Finite Group Databasebeta

← #480954

Subgroups — #480954

Order 512.

823 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36181C211yesmaximal, characteristic
2#17417C212yesmaximal
3#11571C212yesmaximal, complemented
4#11552C212yesmaximal, complemented
5#11551C212yesmaximal, complemented
6#11532C211yesmaximal, characteristic, complemented
7#11513C211yesmaximal, characteristic, complemented
8#11512C211yesmaximal, characteristic, complemented
9#11478C211yesmaximal, characteristic, complemented
10#11447C211yesmaximal, characteristic, complemented
11C2×#1325C211yesmaximal, characteristic, complemented
12C2×#552C411yescharacteristic
13C2×#552C2211yescharacteristic
14#2924—22no—
15#2859C411yescharacteristic
16#2857C411yescharacteristic
17#2857C411yescharacteristic
18#2565C411yescharacteristic
19#2565C2211yescharacteristic
20#2506—22no—
21#2503C411yescharacteristic
22#2503C2211yescharacteristic
23#2499C411yescharacteristic
24#2499C2211yescharacteristic
25#2496—22no—
26#2486—22no—
27#2344—22no—
28C4×C4⋊Q8C2211yescharacteristic
29#2285C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C42×D4C411yescharacteristic
32#1473—22no—
33#1470—22no—
34#1468—22no—
35#1467C2212yes—
36#1464C2212yes—
37#1463C2212yes—
38#1461C2212yes—
39#1458C2212yes—
40C2×#324C2212yes—
41#1325C2211yescharacteristic
42#1325C2211yescharacteristic
43#1325C2212yes—
44#1325C2212yes—
45#1325C2211yescharacteristic
46#1325C2211yescharacteristic
47#1324C2212yes—
48#1324C2212yes—
49#1319C2212yes—
50#1319C2212yes—
51#552—41no—
52#552—41no—
53#552—41no—
54#552—41no—
55#552—41no—
56#552—41no—
57#552—41no—
58#552—41no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#126—21no—
61C2×#126D412yes—
62C2×#125—21no—
63C2×#125C2×C411yescharacteristic
64C2×#125C2×C411yescharacteristic
65C2×#125D412yes—
66C2×#123—21no—
67C2×#123D412yes—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C2×#117C2×C411yescharacteristic
78C2×#117D412yes—
79C2×#117C2311yescharacteristic
80C22×C4⋊C4—22no—
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C4⋊C4C2311yescharacteristic
83C22×C42C2×C411yescharacteristic
84C22×C42C2×C411yescharacteristic
85#433—42no—
86#433—42no—
87#432—22no—
88#432—22no—
89#431—22no—
90#431—22no—
91#424—42no—
92#407—22no—
93C2×C22⋊C8—22no—
94#405—42no—
95#400—22no—
96#400—22no—
97#398—22no—
98#398—22no—
99#397—22no—
100#397—22no—
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#391C2×C411yescharacteristic
106#391C2311yescharacteristic
107#389—22no—
108#389—22no—
109#389—22no—
110#389—22no—
111#389—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#384—22no—
117#384C2×C411yescharacteristic
118#384C2×C411yescharacteristic
119#384C2×C411yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#382C2×C411yescharacteristic
123#382C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#382C2311yescharacteristic
126#381—22no—
127#381—22no—
128#380—22no—
129#380—22no—
130#380—22no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C4×C22⋊C4C2×C411yescharacteristic
138C4×C22⋊C4C2×C411yescharacteristic
139C43C2×C411yescharacteristic
140#324—22no—
141#324—22no—
142#324—22no—
143#324—22no—
144#324C2312yes—
145#324C2312yes—
146#324C2312yes—
147#324C2312yes—
148#323—22no—
149#323—22no—
150C22×Q8C22×C411yescharacteristic
151C22×Q8C22×C411yescharacteristic
152C22×D4C22⋊C411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D412yes—
159C2×M16—42no—
160C22×C8—42no—
161C4⋊Q8—22no—
162C4⋊Q8—41no—
163#126—42no—
164#126—42no—
165#126—42no—
166#126—42no—
167#125—41no—
168#125—41no—
169#125—42no—
170#125—42no—
171#125—41no—
172#125—41no—
173#125—42no—
174#125—42no—
175#123—42no—
176#123—42no—
177#123—42no—
178#123—42no—
179C22⋊Q8—41no—
180C22⋊Q8—41no—
181C22⋊Q8—41no—
182C22⋊Q8—41no—
183C22⋊Q8—41no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—41no—
187C4×Q8—41no—
188C4×Q8—41no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—44no—
196#117—22no—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—41no—
201#117—41no—
202#117—41no—
203#117—41no—
204#117—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
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—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—48no—
226C2×C4⋊C4C22⋊C412yes—
227C2×C4⋊C4C22⋊C412yes—
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C412yes—
230C2×C4⋊C4C22⋊C412yes—
231C2×C4⋊C4C22×C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2411yescharacteristic
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4C22⋊C412yes—
245C2×C22⋊C4C22⋊C411yescharacteristic
246C2×C22⋊C4C22⋊C412yes—
247C2×C22⋊C4C22⋊C411yescharacteristic
248C2×C22⋊C4C2×D412yes—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—41no—
258C2×C42—41no—
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C22×C411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D412yes—
266C4⋊C8—42no—
267C4⋊C8—42no—
268C4⋊C8—42no—
269C4⋊C8—42no—
270C22⋊C8—22no—
271C22⋊C8—22no—
272C22⋊C8—22no—
273C22⋊C8—22no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276#97—42no—
277#97—42no—
278C4×C8—42no—
279C4×C8—42no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293C24C2×C22⋊C411yescharacteristic
294C24C2×C22⋊C411yescharacteristic
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—41no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—42no—
306C22×C4—21no—
307C22×C4—24no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—44no—
333C22×C4—44no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4C2×C22⋊C412yes—
337C22×C4C2×C22⋊C411yescharacteristic
338C22×C4C2×C22⋊C411yescharacteristic
339C22×C4C2×C22⋊C412yes—
340C22×C4C2×C22⋊C412yes—
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D412yes—
345M16—84no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—84no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—44no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—44no—
369C4⋊C4—44no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—44no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—21no—
385C22⋊C4—21no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4#9911yescharacteristic
395C22⋊C4#9911yescharacteristic
396C22⋊C4#9911yescharacteristic
397C22⋊C4#9911yescharacteristic
398C42—21no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—42no—
410C42—42no—
411C42—41no—
412C42—42no—
413C42—42no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42#13112yes—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23C2×#9911yescharacteristic
429C23C2×#9911yescharacteristic
430C23C22×C22⋊C411yescharacteristic
431Q8—81no—
432Q8—81no—
433Q8—81no—
434Q8—81no—
435D4—42no—
436D4—42no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—22no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—48no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—84no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4C2×#9911yescharacteristic
492C2×C4C2×#9911yescharacteristic
493C2×C4C2×#9911yescharacteristic
494C2×C4C2×#9911yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4#56811yescharacteristic
497C8—82no—
498C8—82no—
499C8—82no—
500C8—82no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—42no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—81no—
514C22—81no—
515C22#212612yes—
516C22#288011yescharacteristic
517C22C22×#9911yescharacteristic
518C22#288911yescharacteristic
519C4—24no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—42no—
527C4—41no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—41no—
532C4—42no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C4—81no—
538C4—81no—
539C2—21no—
540C2—21no—
541C2—22no—
542C2—41no—
543C2—81no—
544C2#1741611yescharacteristic
545C2C2×#212611yescharacteristic
546C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#552 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11571C2semidirect
4. #11552C2semidirect
5. #11551C2semidirect
6. #11532C2semidirect
7. #11513C2semidirect
8. #11512C2semidirect
9. #11478C2semidirect
10. #11447C2semidirect
11. C2×#1325C2semidirect

Maximal subgroups

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

subgroupindex
#361812
#174172
#115712
#115522
#115512
#115322
#115132
#115122
#114782
#114472
C2×#13252