Finite Group Databasebeta

← #480915

Subgroups — #480915

Order 512.

866 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#38094C211yesmaximal, characteristic
2#17428C212yesmaximal, complemented
3#11565C212yesmaximal, complemented
4#11554C212yesmaximal, complemented
5#11552C212yesmaximal, complemented
6#11534C211yesmaximal, characteristic, complemented
7#11509C211yesmaximal, characteristic, complemented
8#11507C211yesmaximal, characteristic, complemented
9#11480C211yesmaximal, characteristic, complemented
10#11433C211yesmaximal, characteristic, complemented
11#11431C211yesmaximal, characteristic, complemented
12C2×#556C2211yescharacteristic
13C2×#546C411yescharacteristic
14#2984—22no—
15#2850C411yescharacteristic
16#2691—22no—
17#2684—22no—
18#2640C411yescharacteristic
19#2621C411yescharacteristic
20#2563C411yescharacteristic
21#2545C2211yescharacteristic
22#2500C2211yescharacteristic
23#2499C2211yescharacteristic
24#2480—22no—
25#2458—22no—
26#2435C411yescharacteristic
27#2432C411yescharacteristic
28#2326C2211yescharacteristic
29#2309C2211yescharacteristic
30#2297C2211yescharacteristic
31#2290C411yescharacteristic
32#1473—22no—
33#1473—22no—
34#1469—22no—
35#1467C2212yescomplemented
36#1464C2212yes—
37#1464C2212yescomplemented
38#1461C2212yescomplemented
39#1458C2212yes—
40#1458C2212yescomplemented
41#1325C2212yescomplemented
42#1325C2212yescomplemented
43#1319C2212yescomplemented
44#1319C2212yescomplemented
45#1318C2211yescharacteristic
46#1318C2211yescharacteristic
47#1318C2212yes—
48#1318C2212yes—
49#1318C2211yescharacteristic
50#1318C2211yescharacteristic
51C22×#41C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53#556—42no—
54#556—42no—
55#546—42no—
56#546—42no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#126D412yes—
60C2×#126D412yes—
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×#124D412yes—
65C2×#123—21no—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×#121—21no—
70C2×#121C2×C411yescharacteristic
71C2×#120—21no—
72C2×C4×Q8D412yes—
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×#117C2×C411yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C22⋊C4—21no—
80C22×C42C2×C411yescharacteristic
81#446—42no—
82#446—42no—
83#446—42no—
84C8×Q8—42no—
85#432—22no—
86#432—22no—
87#432—22no—
88#431—22no—
89#431—22no—
90#431—22no—
91#398—22no—
92#397—22no—
93#397—22no—
94#397—22no—
95#396—22no—
96#394—22no—
97#394—22no—
98#394—22no—
99#393—22no—
100#393—22no—
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#391C2×C411yescharacteristic
104#391C2311yescharacteristic
105#390—22no—
106#390C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#390C2×C411yescharacteristic
110#390C2311yescharacteristic
111#390C2311yescharacteristic
112#388—22no—
113#388—21no—
114#388—22no—
115#388—22no—
116#388—22no—
117#387—21no—
118#387—22no—
119#386—21no—
120#386—22no—
121#386—22no—
122#385—22no—
123#385—22no—
124#385—21no—
125#384C2311yescharacteristic
126#382C2×C411yescharacteristic
127#382C2×C411yescharacteristic
128#380—22no—
129#379—22no—
130#379—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#324C2312yes—
139#324C2312yes—
140#324C2312yes—
141#324C2312yes—
142#323—22no—
143#323—22no—
144#323—22no—
145#323—22no—
146C25—21no—
147C2×#41—22no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—41no—
151C2×#41—81no—
152C22×Q8C22⋊C412yes—
153C22×Q8C22×C411yescharacteristic
154C22×Q8C22×C411yescharacteristic
155C22×D4—22no—
156C22×D4—21no—
157C22×D4—21no—
158C22×D4—21no—
159C22×D4—41no—
160C22×D4—42no—
161C22×D4—81no—
162C22×D4C22⋊C411yescharacteristic
163C22×D4C22×C411yescharacteristic
164C22×D4C22×C411yescharacteristic
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—21no—
168C23×C4C22×C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C4⋊Q8—42no—
171C4⋊Q8—81no—
172#126—42no—
173#126—42no—
174#126—42no—
175#126—42no—
176#124—22no—
177#124—41no—
178#124—42no—
179#124—42no—
180#124—42no—
181#124—81no—
182#123—42no—
183#123—42no—
184C22⋊Q8—41no—
185C22⋊Q8—41no—
186C22⋊Q8—42no—
187C22⋊Q8—42no—
188#121—44no—
189#121—41no—
190#121—41no—
191#120—44no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—42no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—42no—
200#117—41no—
201#117—42no—
202#117—41no—
203#117—81no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C22⋊C4C2×D412yes—
241C2×C22⋊C4C2×D412yes—
242C2×C42—21no—
243C2×C42—81no—
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2×D412yes—
252C4⋊C8—42no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C22⋊C8—22no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262#97—42no—
263#97—42no—
264#97—42no—
265C4×C8—42no—
266C4×C8—42no—
267C4×C8—42no—
268#95—21no—
269#95—21no—
270#95—22no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—22no—
278#95—22no—
279C24—21no—
280C24—21no—
281C24—42no—
282C24—42no—
283C24—81no—
284C24—82no—
285C24—82no—
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C411yescharacteristic
288#41—82no—
289#41—82no—
290#41—82no—
291#41—82no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—22no—
295C2×Q8—22no—
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—82no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—42no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—81no—
317C2×D4—84no—
318C2×D4—84no—
319C2×D4—81no—
320C2×D4—81no—
321C2×D4—82no—
322C2×D4#13112yes—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—81no—
343C22×C4—81no—
344C22×C4—81no—
345C22×C4—81no—
346C22×C4—82no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C412yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C23×C411yescharacteristic
355C22×C4C22×D412yes—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—42no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—81no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—81no—
375C4⋊C4#9912yes—
376C4⋊C4#9912yes—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—22no—
380C22⋊C4—42no—
381C22⋊C4—42no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—82no—
392C22⋊C4—81no—
393C22⋊C4—82no—
394C22⋊C4—82no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—81no—
400C22⋊C4—81no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—21no—
409C42—21no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—81no—
414C23—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—42no—
422C23—42no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—84no—
435C23—84no—
436C23—164no—
437C23—162no—
438C23—164no—
439C23C22×C22⋊C411yescharacteristic
440Q8—44no—
441Q8—81no—
442Q8—81no—
443Q8—82no—
444Q8—81no—
445Q8—81no—
446D4—42no—
447D4—42no—
448D4—82no—
449D4—84no—
450D4—81no—
451D4—81no—
452D4—84no—
453D4—162no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
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—42no—
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—82no—
496C2×C4—82no—
497C2×C4—84no—
498C2×C4—162no—
499C2×C4C2×#9912yes—
500C2×C4C2×#9911yescharacteristic
501C2×C4C2×#9911yescharacteristic
502C2×C4C2×#9912yes—
503C2×C4C2×#13111yescharacteristic
504C2×C4#56811yescharacteristic
505C8—82no—
506C8—82no—
507C8—82no—
508C8—82no—
509C22—21no—
510C22—21no—
511C22—44no—
512C22—41no—
513C22—41no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22—81no—
530C22—81no—
531C22—168no—
532C22#288011yescharacteristic
533C22C22×#9911yescharacteristic
534C22#288811yescharacteristic
535C4—21no—
536C4—21no—
537C4—21no—
538C4—21no—
539C4—22no—
540C4—22no—
541C4—42no—
542C4—42no—
543C4—41no—
544C4—42no—
545C4—42no—
546C4—81no—
547C4—81no—
548C4—81no—
549C4—81no—
550C4—81no—
551C4—81no—
552C2—21no—
553C2—21no—
554C2—81no—
555C2—82no—
556C2—81no—
557C2—161no—
558C2#1741111yescharacteristic
559C2#3336211yescharacteristic
560C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#546C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#556 ◁ G
LECSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17428C2semidirect
3. #11565C2semidirect
4. #11554C2semidirect
5. #11552C2semidirect
6. #11534C2semidirect
7. #11509C2semidirect
8. #11507C2semidirect
9. #11480C2semidirect
10. #11433C2semidirect
11. #11431C2semidirect
35. #1467C22semidirect
37. #1464C22semidirect
38. #1461C22semidirect
40. #1458C22semidirect
41. #1325C22semidirect
42. #1325C22semidirect
43. #1319C22semidirect
44. #1319C22semidirect

Maximal subgroups

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

subgroupindex
#380942
#174282
#115652
#115542
#115522
#115342
#115092
#115072
#114802
#114332
#114312