Finite Group Databasebeta

← #480905

Subgroups — #480905

Order 512.

841 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38088C211yesmaximal, characteristic
2#17417C212yesmaximal, complemented
3#11568C212yesmaximal, complemented
4#11555C212yesmaximal, complemented
5#11552C212yesmaximal, complemented
6#11534C211yesmaximal, characteristic, complemented
7#11508C211yesmaximal, characteristic, complemented
8#11507C211yesmaximal, characteristic, complemented
9#11480C211yesmaximal, characteristic, complemented
10#11432C211yesmaximal, characteristic, complemented
11#11431C211yesmaximal, characteristic, complemented
12C2×#552C2211yescharacteristic
13C2×#547C411yescharacteristic
14#2924—22no—
15#2857C411yescharacteristic
16#2708—22no—
17#2695—22no—
18#2642C411yescharacteristic
19#2621C411yescharacteristic
20#2566C411yescharacteristic
21#2546C2211yescharacteristic
22#2502C2211yescharacteristic
23#2499C2211yescharacteristic
24#2473—22no—
25#2470—22no—
26#2432C411yescharacteristic
27#2432C411yescharacteristic
28#2326C2211yescharacteristic
29#2311C2211yescharacteristic
30#2297C2211yescharacteristic
31#2290C411yescharacteristic
32#1473—22no—
33#1470—22no—
34#1468—22no—
35#1467C2212yes—
36#1464C2212yes—
37#1463C2212yes—
38#1461C2212yes—
39#1458C2212yes—
40C2×#324C2212yes—
41#1325C2212yes—
42#1325C2212yes—
43#1319C2212yes—
44#1319C2212yes—
45#1319C2212yes—
46#1319C2212yes—
47#1318C2211yescharacteristic
48#1318C2211yescharacteristic
49#1318C2211yescharacteristic
50#1318C2211yescharacteristic
51C22×#41C2×C411yescharacteristic
52C22×#41C2×C411yescharacteristic
53#552—42no—
54#552—42no—
55#547—42no—
56#547—42no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#126D412yes—
61C2×#125C2×C411yescharacteristic
62C2×#125D412yes—
63C2×#124C2×C411yescharacteristic
64C2×#123—21no—
65C2×#123—21no—
66C2×#123D412yes—
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C22⋊Q8C2×C411yescharacteristic
70C2×C22⋊Q8C2×C411yescharacteristic
71C2×#121—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×#117C2×C411yescharacteristic
76C2×#117D412yes—
77C2×#117C2311yescharacteristic
78C22×C4⋊C4—21no—
79C22×C4⋊C4C2311yescharacteristic
80C22×C42C2×C411yescharacteristic
81#433—42no—
82#433—42no—
83#432—22no—
84#432—22no—
85#431—22no—
86#431—22no—
87#424—42no—
88#407—22no—
89C2×C22⋊C8—22no—
90#405—42no—
91#400—22no—
92#400—22no—
93#398—22no—
94#398—22no—
95#398—22no—
96#397—22no—
97#397—22no—
98#396—22no—
99#394—22no—
100#394—22no—
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2×C411yescharacteristic
105#391C2×C411yescharacteristic
106#391C2311yescharacteristic
107#391C2311yescharacteristic
108#390C2×C411yescharacteristic
109#390C2×C411yescharacteristic
110#390C2311yescharacteristic
111#389—22no—
112#389—22no—
113#388—21no—
114#388—22no—
115#387—21no—
116#387—22no—
117#386—21no—
118#386—22no—
119#386—22no—
120#385—22no—
121#385—22no—
122#385—22no—
123#385—22no—
124#385—22no—
125#385—21no—
126#384C2×C411yescharacteristic
127#384C2311yescharacteristic
128#382C2×C411yescharacteristic
129#380—22no—
130#380—22no—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C22⋊C4—22no—
133C4×C22⋊C4C2×C411yescharacteristic
134C4×C22⋊C4C2×C411yescharacteristic
135#376C2×C411yescharacteristic
136#324—22no—
137#324—22no—
138#324—22no—
139#324—22no—
140#324C2312yes—
141#324C2312yes—
142#324C2312yes—
143#324C2312yes—
144#323—22no—
145#323—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—42no—
151C2×#41—41no—
152C2×#41—41no—
153C2×#41—81no—
154C2×#41—81no—
155C22×Q8C22×C411yescharacteristic
156C22×Q8C22×C411yescharacteristic
157C22×Q8C22×C411yescharacteristic
158C22×D4—21no—
159C22×D4—21no—
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D412yes—
169C2×M16—42no—
170C22×C8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—81no—
173C4⋊Q8—81no—
174#126—42no—
175#126—42no—
176#125—42no—
177#125—42no—
178#125—42no—
179#124—22no—
180#124—41no—
181#123—42no—
182#123—42no—
183#123—44no—
184#123—42no—
185#123—42no—
186C22⋊Q8—42no—
187C22⋊Q8—41no—
188C22⋊Q8—41no—
189C22⋊Q8—41no—
190C22⋊Q8—41no—
191C22⋊Q8—42no—
192#121—44no—
193C4×Q8—42no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—42no—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—22no—
201#117—41no—
202#117—41no—
203#117—81no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4C22⋊C411yescharacteristic
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2411yescharacteristic
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C22⋊C4C22⋊C412yes—
241C2×C22⋊C4C22⋊C411yescharacteristic
242C2×C22⋊C4C2×D412yes—
243C2×C42—21no—
244C2×C42—81no—
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C4⋊C8—42no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C22⋊C8—22no—
257C22⋊C8—22no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262#97—42no—
263#97—42no—
264C4×C8—42no—
265C4×C8—42no—
266#95—21no—
267#95—21no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279C24—21no—
280C24C2×C22⋊C411yescharacteristic
281C24C2×C22⋊C411yescharacteristic
282#41—82no—
283#41—82no—
284#41—84no—
285#41—84no—
286#41—82no—
287#41—82no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—22no—
301C2×D4—21no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4—81no—
311C2×D4#13112yes—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
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—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—81no—
338C22×C4—81no—
339C22×C4—82no—
340C22×C4—81no—
341C22×C4—81no—
342C22×C4—82no—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C23×C411yescharacteristic
351C22×C4C22×D412yes—
352M16—84no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—84no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—42no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4—82no—
382C4⋊C4—82no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—44no—
389C22⋊C4—44no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—81no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—81no—
398C22⋊C4#9912yes—
399C22⋊C4#9912yes—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—21no—
406C42—21no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—81no—
411C23—21no—
412C23—21no—
413C23—22no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—41no—
418C23—41no—
419C23—42no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23C2×#9912yes—
426C23C22×C22⋊C411yescharacteristic
427Q8—81no—
428Q8—81no—
429Q8—81no—
430Q8—81no—
431Q8—81no—
432Q8—81no—
433D4—42no—
434D4—42no—
435D4—82no—
436D4—84no—
437D4—84no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—21no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—21no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—44no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—84no—
488C2×C4—84no—
489C2×C4—84no—
490C2×C4C2×#9912yes—
491C2×C4C2×#9911yescharacteristic
492C2×C4C2×#9911yescharacteristic
493C2×C4C2×#13111yescharacteristic
494C2×C4#56811yescharacteristic
495C8—82no—
496C8—82no—
497C8—82no—
498C8—82no—
499C22—21no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—44no—
505C22—41no—
506C22—42no—
507C22—41no—
508C22—41no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22#288011yescharacteristic
517C22C22×#9911yescharacteristic
518C22#288811yescharacteristic
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—22no—
524C4—42no—
525C4—41no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—41no—
530C4—42no—
531C4—81no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4—81no—
537C2—21no—
538C2—21no—
539C2—42no—
540C2—81no—
541C2—82no—
542C2#1741111yescharacteristic
543C2#3335811yescharacteristic
544C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#547C4
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
2. #17417C2semidirect
3. #11568C2semidirect
4. #11555C2semidirect
5. #11552C2semidirect
6. #11534C2semidirect
7. #11508C2semidirect
8. #11507C2semidirect
9. #11480C2semidirect
10. #11432C2semidirect
11. #11431C2semidirect

Maximal subgroups

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

subgroupindex
#380882
#174172
#115682
#115552
#115522
#115342
#115082
#115072
#114802
#114322
#114312