Finite Group Databasebeta

← #6528598

Subgroups — #6528598

Order 512.

1474 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37640C211yesmaximal, characteristic
2#35181C211yesmaximal, characteristic, complemented
3#29345C214yesmaximal, complemented
4#27984C214yesmaximal, complemented
5#27764C214yesmaximal, complemented
6#27758C212yesmaximal, complemented
7#27750C212yesmaximal, complemented
8#27744C214yesmaximal, complemented
9#27738C212yesmaximal, complemented
10#27729C212yesmaximal, complemented
11#21870C211yesmaximal, characteristic, complemented
12#21863C211yesmaximal, characteristic
13#21395C211yesmaximal, characteristic
14#21385C211yesmaximal, characteristic, complemented
15#20537C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#532C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2851—22no—
21#2820—22no—
22#2646—22no—
23#2624—22no—
24#2620C2214yes—
25#2616—22no—
26#2571—21no—
27#2571C2212yes—
28#2570—22no—
29#2570C2214yes—
30#2567C2212yes—
31#2552—21no—
32#2552C2212yes—
33#2548C2212yes—
34#2547C2212yes—
35#2542C2212yes—
36#2538—21no—
37#2538—21no—
38#2317C2212yes—
39#2317C2212yes—
40#2285C2211yescharacteristic
41#2285C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43#2100C2214yes—
44#2099C2214yes—
45#2074C2214yes—
46#2073C2214yes—
47#2072C2214yes—
48#2071C2214yes—
49#2042C2214yes—
50#2042C2214yes—
51#1993C2212yes—
52#1990C2211yescharacteristic
53#1990C2211yescharacteristic
54#1987C2212yes—
55#1987C2212yes—
56#1982C2212yes—
57#1981C2212yes—
58#1960C2212yes—
59#1959C2214yes—
60#1958C2214yes—
61#1957C2211yescharacteristic
62#1957C2211yescharacteristic
63#1936C2214yes—
64#1935C2214yes—
65#1934C2214yes—
66#1933C2214yes—
67#1875C2214yes—
68#1874C2212yes—
69#1874C2212yes—
70#1873C2214yes—
71#1872C2212yes—
72#1872C2212yes—
73#1836C2214yes—
74#1834C2214yes—
75#1804C2214yes—
76#1803C2212yes—
77#1803C2212yes—
78#1771C2214yes—
79#1770C2212yes—
80C8×C4⋊C4C2212yes—
81#1762C2214yes—
82C4×#103C2212yes—
83C4×#103C2212yes—
84#1750C2214yes—
85C22×#41—21no—
86#533—24no—
87#533—24no—
88#532—24no—
89#532—24no—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#127C2311yescharacteristic
94C2×#125—22no—
95C2×#125—21no—
96C2×#124—21no—
97C2×#124C2314yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8C2312yes—
102C2×#121—21no—
103C2×#121C2312yes—
104C2×C4×Q8—21no—
105C2×C4×Q8C2312yes—
106C2×C4×D4—21no—
107C2×C4×D4C2312yes—
108C2×#117—21no—
109C2×#117C2312yes—
110C2×#117C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C42—21no—
114C22×C42C2311yescharacteristic
115#432—24no—
116#431—24no—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2312yes—
122C2×#103C2312yes—
123C2×#103C2312yes—
124C2×#103C2312yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2312yes—
132C2×#97C2314yes—
133C2×C4×C8C2311yescharacteristic
134C2×C4×C8C2311yescharacteristic
135C2×C4×C8C2314yes—
136#400—24no—
137#400—24no—
138#399—24no—
139#398—24no—
140#398—24no—
141#397—24no—
142#396—24no—
143#396—24no—
144#395—24no—
145#394—24no—
146#394—24no—
147#394—24no—
148#391C2314yes—
149#390—22no—
150#390—22no—
151#390C2314yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389—22no—
158#389C2314yes—
159#389C2314yes—
160#387—24no—
161#387—24no—
162#385—24no—
163#385—24no—
164#384—22no—
165#384D411yescharacteristic
166#384D411yescharacteristic
167#384C2314yes—
168#384C2314yes—
169#384C2311yescharacteristic
170#384C2311yescharacteristic
171#382—22no—
172#382—22no—
173#382D412yes—
174#382C2312yes—
175#382C2314yes—
176#380—21no—
177#380—21no—
178#380—22no—
179#380—22no—
180#380D412yes—
181#380D412yes—
182C4×C4⋊C4—21no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4—21no—
185C4×C4⋊C4—21no—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C4⋊C4C2312yes—
190C2×#95—21no—
191#340C2314yes—
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2312yes—
196#340C2314yes—
197#340C2312yes—
198#340C2312yes—
199#340C2314yes—
200#340C2312yes—
201#336C2314yes—
202#336C2314yes—
203C2×#41—44no—
204C2×#41—82no—
205C2×#41—82no—
206C22×Q8—21no—
207C22×Q8—21no—
208C22×Q8C2412yes—
209C22×D4—22no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C2412yes—
213C23×C4—21no—
214C23×C4—21no—
215C23×C4—22no—
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C23×C4C2411yescharacteristic
219C22×C8C2412yes—
220C22×C8C2412yes—
221C22×C8C2414yes—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227C4⋊Q8—22no—
228#127—22no—
229#127—22no—
230#125—44no—
231#125—84no—
232#124—44no—
233#124—44no—
234#124—44no—
235C22⋊Q8—24no—
236C22⋊Q8—24no—
237C22⋊Q8—44no—
238C22⋊Q8—84no—
239C22⋊Q8—84no—
240#121—24no—
241#121—24no—
242#121—44no—
243C4×Q8—24no—
244C4×Q8—24no—
245C4×Q8—44no—
246C4×D4—24no—
247C4×D4—24no—
248C4×D4—44no—
249#117—44no—
250#117—44no—
251#117—44no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4C2×D412yes—
273C2×C4⋊C4C2×D412yes—
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D412yes—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4C2×D414yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—82no—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42#4114yes—
301C2×C42#4114yes—
302C2×C42C2414yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C4⋊C8—24no—
306C4⋊C8—24no—
307#103—44no—
308#103—44no—
309#103—42no—
310#103—42no—
311#103—42no—
312#103—42no—
313#102—44no—
314#102—44no—
315#102—44no—
316#102—44no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319#97—24no—
320#97—44no—
321C4×C8—22no—
322C4×C8—22no—
323C4×C8—44no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95C2×D414yes—
335#95C2×D414yes—
336C24—22no—
337C24C22×D411yescharacteristic
338#41—816no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—81no—
350C2×Q8—81no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—44no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—81no—
361C2×D4—81no—
362C2×D4—82no—
363C2×D4—82no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4#12414yes—
387C22×C4#12414yes—
388C22×C4C22×D414yes—
389C22×C4C22×D414yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2511yescharacteristic
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—22no—
402C2×C8—24no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—42no—
431C4⋊C4—42no—
432C4⋊C4—42no—
433C4⋊C4—81no—
434C4⋊C4—81no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4—84no—
440C4⋊C4—84no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—84no—
447C22⋊C4—84no—
448C22⋊C4—84no—
449C22⋊C4—84no—
450C42—24no—
451C42—24no—
452C42—44no—
453C42—44no—
454C42—44no—
455C42—82no—
456C42#12414yes—
457C42#12414yes—
458C42C2×#4112yes—
459C42C2×#4112yes—
460C42C2×#4112yes—
461C42C2×#4112yes—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—42no—
475C23C2×#12412yes—
476C23C23×D411yescharacteristic
477Q8—42no—
478Q8—42no—
479Q8—42no—
480Q8—42no—
481Q8—84no—
482Q8—84no—
483D4—44no—
484D4—44no—
485D4—84no—
486D4—88no—
487D4—84no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—28no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—24no—
500C2×C4—24no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—88no—
528C2×C4—84no—
529C2×C4—84no—
530C2×C4C2×#12412yes—
531C2×C4C2×#12412yes—
532C2×C4C2×#12412yes—
533C2×C4C2×#12412yes—
534C2×C4C2×#12412yes—
535C2×C4#57712yes—
536C2×C4#57812yes—
537C2×C4C22×#4111yescharacteristic
538C2×C4C22×#4111yescharacteristic
539C2×C4#58512yes—
540C2×C4#58512yes—
541C8—42no—
542C8—42no—
543C8—44no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22C22×#12411yescharacteristic
559C22#348511yescharacteristic
560C22#348711yescharacteristic
561C22C2×#57711yescharacteristic
562C22C2×#57811yescharacteristic
563C22#358511yescharacteristic
564C22#358511yescharacteristic
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—44no—
570C4—44no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4#313714yes—
579C4#351212yes—
580C4#351212yes—
581C2—42no—
582C2—162no—
583C2#3564911yescharacteristic
584C2C2×#313711yescharacteristic
585C2#6061911yescharacteristic
586C2#6180811yescharacteristic
587C2#6181011yescharacteristic
588C2#6191311yescharacteristic
589C2#6191711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #35181C2semidirect
3. #29345C2semidirect
4. #27984C2semidirect
5. #27764C2semidirect
6. #27758C2semidirect
7. #27750C2semidirect
8. #27744C2semidirect
9. #27738C2semidirect
10. #27729C2semidirect
11. #21870C2semidirect
14. #21385C2semidirect
15. #20537C2semidirect

Maximal subgroups

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

subgroupindex
#376402
#351812
#293452
#279842
#277642
#277582
#277502
#277442
#277382
#277292
#218702
#218632
#213952
#213852
#205372