206#336—21no—
207C22×Q8—21no—
208C22×Q8—21no—
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C4⋊Q8—81no—
223C4⋊Q8—81no—
224#125—81no—
225#125—81no—
226C22⋊Q8—44no—
227C22⋊Q8—44no—
228#117—24no—
229#117—42no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
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—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—81no—
251C2×C4⋊C4—81no—
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22⋊C411yescharacteristic
254C2×C4⋊C4C22⋊C411yescharacteristic
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22×C411yescharacteristic
257C2×C4⋊C4C22×C411yescharacteristic
258C2×C4⋊C4C22×C411yescharacteristic
259C2×C4⋊C4C22×C411yescharacteristic
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2411yescharacteristic
292C4⋊C8—24no—
293C4⋊C8—24no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C22⋊C8—24no—
297C22⋊C8—24no—
298C22⋊C8—42no—
299C22⋊C8—42no—
300#97—24no—
301#97—21no—
302#97—21no—
303#97—21no—
304#97—21no—
305#97—21no—
306#97—21no—
307#97—21no—
308#97—21no—
309#97—21no—
310#97—21no—
311#97—21no—
312#97—21no—
313#97—21no—
314#97—21no—
315#97—21no—
316#97—21no—
317#97—42no—
318#97—42no—
319C4×C8—24no—
320C4×C8—42no—
321C4×C8—42no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356#95C22⋊C411yescharacteristic
357#95C22⋊C411yescharacteristic
358C24C2×C22⋊C411yescharacteristic
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8—81no—
362C2×Q8—81no—
363C2×Q8—81no—
364C2×Q8—81no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4—81no—
390C22×C4—81no—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C2×C8—22no—
406C2×C8—22no—
407C2×C8—22no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—41no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4#10414yes—
467C42—21no—
468C42—21no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—22no—
474C42—22no—
475C42—22no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—81no—
486C42—81no—
487C422−1+411yescharacteristic
488C422−1+411yescharacteristic
489C422−1+411yescharacteristic
490C422−1+411yescharacteristic
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23C2×#10412yes—
498C23C22×C22⋊C411yescharacteristic
499Q8—162no—
500Q8—162no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
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—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—81no—
566C2×C4—81no—
567C2×C4—162no—
568C2×C4—162no—
569C2×C4C2×#10412yes—
570C2×C4C2×#10412yes—
571C2×C4#42112yes—
572C2×C4#51911yescharacteristic
573C2×C4#51911yescharacteristic
574C2×C4#53611yescharacteristic
575C2×C4#53611yescharacteristic
576C2×C4#53611yescharacteristic
577C2×C4#53611yescharacteristic
578C8—42no—
579C8—42no—
580C8—42no—
581C8—42no—
582C8—42no—
583C8—42no—
584C8—81no—
585C8—81no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22#231011yescharacteristic
594C22#288811yescharacteristic
595C22#288911yescharacteristic
596C22C22×#10411yescharacteristic
597C22C2×#42111yescharacteristic
598C22#290111yescharacteristic
599C22#290111yescharacteristic
600C4—22no—
601C4—21no—
602C4—21no—
603C4—21no—
604C4—21no—
605C4—21no—
606C4—21no—
607C4—22no—
608C4—22no—
609C4—41no—
610C4—41no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C2—42no—
626C2#1765211yescharacteristic
627C2#1765311yescharacteristic
628C2#1767911yescharacteristic
629C2#1768211yescharacteristic
630C2#2183211yescharacteristic
631C2#2184411yescharacteristic
632C2#2184611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9894C2semidirect
5. #9893C2semidirect
6. #9892C2semidirect
7. #9892C2semidirect
8. #9892C2semidirect
9. #9891C2semidirect
10. #9890C2semidirect
11. #9890C2semidirect

Maximal subgroups

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

subgroupindex
#351322
#214352
#204652
#98942
#98932
#98922
#98912
#98902
#87202
#87192
#87182