Finite Group Databasebeta

← #481000

Subgroups — #481000

Order 512.

843 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37862C211yesmaximal, characteristic
2#17427C212yesmaximal
3#11570C212yesmaximal, complemented
4#11555C212yesmaximal, complemented
5#11552C212yesmaximal, complemented
6#11535C211yesmaximal, characteristic, complemented
7#11516C211yesmaximal, characteristic, complemented
8#11512C211yesmaximal, characteristic, complemented
9#11481C211yesmaximal, characteristic, complemented
10#11450C211yesmaximal, characteristic, complemented
11C2×#1325C211yesmaximal, characteristic, complemented
12C2×#554C411yescharacteristic
13C2×#553C2211yescharacteristic
14#2974—22no—
15#2850C411yescharacteristic
16#2721—22no—
17#2680—22no—
18#2660C411yescharacteristic
19#2620C411yescharacteristic
20#2565C411yescharacteristic
21#2542C2211yescharacteristic
22#2504C411yescharacteristic
23#2502C2211yescharacteristic
24#2501C411yescharacteristic
25#2499C2211yescharacteristic
26#2449—22no—
27#2443—22no—
28#2326C2211yescharacteristic
29#2317C2211yescharacteristic
30#2290C411yescharacteristic
31C2×#384C2211yescharacteristic
32#1473—22no—
33#1471—22no—
34#1468—22no—
35#1465C2212yes—
36#1464C2212yes—
37#1463C2212yes—
38#1459C2212yes—
39#1458C2212yes—
40C2×#324C2212yes—
41#1325C2211yescharacteristic
42#1325C2211yescharacteristic
43#1325C2212yes—
44#1325C2212yes—
45#1325C2212yes—
46#1325C2212yes—
47#1325C2211yescharacteristic
48#1325C2211yescharacteristic
49#1319C2212yes—
50#1319C2212yes—
51#554—42no—
52#554—42no—
53#553—42no—
54#553—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126D412yes—
57C2×#125—21no—
58C2×#125C2×C411yescharacteristic
59C2×#124C2×C411yescharacteristic
60C2×#123—22no—
61C2×#123D412yes—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×#121C2×C411yescharacteristic
69C2×#121D412yes—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D412yes—
74C2×C4×D4C2311yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4—22no—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C42C2×C411yescharacteristic
82#436—42no—
83#435—42no—
84#435—42no—
85C8×D4—42no—
86#432—22no—
87#431—22no—
88#407—22no—
89#407—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92#399—22no—
93#399—22no—
94#398—22no—
95#398—22no—
96#398—22no—
97#397—22no—
98#397—22no—
99#396—22no—
100#395—22no—
101#394—22no—
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2311yescharacteristic
105#390C2×C411yescharacteristic
106#390C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#389—22no—
109#388—22no—
110#388—21no—
111#388—22no—
112#387—22no—
113#387—21no—
114#387—22no—
115#386—22no—
116#386—21no—
117#386—22no—
118#385—22no—
119#385—22no—
120#385—21no—
121#385—22no—
122#384—22no—
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125#384C2×C411yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#382C2×C411yescharacteristic
130#382C2×C411yescharacteristic
131#382C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4C2×C411yescharacteristic
134C4×C22⋊C4C2×C411yescharacteristic
135#376C2×C411yescharacteristic
136#324—22no—
137#324—22no—
138#324C2312yes—
139#324C2312yes—
140#324C2312yes—
141#324C2312yes—
142#323—22no—
143#323—22no—
144#323—22no—
145#323—22no—
146C22×Q8—21no—
147C22×Q8C22×C411yescharacteristic
148C22×D4C22⋊C411yescharacteristic
149C22×D4C22⋊C412yes—
150C22×D4C22⋊C412yes—
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C2×M16—42no—
160C2×M16—42no—
161C22×C8—42no—
162C22×C8—42no—
163C4⋊Q8—42no—
164#126—42no—
165#126—42no—
166#125—42no—
167#125—44no—
168#124—22no—
169#124—41no—
170#123—42no—
171#123—42no—
172#123—84no—
173C22⋊Q8—44no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C22⋊Q8—84no—
180#121—42no—
181#121—41no—
182#121—41no—
183#121—42no—
184#121—42no—
185C4×Q8—42no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—22no—
192C4×D4—42no—
193C4×D4—42no—
194C4×D4—42no—
195#117—22no—
196#117—41no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—84no—
218C2×C4⋊C4C22⋊C411yescharacteristic
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C2×D412yes—
242C2×C22⋊C4C2×D412yes—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—41no—
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D412yes—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263#97—42no—
264C4×C8—42no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276C24C2×C22⋊C411yescharacteristic
277C24C2×C22⋊C412yes—
278C24C2×C22⋊C411yescharacteristic
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—42no—
284C2×Q8—82no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—44no—
296C2×D4—82no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—82no—
325C22×C4—82no—
326C22×C4—82no—
327C22×C4—82no—
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C412yes—
333C22×C4C2×C22⋊C411yescharacteristic
334C22×C4C23×C411yescharacteristic
335C22×C4C22×D412yes—
336M16—162no—
337M16—162no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—162no—
343C2×C8—162no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—41no—
350C4⋊C4—42no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4—82no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—44no—
377C22⋊C4—44no—
378C22⋊C4—42no—
379C22⋊C4—42no—
380C22⋊C4—42no—
381C22⋊C4—41no—
382C22⋊C4—41no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4#9912yes—
386C22⋊C4#9912yes—
387C42—21no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—42no—
396C42—41no—
397C42—42no—
398C42#13112yes—
399C23—21no—
400C23—21no—
401C23—22no—
402C23—21no—
403C23—22no—
404C23—21no—
405C23—22no—
406C23—21no—
407C23—22no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—41no—
412C23—42no—
413C23C2×#9911yescharacteristic
414C23C2×#9911yescharacteristic
415C23C22×C22⋊C411yescharacteristic
416Q8—81no—
417Q8—81no—
418Q8—162no—
419D4—42no—
420D4—42no—
421D4—44no—
422D4—81no—
423D4—81no—
424D4—82no—
425D4—162no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—48no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—41no—
454C2×C4—42no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—161no—
475C2×C4—161no—
476C2×C4—162no—
477C2×C4C2×#9912yes—
478C2×C4C2×#9912yes—
479C2×C4C2×#13111yescharacteristic
480C2×C4#56811yescharacteristic
481C8—82no—
482C8—82no—
483C8—82no—
484C8—82no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—42no—
493C22—42no—
494C22—41no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—81no—
501C22—82no—
502C22—81no—
503C22—82no—
504C22#213512yes—
505C22#288011yescharacteristic
506C22C22×#9911yescharacteristic
507C22#288911yescharacteristic
508C4—21no—
509C4—21no—
510C4—22no—
511C4—22no—
512C4—42no—
513C4—42no—
514C4—41no—
515C4—42no—
516C4—42no—
517C4—41no—
518C4—42no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—81no—
524C4—81no—
525C2—21no—
526C2—21no—
527C2—22no—
528C2—41no—
529C2—81no—
530C2—82no—
531C2#1741611yescharacteristic
532C2#3335911yescharacteristic
533C2C2×#213511yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#553C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#553 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11570C2semidirect
4. #11555C2semidirect
5. #11552C2semidirect
6. #11535C2semidirect
7. #11516C2semidirect
8. #11512C2semidirect
9. #11481C2semidirect
10. #11450C2semidirect
11. C2×#1325C2semidirect

Maximal subgroups

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

subgroupindex
#378622
#174272
#115702
#115552
#115522
#115352
#115162
#115122
#114812
#114502
C2×#13252