Finite Group Databasebeta

← #480993

Subgroups — #480993

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37871C211yesmaximal, characteristic
2#17418C212yesmaximal
3#11572C212yesmaximal, complemented
4#11556C212yesmaximal, complemented
5#11552C212yesmaximal, complemented
6#11535C211yesmaximal, characteristic, complemented
7#11516C211yesmaximal, characteristic, complemented
8#11513C211yesmaximal, characteristic, complemented
9#11481C211yesmaximal, characteristic, complemented
10#11450C211yesmaximal, characteristic, complemented
11#11447C211yesmaximal, characteristic, complemented
12C2×#555C2211yescharacteristic
13C2×#552C411yescharacteristic
14#2924—22no—
15#2850C411yescharacteristic
16#2715—22no—
17#2708—22no—
18#2660C411yescharacteristic
19#2620C411yescharacteristic
20#2568C411yescharacteristic
21#2566C2211yescharacteristic
22#2504C411yescharacteristic
23#2500C411yescharacteristic
24#2499C2211yescharacteristic
25#2499C2211yescharacteristic
26#2475—22no—
27#2474—22no—
28#2326C2211yescharacteristic
29#2317C2211yescharacteristic
30#2290C411yescharacteristic
31#2285C2211yescharacteristic
32#1471—22no—
33#1469—22no—
34#1469—22no—
35#1465C2212yes—
36#1464C2212yes—
37#1464C2212yes—
38#1459C2212yes—
39#1458C2212yes—
40#1458C2212yes—
41#1325C2211yescharacteristic
42#1325C2211yescharacteristic
43#1325C2212yes—
44#1325C2212yes—
45#1325C2211yescharacteristic
46#1325C2211yescharacteristic
47#1324C2212yes—
48#1324C2212yes—
49#1319C2212yes—
50#1319C2212yes—
51#555—42no—
52#555—42no—
53#552—42no—
54#552—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—21no—
57C2×#125—21no—
58C2×#125C2×C411yescharacteristic
59C2×#124C2×C411yescharacteristic
60C2×#124D412yes—
61C2×#124D412yes—
62C2×#123—21no—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×#121C2×C411yescharacteristic
66C2×#121C2×C411yescharacteristic
67C2×#121D412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117D412yes—
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79#433—42no—
80#433—42no—
81#432—22no—
82#432—22no—
83#431—22no—
84#431—22no—
85#424—42no—
86#407—22no—
87C2×C22⋊C8—22no—
88#405—42no—
89#400—22no—
90#398—22no—
91#397—22no—
92#397—22no—
93#397—22no—
94#396—22no—
95#396—22no—
96#395—22no—
97#394—22no—
98#394—22no—
99#391—22no—
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#391C2311yescharacteristic
103#390C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#390C2×C411yescharacteristic
106#389—22no—
107#388—22no—
108#388—21no—
109#388—22no—
110#388—22no—
111#387—21no—
112#387—22no—
113#387—22no—
114#386—22no—
115#386—21no—
116#386—22no—
117#385—22no—
118#385—22no—
119#385—21no—
120#385—22no—
121#384C2×C411yescharacteristic
122#384C2×C411yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#382C2×C411yescharacteristic
128#382C2×C411yescharacteristic
129#382C2311yescharacteristic
130#380—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4C2×C411yescharacteristic
134C4×C22⋊C4C2×C411yescharacteristic
135#376C2×C411yescharacteristic
136#324C2312yes—
137#324C2312yes—
138#324C2312yes—
139#324C2312yes—
140#323—22no—
141#323—22no—
142#323—22no—
143#323—22no—
144#323—22no—
145#323—22no—
146C22×Q8C22⋊C412yes—
147C22×Q8C22×C411yescharacteristic
148C22×D4C22⋊C411yescharacteristic
149C22×D4C22⋊C412yes—
150C22×D4C22⋊C412yes—
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4C22×C411yescharacteristic
155C23×C4C22×C411yescharacteristic
156C23×C4C2×D412yes—
157C2×M16—42no—
158C22×C8—42no—
159C4⋊Q8—42no—
160#126—44no—
161#125—42no—
162#125—42no—
163#125—42no—
164#124—22no—
165#124—42no—
166#124—42no—
167#124—42no—
168#124—42no—
169#124—41no—
170#123—44no—
171C22⋊Q8—42no—
172C22⋊Q8—41no—
173C22⋊Q8—41no—
174#121—42no—
175#121—41no—
176#121—41no—
177#121—42no—
178#121—42no—
179C4×Q8—42no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—42no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—41no—
189#117—41no—
190#117—42no—
191#117—41no—
192#117—42no—
193#117—42no—
194#117—41no—
195#117—41no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C2411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4C22⋊C411yescharacteristic
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C411yescharacteristic
231C2×C22⋊C4C2×D412yes—
232C2×C22⋊C4C2×D412yes—
233C2×C22⋊C4C2×D412yes—
234C2×C42—22no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—41no—
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C22×C411yescharacteristic
242C2×C42C22×C411yescharacteristic
243C2×C42C22×C411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254C22⋊C8—42no—
255C22⋊C8—42no—
256#97—42no—
257#97—42no—
258C4×C8—42no—
259C4×C8—42no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—22no—
265#95—21no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273C24C2×C22⋊C412yes—
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—42no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—82no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—22no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—81no—
290C2×D4—82no—
291C2×D4—82no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—82no—
314C22×C4—81no—
315C22×C4—81no—
316C22×C4—82no—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C412yes—
320C22×C4C2×C22⋊C412yes—
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D412yes—
325M16—84no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—84no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—81no—
352C4⋊C4#9912yes—
353C4⋊C4#9912yes—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—21no—
359C22⋊C4—21no—
360C22⋊C4—22no—
361C22⋊C4—42no—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—42no—
365C22⋊C4—44no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—81no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C22⋊C4—82no—
373C22⋊C4—81no—
374C42—21no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—42no—
385C42—41no—
386C42—42no—
387C42—41no—
388C42—41no—
389C42#13112yes—
390C23—22no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—22no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—22no—
399C23—42no—
400C23—41no—
401C23—42no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23C22×C22⋊C411yescharacteristic
406Q8—81no—
407Q8—81no—
408Q8—162no—
409D4—81no—
410D4—81no—
411D4—82no—
412D4—162no—
413D4—162no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4C2×#9911yescharacteristic
461C2×C4C2×#9911yescharacteristic
462C2×C4C2×#9912yes—
463C2×C4C2×#9912yes—
464C2×C4C2×#13111yescharacteristic
465C2×C4#56811yescharacteristic
466C8—82no—
467C8—82no—
468C8—82no—
469C8—82no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—42no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—42no—
479C22—41no—
480C22—41no—
481C22—81no—
482C22—81no—
483C22—82no—
484C22—81no—
485C22—81no—
486C22—82no—
487C22#288011yescharacteristic
488C22C22×#9911yescharacteristic
489C22#288911yescharacteristic
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—22no—
495C4—22no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—42no—
500C4—41no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—81no—
508C2—21no—
509C2—21no—
510C2—81no—
511C2—81no—
512C2—82no—
513C2#1741611yescharacteristic
514C2#3335811yescharacteristic
515C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#555 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11572C2semidirect
4. #11556C2semidirect
5. #11552C2semidirect
6. #11535C2semidirect
7. #11516C2semidirect
8. #11513C2semidirect
9. #11481C2semidirect
10. #11450C2semidirect
11. #11447C2semidirect

Maximal subgroups

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

subgroupindex
#378712
#174182
#115722
#115562
#115522
#115352
#115162
#115132
#114812
#114502
#114472