Finite Group Databasebeta

← #481999

Subgroups — #481999

Order 512.

945 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35693C211yesmaximal, characteristic
2#20725C211yesmaximal, characteristic, complemented
3#17372C211yesmaximal, characteristic, complemented
4#11826C212yesmaximal, complemented
5#11820C212yesmaximal, complemented
6#11812C212yesmaximal, complemented
7#11798C212yesmaximal, complemented
8#11731C212yesmaximal, complemented
9#11725C212yesmaximal, complemented
10C2×#547C411yescharacteristic, complemented
11C2×#520C2211yescharacteristic
12#2983—21no—
13#2638C2211yescharacteristic
14#2638C2211yescharacteristic
15#2628C2211yescharacteristic
16#2583C411yescharacteristic, complemented
17#2583C411yescharacteristic, complemented
18#2567C411yescharacteristic, complemented
19#2536C411yescharacteristic, complemented
20#2470—21no—
21#2470—21no—
22#2455—21no—
23#2455—21no—
24#2422—21no—
25#2407—21no—
26#2406C411yescharacteristic, complemented
27#2406C411yescharacteristic, complemented
28#2392C2211yescharacteristic
29#2388C2211yescharacteristic
30#2388C2211yescharacteristic
31#2360—21no—
32#2357C411yescharacteristic, complemented
33D4×C4⋊C4—21no—
34#2313—21no—
35#1820C2212yes—
36#1799—21no—
37#1799C2212yes—
38#1782—21no—
39C4×#99—21no—
40C4×#99C2212yes—
41#1466—21no—
42#1466C2212yes—
43#1464—21no—
44#1464C2212yes—
45#1460C2212yes—
46#1458—21no—
47#1351C2214yes—
48#1351C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C22×#41C2×C411yescharacteristic
52C22×#41C2×C411yescharacteristic
53C22×#41D411yescharacteristic
54#547—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#123—21no—
57C2×#123—21no—
58C2×#123—21no—
59C2×#123—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8C2×C411yescharacteristic
62C2×C22⋊Q8C2×C411yescharacteristic
63C2×#121—21no—
64#520—44no—
65C2×C4×Q8C2×C411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C2×C4×D4C2311yescharacteristic
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2×C411yescharacteristic
83C22×C4⋊C4D411yescharacteristic
84#446—44no—
85#432—22no—
86#432—22no—
87#431—22no—
88C2×#99—22no—
89C2×#99—22no—
90#400—21no—
91#399—21no—
92#398C2×C411yescharacteristic
93#398C2×C411yescharacteristic
94#398C2×C411yescharacteristic
95#397—21no—
96#397—21no—
97#397C2311yescharacteristic
98#396—21no—
99#395C2×C411yescharacteristic
100#395C2×C411yescharacteristic
101#395C2×C411yescharacteristic
102#395C2×C411yescharacteristic
103#395C2×C411yescharacteristic
104#395C2×C411yescharacteristic
105#394—21no—
106#394—21no—
107#394—21no—
108#394—21no—
109#394—21no—
110#394C2×C411yescharacteristic
111#394C2×C411yescharacteristic
112#394C2×C411yescharacteristic
113#393C2311yescharacteristic
114#393C2311yescharacteristic
115#392—21no—
116#392—21no—
117#392C2311yescharacteristic
118#391—21no—
119#390—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#387—21no—
132#387—21no—
133#387—44no—
134#386—21no—
135#386—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#384—21no—
147#382C2×C411yescharacteristic
148#381—22no—
149#381C2×C411yescharacteristic
150#381C2×C411yescharacteristic
151#380—21no—
152#380—21no—
153#380—22no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4—21no—
159C4×C22⋊C4—22no—
160C4×C22⋊C4C2×C411yescharacteristic
161C4×C22⋊C4C2×C411yescharacteristic
162#342—22no—
163#342C2314yes—
164#324—22no—
165#323—22no—
166#323—22no—
167#323C2314yes—
168C2×#41—42no—
169C2×#41—42no—
170C2×#41—42no—
171C2×#41—42no—
172C2×#41—42no—
173C2×#41—82no—
174C2×#41—82no—
175C22×Q8C22⋊C411yescharacteristic
176C22×Q8C22×C411yescharacteristic
177C22×Q8C22×C411yescharacteristic
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4C22⋊C411yescharacteristic
182C22×D4C22×C411yescharacteristic
183C22×D4C2×D411yescharacteristic
184C22×D4C2×D411yescharacteristic
185C23×C4—21no—
186C23×C4—21no—
187C23×C4—21no—
188C23×C4C22⋊C411yescharacteristic
189C23×C4C22⋊C411yescharacteristic
190C23×C4C22⋊C411yescharacteristic
191C23×C4C22⋊C411yescharacteristic
192C23×C4C22×C411yescharacteristic
193C23×C4C22×C411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2×D411yescharacteristic
196C23×C4C2×D411yescharacteristic
197C4⋊Q8—42no—
198#123—44no—
199#123—44no—
200#123—82no—
201#123—82no—
202C22⋊Q8—44no—
203C22⋊Q8—42no—
204C22⋊Q8—42no—
205#121—44no—
206C4×Q8—24no—
207C4×Q8—42no—
208C4×D4—24no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214C4×D4—42no—
215C4×D4—44no—
216C4×D4—82no—
217C4×D4—82no—
218#117—42no—
219#117—42no—
220#117—42no—
221C2×C4⋊C4—24no—
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—21no—
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—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C4⋊C4C22×C411yescharacteristic
250C2×C4⋊C4C22×C411yescharacteristic
251C2×C4⋊C4C2×D411yescharacteristic
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⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C2×D411yescharacteristic
273C2×C22⋊C4C2×D411yescharacteristic
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C22⋊C4C2411yescharacteristic
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—42no—
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C4⋊C8—42no—
287C4⋊C8—42no—
288C4⋊C8—42no—
289#99—44no—
290#99—44no—
291C22⋊C8—24no—
292#97—42no—
293#97—42no—
294C4×C8—42no—
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—21no—
310#95—42no—
311#95—42no—
312#95—44no—
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317C24—21no—
318C24C2×C22⋊C411yescharacteristic
319C24C2×C22⋊C411yescharacteristic
320C24C22×D411yescharacteristic
321#41—416no—
322#41—84no—
323#41—84no—
324#41—84no—
325#41—84no—
326C2×Q8—22no—
327C2×Q8—22no—
328C2×Q8—41no—
329C2×Q8—41no—
330C2×Q8—41no—
331C2×Q8—81no—
332C2×Q8—81no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—42no—
345C2×D4—42no—
346C2×D4—42no—
347C2×D4—42no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—81no—
352C2×D4—81no—
353C22×C4—21no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
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×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C2×C8—44no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—42no—
421C4⋊C4—42no—
422C4⋊C4—42no—
423C4⋊C4—81no—
424C4⋊C4—81no—
425C4⋊C4—81no—
426C4⋊C4—81no—
427C4⋊C4—81no—
428C4⋊C4—81no—
429C4⋊C4—162no—
430C4⋊C4#9914yes—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—42no—
446C22⋊C4—42no—
447C22⋊C4—42no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—44no—
452C22⋊C4—81no—
453C22⋊C4—81no—
454C22⋊C4—81no—
455C22⋊C4—81no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—41no—
461C42—41no—
462C42—42no—
463C42—81no—
464C42—81no—
465C42—162no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23C22×C22⋊C411yescharacteristic
488C23#57512yes—
489Q8—44no—
490Q8—82no—
491Q8—82no—
492D4—44no—
493D4—44no—
494D4—44no—
495D4—42no—
496D4—48no—
497D4—42no—
498D4—84no—
499D4—84no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
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—41no—
518C2×C4—41no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—41no—
522C2×C4—44no—
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—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—84no—
553C2×C4—84no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—84no—
558C2×C4—161no—
559C2×C4—161no—
560C2×C4C2×#9912yes—
561C2×C4C2×#9912yes—
562C2×C4C2×#9912yes—
563C2×C4#42112yes—
564C8—84no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—41no—
573C22—42no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C22×#9911yescharacteristic
590C22#289611yescharacteristic
591C22C2×#42111yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—42no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—164no—
612C2—21no—
613C2—21no—
614C2—42no—
615C2—81no—
616C2—81no—
617C2—82no—
618C2#1755911yescharacteristic
619C2#3334811yescharacteristic
620C2#3336011yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#35693C2
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. #20725C2semidirect
3. #17372C2semidirect
4. #11826C2semidirect
5. #11820C2semidirect
6. #11812C2semidirect
7. #11798C2semidirect
8. #11731C2semidirect
9. #11725C2semidirect
10. C2×#547C4semidirect
16. #2583C4semidirect
17. #2583C4semidirect
18. #2567C4semidirect
19. #2536C4semidirect
26. #2406C4semidirect
27. #2406C4semidirect
32. #2357C4semidirect

Maximal subgroups

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

subgroupindex
#356932
#207252
#173722
#118262
#118202
#118122
#117982
#117312
#117252