Finite Group Databasebeta

← #482600

Subgroups — #482600

Order 512.

854 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35738C211yesmaximal, characteristic
2#20691C211yesmaximal, characteristic, complemented
3#17371C211yesmaximal, characteristic, complemented
4#11846C212yesmaximal, complemented
5#11840C212yesmaximal
6#11813C212yesmaximal, complemented
7#11798C212yesmaximal
8#11760C212yesmaximal, complemented
9#11754C212yesmaximal
10C2×#554C411yescharacteristic, complemented
11C2×#519C2211yescharacteristic
12#2983—21no—
13#2634C411yescharacteristic, complemented
14#2634C411yescharacteristic, complemented
15#2628C2211yescharacteristic
16#2611C2211yescharacteristic
17#2611C2211yescharacteristic
18#2598C2211yescharacteristic
19#2598C2211yescharacteristic
20#2568C411yescharacteristic, complemented
21#2536C411yescharacteristic, complemented
22#2477—21no—
23#2477—21no—
24#2465—21no—
25#2465—21no—
26#2421—21no—
27#2409—21no—
28#2398C411yescharacteristic, complemented
29#2398C411yescharacteristic, complemented
30#2392C2211yescharacteristic
31#2360—21no—
32#2356C411yescharacteristic, complemented
33#2352—21no—
34#2310—21no—
35#1820—21no—
36#1820—21no—
37#1820C2212yes—
38#1820C2212yes—
39#1740—21no—
40#1740C2212yes—
41#1467—21no—
42#1467—21no—
43#1464C2212yes—
44#1464C2212yes—
45#1461—21no—
46#1458C2212yes—
47#1352C2214yes—
48#1352C2214yes—
49#1344C2214yes—
50#1343C2214yes—
51C23×Q8D411yescharacteristic
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C411yescharacteristic
55C2×#125C2×C411yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60#519—44no—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C411yescharacteristic
64C2×C4×Q8C2×C411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×Q8D411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×#117—22no—
74C2×#117D411yescharacteristic
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—21no—
81#446—44no—
82#432—22no—
83#432—22no—
84#431—22no—
85#400—21no—
86#400—21no—
87#400C2×C411yescharacteristic
88#400C2×C411yescharacteristic
89#400C2×C411yescharacteristic
90#400C2×C411yescharacteristic
91#400C2×C411yescharacteristic
92#400C2×C411yescharacteristic
93#399C2311yescharacteristic
94#399C2311yescharacteristic
95#398C2×C411yescharacteristic
96#398C2×C411yescharacteristic
97#398C2×C411yescharacteristic
98#397—21no—
99#397—21no—
100#397C2311yescharacteristic
101#396—21no—
102#395—21no—
103#395—21no—
104#395—21no—
105#394—21no—
106#394C2×C411yescharacteristic
107#394C2×C411yescharacteristic
108#394C2×C411yescharacteristic
109#393—21no—
110#393—21no—
111#393—21no—
112#392C2311yescharacteristic
113#391—21no—
114#391—21no—
115#391—21no—
116#391—21no—
117#391—44no—
118#389—21no—
119#389—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#387—21no—
134#387—21no—
135#386—21no—
136#386—21no—
137#386—21no—
138#386—21no—
139#385—21no—
140#385—21no—
141#385—21no—
142#385—21no—
143#383C2×C411yescharacteristic
144#383C2×C411yescharacteristic
145#382—21no—
146#382C2×C411yescharacteristic
147#381—22no—
148#381—22no—
149#380—21no—
150#380—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4C2×C411yescharacteristic
153C4×C22⋊C4—21no—
154C4×C22⋊C4—21no—
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158#376C2×C411yescharacteristic
159#376C2×C411yescharacteristic
160#342—22no—
161#342—22no—
162#342—22no—
163#342C2314yes—
164#324—22no—
165#324—22no—
166#324—22no—
167#323C2314yes—
168C22×Q8—24no—
169C22×Q8—21no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8C22⋊C411yescharacteristic
173C22×Q8C22⋊C411yescharacteristic
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22⋊C411yescharacteristic
176C22×D4—21no—
177C22×D4C22×C411yescharacteristic
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4C2×D411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C4⋊Q8—42no—
185#125—42no—
186#125—42no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C22⋊Q8—44no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—42no—
194C4×Q8—42no—
195C4×Q8—42no—
196C4×Q8—42no—
197C4×Q8—42no—
198C4×Q8—42no—
199C4×D4—42no—
200C4×D4—42no—
201C4×D4—42no—
202C4×D4—44no—
203#117—42no—
204#117—42no—
205#117—42no—
206#117—42no—
207#117—42no—
208#117—42no—
209#117—44no—
210#117—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—44no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
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—24no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4C2×D411yescharacteristic
255C2×C22⋊C4C2×D411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2411yescharacteristic
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—42no—
262C2×C42C22⋊C411yescharacteristic
263C2×C42C22⋊C411yescharacteristic
264C2×C42C22⋊C411yescharacteristic
265C2×C42C22⋊C411yescharacteristic
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C22⋊C8—24no—
278#97—42no—
279#97—42no—
280C4×C8—42no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
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—44no—
305#95—44no—
306#95C22×C411yescharacteristic
307#95C22×C411yescharacteristic
308#95C22×C411yescharacteristic
309#95C22×C411yescharacteristic
310C24—21no—
311C24C22×D411yescharacteristic
312C2×Q8—22no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—22no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—41no—
321C2×Q8—41no—
322C2×Q8—41no—
323C2×Q8—48no—
324C2×Q8—48no—
325C2×Q8—42no—
326C2×Q8—81no—
327C2×Q8—81no—
328C2×Q8#9914yes—
329C2×D4—41no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—42no—
333C2×D4—81no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C2×C8—44no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—84no—
400C4⋊C4—81no—
401C4⋊C4—81no—
402C4⋊C4—81no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—81no—
420C22⋊C4—81no—
421C22⋊C4—81no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—42no—
435C42—84no—
436C42—81no—
437C42—81no—
438C23—21no—
439C23—21no—
440C23—22no—
441C23—41no—
442C23—41no—
443C23—41no—
444C23—41no—
445C23—41no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23C22×C22⋊C411yescharacteristic
450C23#57512yes—
451Q8—44no—
452Q8—44no—
453Q8—44no—
454Q8—44no—
455Q8—162no—
456D4—42no—
457D4—42no—
458D4—84no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—162no—
519C2×C4—162no—
520C2×C4C2×#9912yes—
521C2×C4C2×#9912yes—
522C2×C4C2×#9912yes—
523C2×C4#42112yes—
524C8—84no—
525C22—21no—
526C22—21no—
527C22—22no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—41no—
532C22—41no—
533C22—42no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22C22×#9911yescharacteristic
540C22#289611yescharacteristic
541C22C2×#42111yescharacteristic
542C4—22no—
543C4—22no—
544C4—22no—
545C4—22no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4—81no—
557C4—81no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4—164no—
564C2—21no—
565C2—21no—
566C2—42no—
567C2—82no—
568C2#1758711yescharacteristic
569C2#3334911yescharacteristic
570C2#3336211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radical#392C23
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×#519 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20691C2semidirect
3. #17371C2semidirect
4. #11846C2semidirect
6. #11813C2semidirect
8. #11760C2semidirect
10. C2×#554C4semidirect
13. #2634C4semidirect
14. #2634C4semidirect
20. #2568C4semidirect
21. #2536C4semidirect
28. #2398C4semidirect
29. #2398C4semidirect
32. #2356C4semidirect

Maximal subgroups

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

subgroupindex
#357382
#206912
#173712
#118462
#118402
#118132
#117982
#117602
#117542