Finite Group Databasebeta

← #481786

Subgroups — #481786

Order 512.

949 conjugacy classes of subgroups.

The subgroup lattice has 611 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36019C211yesmaximal, characteristic
2#20393C211yesmaximal, characteristic, complemented
3#17368C211yesmaximal, characteristic, complemented
4#11787C212yesmaximal, complemented
5#11774C212yesmaximal, complemented
6#11758C212yesmaximal, complemented
7#11756C212yesmaximal, complemented
8#11730C212yesmaximal, complemented
9#11728C212yesmaximal, complemented
10C2×#556C411yescharacteristic, complemented
11C2×#520C2211yescharacteristic
12#2973—21no—
13C2×#410—21no—
14#2631C411yescharacteristic, complemented
15#2631C2211yescharacteristic
16#2628C411yescharacteristic, complemented
17#2610C411yescharacteristic, complemented
18#2610C2211yescharacteristic
19#2599C411yescharacteristic, complemented
20#2599C2211yescharacteristic
21#2563C2211yescharacteristic
22#2535C2211yescharacteristic
23#2481—21no—
24#2481—21no—
25#2467—21no—
26#2467—21no—
27#2421—21no—
28#2407—21no—
29#2398C411yescharacteristic, complemented
30#2398C2211yescharacteristic
31#2392C411yescharacteristic, complemented
32#2363—21no—
33#2356C411yescharacteristic, complemented
34#2352—21no—
35#2274—21no—
36C4×#99—21no—
37C4×#99—21no—
38C4×#99C2212yes—
39C4×#99C2212yes—
40#1740C2212yes—
41#1467—21no—
42#1465C2212yes—
43#1464C2212yes—
44#1463—21no—
45#1459C2212yes—
46C2×#324—21no—
47#1351C2214yes—
48#1351C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C22×#41D411yescharacteristic
52#556—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#126—21no—
55C2×#126—21no—
56C2×#124C2×C411yescharacteristic
57C2×#124C2×C411yescharacteristic
58C2×C22⋊Q8—21no—
59C2×#121—21no—
60#520—44no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—22no—
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2311yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81#435—44no—
82#432—22no—
83#410—28no—
84C2×#99—22no—
85C2×#99—22no—
86#407—22no—
87C2×C22⋊C8—22no—
88#400—21no—
89#400—21no—
90#400—21no—
91#399—21no—
92#398C2×C411yescharacteristic
93#397—21no—
94#397—21no—
95#397C2×C411yescharacteristic
96#397C2×C411yescharacteristic
97#397C2311yescharacteristic
98#396—21no—
99#395C2×C411yescharacteristic
100#395C2×C411yescharacteristic
101#394—21no—
102#394—21no—
103#394—21no—
104#394C2×C411yescharacteristic
105#393C2×C411yescharacteristic
106#393C2×C411yescharacteristic
107#393C2×C411yescharacteristic
108#393C2×C411yescharacteristic
109#393C2311yescharacteristic
110#393C2311yescharacteristic
111#392—21no—
112#392—21no—
113#392C2×C411yescharacteristic
114#392C2×C411yescharacteristic
115#392C2311yescharacteristic
116#391—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#387—21no—
133#386—21no—
134#386—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#386—21no—
140#386—21no—
141#385—21no—
142#385—21no—
143#383C2×C411yescharacteristic
144#383C2×C411yescharacteristic
145#382—21no—
146#382C2×C411yescharacteristic
147#380—21no—
148#380—21no—
149#380—21no—
150#380—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—44no—
153C4×C4⋊C4C2×C411yescharacteristic
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4—22no—
161#376C2×C411yescharacteristic
162#376C2×C411yescharacteristic
163#342—22no—
164#342C2314yes—
165#324—22no—
166#324—22no—
167#324—22no—
168#323C2314yes—
169C2×#41—24no—
170C2×#41—42no—
171C2×#41—42no—
172C22×Q8C22⋊C411yescharacteristic
173C22×Q8C22×C411yescharacteristic
174C22×Q8C22×C411yescharacteristic
175C22×D4—21no—
176C22×D4—21no—
177C22×D4—21no—
178C22×D4C22⋊C411yescharacteristic
179C22×D4C22×C411yescharacteristic
180C22×D4C2×D411yescharacteristic
181C22×D4C2×D411yescharacteristic
182C23×C4—21no—
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C2×M16—42no—
193C22×C8—42no—
194C4⋊Q8—42no—
195#126—44no—
196#126—44no—
197#124—42no—
198#124—42no—
199C22⋊Q8—44no—
200#121—44no—
201C4×Q8—24no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205C4×D4—24no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—44no—
213#117—28no—
214#117—42no—
215#117—42no—
216#117—42no—
217#117—44no—
218#117—44no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C22⋊C4—24no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22×C411yescharacteristic
266C2×C22⋊C4C22×C411yescharacteristic
267C2×C22⋊C4C2×D411yescharacteristic
268C2×C22⋊C4C2411yescharacteristic
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—42no—
275C2×C42—42no—
276C2×C42—42no—
277C2×C42C22⋊C411yescharacteristic
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22×C411yescharacteristic
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C4⋊C8—42no—
284#99—28no—
285#99—28no—
286C22⋊C8—24no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289#97—42no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—42no—
310#95—42no—
311#95C22×C411yescharacteristic
312#95C22×C411yescharacteristic
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315C24—21no—
316C24C2×C22⋊C411yescharacteristic
317C24C2×C22⋊C411yescharacteristic
318C24C22×D411yescharacteristic
319#41—48no—
320#41—48no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—41no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—81no—
327C2×Q8—81no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—42no—
341C2×D4—81no—
342C2×D4—81no—
343C2×D4—81no—
344C2×D4—81no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D4—81no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—44no—
372C22×C4—42no—
373C22×C4—81no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4#10414yes—
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C23×C411yescharacteristic
391C22×C4C22×D411yescharacteristic
392M16—162no—
393C2×C8—44no—
394C2×C8—162no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—48no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4#9914yes—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C22⋊C4—28no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—44no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—42no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C22⋊C4—81no—
436C22⋊C4—81no—
437C22⋊C4—81no—
438C22⋊C4—81no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—81no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—42no—
450C42—48no—
451C42—42no—
452C42—42no—
453C42—81no—
454C42—81no—
455C42—162no—
456C42—162no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23C2×#10412yes—
478C23C22×C22⋊C411yescharacteristic
479C23#57512yes—
480Q8—44no—
481Q8—82no—
482Q8—82no—
483D4—44no—
484D4—44no—
485D4—44no—
486D4—82no—
487D4—84no—
488D4—162no—
489D4—162no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—48no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—84no—
544C2×C4—84no—
545C2×C4—84no—
546C2×C4—161no—
547C2×C4—161no—
548C2×C4C2×#9912yes—
549C2×C4C2×#9912yes—
550C2×C4C2×#9912yes—
551C2×C4C2×#10412yes—
552C2×C4C2×#10412yes—
553C8—84no—
554C22—21no—
555C22—21no—
556C22—22no—
557C22—22no—
558C22—42no—
559C22—42no—
560C22—42no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22—81no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22C22×#9911yescharacteristic
579C22#289611yescharacteristic
580C22C22×#10411yescharacteristic
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—22no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—88no—
598C4—81no—
599C4—81no—
600C4—81no—
601C2—21no—
602C2—21no—
603C2—42no—
604C2—42no—
605C2—81no—
606C2—161no—
607C2#1754211yescharacteristic
608C2#3335111yescharacteristic
609C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36019C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20393C2semidirect
3. #17368C2semidirect
4. #11787C2semidirect
5. #11774C2semidirect
6. #11758C2semidirect
7. #11756C2semidirect
8. #11730C2semidirect
9. #11728C2semidirect
10. C2×#556C4semidirect
14. #2631C4semidirect
16. #2628C4semidirect
17. #2610C4semidirect
19. #2599C4semidirect
29. #2398C4semidirect
31. #2392C4semidirect
33. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#360192
#203932
#173682
#117872
#117742
#117582
#117562
#117302
#117282