Finite Group Databasebeta

← #482686

Subgroups — #482686

Order 512.

907 conjugacy classes of subgroups.

The subgroup lattice has 623 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36032C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic, complemented
3#17372C211yesmaximal, characteristic, complemented
4#11899C212yesmaximal, complemented
5#11876C212yesmaximal, complemented
6#11824C212yesmaximal, complemented
7#11822C212yesmaximal, complemented
8#11760C212yesmaximal, complemented
9#11754C212yesmaximal, complemented
10C2×#553C411yescharacteristic, complemented
11C2×#520C2211yescharacteristic
12#2983—21no—
13C2×#410—21no—
14#2635C2211yescharacteristic
15#2634C411yescharacteristic, complemented
16#2618C411yescharacteristic, complemented
17#2611C2211yescharacteristic
18#2598C2211yescharacteristic
19#2591C411yescharacteristic, complemented
20#2543C2211yescharacteristic
21#2487C2211yescharacteristic
22#2484C411yescharacteristic, complemented
23#2481—21no—
24#2470—21no—
25#2469—21no—
26#2457—21no—
27#2444—21no—
28#2430—21no—
29#2398C411yescharacteristic, complemented
30#2395C2211yescharacteristic
31C2×#400C411yescharacteristic, complemented
32#2364—21no—
33#2359—21no—
34#2353C411yescharacteristic, complemented
35#2316—21no—
36#1799—21no—
37#1799—21no—
38#1799C2212yes—
39#1799C2212yes—
40#1740C2212yes—
41#1466—21no—
42#1466C2212yes—
43#1464—21no—
44#1464C2212yes—
45#1460C2212yes—
46#1458—21no—
47#1352C2214yes—
48#1352C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C22×#41D411yescharacteristic
52#553—44no—
53C2×#126—21no—
54C2×#125C2×C411yescharacteristic
55C2×#123—21no—
56C2×#123—21no—
57C2×#121—21no—
58C2×#121—21no—
59C2×#121C2×C411yescharacteristic
60C2×#121C2×C411yescharacteristic
61#520—44no—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8D411yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—22no—
73C2×#117C2×C411yescharacteristic
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81#446—44no—
82#432—22no—
83#432—22no—
84#431—22no—
85#410—44no—
86C2×#99—22no—
87C2×#99—22no—
88#400—21no—
89#400—21no—
90#400—21no—
91#400C2×C411yescharacteristic
92#400C2×C411yescharacteristic
93#400C2×C411yescharacteristic
94#400C2×C411yescharacteristic
95#400C2×C411yescharacteristic
96#400C2×C411yescharacteristic
97#399C2311yescharacteristic
98#399C2311yescharacteristic
99#398C2×C411yescharacteristic
100#398C2×C411yescharacteristic
101#398C2×C411yescharacteristic
102#397—21no—
103#397—21no—
104#397C2311yescharacteristic
105#396—21no—
106#396—21no—
107#395—21no—
108#394—21no—
109#394—21no—
110#394C2×C411yescharacteristic
111#394C2×C411yescharacteristic
112#394C2×C411yescharacteristic
113#393—21no—
114#392—21no—
115#392C2311yescharacteristic
116#389—21no—
117#389—21no—
118#389—44no—
119#388—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#387—21no—
127#387—21no—
128#387—21no—
129#387—21no—
130#387—21no—
131#387—21no—
132#386—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#386—21no—
142#385—21no—
143#385—21no—
144#385—21no—
145#385—21no—
146#383C2×C411yescharacteristic
147#381—21no—
148#381C2×C411yescharacteristic
149#380—21no—
150#380—22no—
151#380—22no—
152#380C2×C411yescharacteristic
153C4×C4⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4C2×C411yescharacteristic
159#376C2×C411yescharacteristic
160C2×#95—21no—
161C2×#95—21no—
162C2×#95C2×C411yescharacteristic
163#342—22no—
164#342C2314yes—
165#324—22no—
166#323—22no—
167#323—22no—
168#323C2314yes—
169C2×#41—22no—
170C2×#41—22no—
171C2×#41—42no—
172C2×#41—42no—
173C22×Q8C22⋊C411yescharacteristic
174C22×D4—21no—
175C22×D4—21no—
176C22×D4—21no—
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22×C411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C22×D4C2×D411yescharacteristic
181C23×C4—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22⋊C411yescharacteristic
186C23×C4C22⋊C411yescharacteristic
187C23×C4C22⋊C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C22×C411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193#126—44no—
194#125—42no—
195#123—44no—
196#123—44no—
197#121—44no—
198#121—42no—
199#121—42no—
200#121—42no—
201#121—42no—
202C4×Q8—24no—
203C4×Q8—42no—
204C4×D4—24no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—44no—
212#117—42no—
213#117—42no—
214#117—42no—
215#117—42no—
216#117—44no—
217#117—44no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
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—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
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⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C411yescharacteristic
243C2×C4⋊C4C22×C411yescharacteristic
244C2×C4⋊C4C22×C411yescharacteristic
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
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—24no—
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⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C2×D411yescharacteristic
268C2×C22⋊C4C2×D411yescharacteristic
269C2×C22⋊C4C2×D411yescharacteristic
270C2×C22⋊C4C2411yescharacteristic
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—42no—
275C2×C42C22⋊C411yescharacteristic
276C2×C42C22⋊C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C4⋊C8—42no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284#99—44no—
285#99—44no—
286C22⋊C8—24no—
287#97—42no—
288#97—42no—
289C4×C8—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—21no—
310#95—21no—
311#95—42no—
312#95—42no—
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317C24—21no—
318C24C2×C22⋊C411yescharacteristic
319C24C2×C22⋊C411yescharacteristic
320C24C22×D411yescharacteristic
321#41—48no—
322#41—48no—
323C2×Q8—22no—
324C2×Q8—22no—
325C2×Q8—41no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—81no—
343C2×D4—81no—
344C2×D4—81no—
345C2×D4—81no—
346C2×D4—81no—
347C2×D4AGL(1,ℤ/8)12yes—
348C2×D4#13712yes—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
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—21no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—81no—
384C22×C4—81no—
385C22×C4—81no—
386C22×C4—81no—
387C22×C4—81no—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C2×C8—44no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—42no—
414C4⋊C4—42no—
415C4⋊C4—42no—
416C4⋊C4—42no—
417C4⋊C4—42no—
418C4⋊C4—42no—
419C4⋊C4—84no—
420C4⋊C4—81no—
421C4⋊C4—81no—
422C4⋊C4—81no—
423C4⋊C4—162no—
424C4⋊C4—162no—
425C4⋊C4#9914yes—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—81no—
445C22⋊C4—81no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C22⋊C4—81no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—42no—
459C42—84no—
460C42—81no—
461C42—81no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23#41712yes—
483C23C22×C22⋊C411yescharacteristic
484C23C2×AGL(1,ℤ/8)11yescharacteristic
485C23C2×#13711yescharacteristic
486Q8—44no—
487D4—42no—
488D4—42no—
489D4—44no—
490D4—42no—
491D4—42no—
492D4—82no—
493D4—84no—
494D4—161no—
495D4—161no—
496D4—162no—
497C2×C4—24no—
498C2×C4—24no—
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—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—44no—
517C2×C4—44no—
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—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—82no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4—161no—
553C2×C4—161no—
554C2×C4—162no—
555C2×C4—162no—
556C2×C4—162no—
557C2×C4C2×#9912yes—
558C2×C4C2×#9912yes—
559C2×C4C2×#9912yes—
560C2×C4#41712yes—
561C2×C4#42112yes—
562C8—84no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—42no—
569C22—41no—
570C22—41no—
571C22—42no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22C22×#9911yescharacteristic
590C22C2×#41711yescharacteristic
591C22C2×#42111yescharacteristic
592C4—22no—
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—164no—
612C2—21no—
613C2—21no—
614C2—41no—
615C2—42no—
616C2—41no—
617C2—81no—
618C2—161no—
619C2#1759211yescharacteristic
620C2#3335011yescharacteristic
621C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#36032C2
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. #20634C2semidirect
3. #17372C2semidirect
4. #11899C2semidirect
5. #11876C2semidirect
6. #11824C2semidirect
7. #11822C2semidirect
8. #11760C2semidirect
9. #11754C2semidirect
10. C2×#553C4semidirect
15. #2634C4semidirect
16. #2618C4semidirect
19. #2591C4semidirect
22. #2484C4semidirect
29. #2398C4semidirect
31. C2×#400C4semidirect
34. #2353C4semidirect

Maximal subgroups

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

subgroupindex
#360322
#206342
#173722
#118992
#118762
#118242
#118222
#117602
#117542