Finite Group Databasebeta

← #481001

Subgroups — #481001

Order 512.

776 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37870C211yesmaximal, characteristic
2#17428C212yesmaximal, complemented
3#11571C212yesmaximal, complemented
4#11554C212yesmaximal, complemented
5#11550C212yesmaximal, complemented
6#11534C211yesmaximal, characteristic, complemented
7#11516C211yesmaximal, characteristic, complemented
8#11513C211yesmaximal, characteristic, complemented
9#11480C211yesmaximal, characteristic, complemented
10#11450C211yesmaximal, characteristic, complemented
11#11447C211yesmaximal, characteristic, complemented
12C2×#556C2211yescharacteristic
13C2×#553C411yescharacteristic
14#2984—22no—
15#2850C411yescharacteristic
16#2695—22no—
17#2686—22no—
18#2660C411yescharacteristic
19#2620C411yescharacteristic
20#2565C2211yescharacteristic
21#2563C411yescharacteristic
22#2503C411yescharacteristic
23#2500C2211yescharacteristic
24#2500C2211yescharacteristic
25#2499C411yescharacteristic
26#2469—22no—
27#2468—22no—
28#2326C2211yescharacteristic
29#2317C2211yescharacteristic
30#2290C411yescharacteristic
31#2285C2211yescharacteristic
32#1473—22no—
33#1473—22no—
34#1469—22no—
35#1467C2212yes—
36#1464C2212yes—
37#1464C2212yes—
38#1461C2212yes—
39#1458C2212yes—
40#1458C2212yes—
41#1325C2211yescharacteristic
42#1325C2211yescharacteristic
43#1325C2212yes—
44#1325C2212yes—
45#1325C2211yescharacteristic
46#1325C2211yescharacteristic
47#1324C2212yes—
48#1324C2212yes—
49#1318C2212yes—
50#1318C2212yes—
51#556—42no—
52#556—42no—
53#553—42no—
54#553—42no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—21no—
57C2×#126D412yes—
58C2×#126D412yes—
59C2×#125C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#124D412yes—
62C2×#123—21no—
63C2×C22⋊Q8C2×C411yescharacteristic
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×#121—21no—
66C2×#121C2×C411yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×C4×Q8D412yes—
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C42C2×C411yescharacteristic
79#446—42no—
80#446—42no—
81#446—42no—
82C8×Q8—42no—
83#432—22no—
84#432—22no—
85#432—22no—
86#431—22no—
87#431—22no—
88#431—22no—
89#400—22no—
90#398—22no—
91#398—22no—
92#397—22no—
93#396—22no—
94#395—22no—
95#394—22no—
96#394—22no—
97#394—22no—
98#393—22no—
99#391C2×C411yescharacteristic
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2311yescharacteristic
105#389—22no—
106#388—22no—
107#388—21no—
108#388—22no—
109#388—22no—
110#388—22no—
111#387—22no—
112#387—21no—
113#387—22no—
114#386—21no—
115#386—22no—
116#385—22no—
117#385—21no—
118#385—22no—
119#384C2×C411yescharacteristic
120#384C2×C411yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382C2×C411yescharacteristic
124#382C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126#382C2×C411yescharacteristic
127#382C2311yescharacteristic
128#381—22no—
129#380—22no—
130#380—22no—
131C4×C4⋊C4—22no—
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×Q8C22⋊C412yes—
147C22×Q8C22×C411yescharacteristic
148C22×D4—21no—
149C22×D4—21no—
150C22×D4C22⋊C411yescharacteristic
151C22×D4C22×C411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4C22×C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C4⋊Q8—42no—
158#126—42no—
159#126—42no—
160#126—42no—
161#126—42no—
162#126—42no—
163#126—42no—
164#125—42no—
165#124—22no—
166#124—41no—
167#124—42no—
168#124—42no—
169#123—44no—
170C22⋊Q8—42no—
171C22⋊Q8—41no—
172C22⋊Q8—41no—
173#121—44no—
174#121—41no—
175#121—41no—
176#121—42no—
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—42no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—42no—
187#117—22no—
188#117—41no—
189#117—41no—
190#117—42no—
191#117—41no—
192#117—41no—
193#117—41no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22⋊C412yes—
205C2×C4⋊C4C22⋊C412yes—
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4C22⋊C411yescharacteristic
227C2×C22⋊C4C22⋊C411yescharacteristic
228C2×C22⋊C4C22⋊C412yes—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C2×D412yes—
231C2×C22⋊C4C2×D412yes—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—41no—
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C22×C411yescharacteristic
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—22no—
253C22⋊C8—22no—
254#97—42no—
255#97—42no—
256#97—42no—
257C4×C8—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—
273C24—21no—
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—82no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—22no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—82no—
291C2×D4—81no—
292C2×D4—82no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4C2×C22⋊C412yes—
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—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—81no—
350C4⋊C4#9912yes—
351C4⋊C4#9912yes—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—22no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C22⋊C4—42no—
358C22⋊C4—42no—
359C22⋊C4—44no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—81no—
370C22⋊C4—81no—
371C42—21no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—42no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42#13112yes—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C22×C22⋊C411yescharacteristic
404Q8—44no—
405Q8—81no—
406Q8—81no—
407Q8—82no—
408D4—82no—
409D4—81no—
410D4—81no—
411D4—84no—
412D4—162no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—84no—
456C2×C4—162no—
457C2×C4—162no—
458C2×C4C2×#9912yes—
459C2×C4C2×#9911yescharacteristic
460C2×C4C2×#9911yescharacteristic
461C2×C4C2×#9912yes—
462C2×C4C2×#13111yescharacteristic
463C2×C4#56811yescharacteristic
464C8—82no—
465C8—82no—
466C8—82no—
467C8—82no—
468C22—21no—
469C22—21no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—82no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22#288011yescharacteristic
486C22C22×#9911yescharacteristic
487C22#288911yescharacteristic
488C4—21no—
489C4—21no—
490C4—21no—
491C4—21no—
492C4—22no—
493C4—22no—
494C4—42no—
495C4—42no—
496C4—42no—
497C4—41no—
498C4—42no—
499C4—42no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C2—21no—
507C2—21no—
508C2—81no—
509C2—81no—
510C2—161no—
511C2#1741611yescharacteristic
512C2#3336211yescharacteristic
513C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C4⋊C4C24
prime radicalC2×#553C4
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. #11571C2semidirect
4. #11554C2semidirect
5. #11550C2semidirect
6. #11534C2semidirect
7. #11516C2semidirect
8. #11513C2semidirect
9. #11480C2semidirect
10. #11450C2semidirect
11. #11447C2semidirect

Maximal subgroups

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

subgroupindex
#378702
#174282
#115712
#115542
#115502
#115342
#115162
#115132
#114802
#114502
#114472