Finite Group Databasebeta

← #182941

Subgroups — #182941

Order 512.

902 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60290C211yesmaximal, characteristic
2#20482C211yesmaximal, characteristic, complemented
3#17642C211yesmaximal, characteristic
4#17567C211yesmaximal, characteristic
5#17326C211yesmaximal, characteristic, complemented
6#8250C212yesmaximal, complemented
7#8249C212yesmaximal, complemented
8#8248C212yesmaximal, complemented
9#8247C212yesmaximal, complemented
10#8084C211yesmaximal, characteristic, complemented
11#8070C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13#2994—21no—
14#2994—21no—
15#2993C412yescomplemented
16#2993C412yescomplemented
17#2983C411yescharacteristic, complemented
18#2982C411yescharacteristic, complemented
19#2975C411yescharacteristic, complemented
20#2974—22no—
21#2972C411yescharacteristic, complemented
22#2969—21no—
23#2969—21no—
24#2966C2212yes—
25#2966C2212yes—
26#2924—22no—
27#2920C2211yescharacteristic
28#2920C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#431C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2283C2211yescharacteristic
33#1770—21no—
34C8×C4⋊C4—21no—
35#1753—22no—
36#1746C2212yescomplemented
37#1726C2212yescomplemented
38#1529C414yescomplemented
39#1528C412yescomplemented
40#1527C412yescomplemented
41#1524C2212yes—
42#1524C2212yes—
43#1521C414yes—
44#1520C412yes—
45#1519C412yes—
46#1500C2212yes—
47#1500C2212yes—
48#1454—22no—
49#1452—21no—
50#1451C2212yescomplemented
51#1450—21no—
52C8×M16C2212yescomplemented
53#1302C2211yescharacteristic, complemented
54#1302C2211yescharacteristic, complemented
55#1302C2211yescharacteristic, complemented
56#1302C2211yescharacteristic, complemented
57#1300C2212yescomplemented
58#1300C2212yescomplemented
59C22×M16—21no—
60#529—21no—
61#529—21no—
62#529—21no—
63#529—21no—
64#529C2×C412yes—
65#529C2×C412yes—
66#529C2×C412yes—
67#529C2312yes—
68C2×#126—21no—
69C2×#125C2×C411yescharacteristic
70C2×#124C2×C411yescharacteristic
71C2×#123—21no—
72C2×C22⋊Q8C2×C411yescharacteristic
73C2×#121C2×C411yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4C2×C411yescharacteristic
77C2×#117—21no—
78C22×C42C2311yescharacteristic
79#446—21no—
80#446—21no—
81#446C2×C412yes—
82#446C2×C412yes—
83C8×Q8—21no—
84C8×Q8—21no—
85C8×Q8C2×C412yes—
86C8×Q8C2×C412yes—
87#436—21no—
88#436—21no—
89#436—22no—
90#436—22no—
91#436C2×C412yes—
92#436C2×C412yes—
93#435—22no—
94#435—22no—
95#435—22no—
96#435—22no—
97C8×D4—22no—
98C8×D4—22no—
99C8×D4—21no—
100C8×D4—21no—
101C8×D4C2×C412yes—
102C8×D4C2×C412yes—
103#433—22no—
104#433—22no—
105#433—22no—
106#433—22no—
107#432—21no—
108#432—22no—
109#432C2×C412yes—
110#432C2×C412yes—
111#432C2311yescharacteristic
112#432C2311yescharacteristic
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431—22no—
119#431C2×C411yescharacteristic
120#431C2×C411yescharacteristic
121#431C2×C412yes—
122#431C2×C411yescharacteristic
123#431C2×C411yescharacteristic
124#431C2×C412yes—
125#431D412yes—
126#431D411yescharacteristic
127#431D411yescharacteristic
128#431C2311yescharacteristic
129#431C2311yescharacteristic
130#424—22no—
131#424—22no—
132#423—21no—
133#423C2×C411yescharacteristic
134#423C2×C411yescharacteristic
135C2×C4⋊C8—21no—
136C2×C4⋊C8D411yescharacteristic
137C2×C4⋊C8D411yescharacteristic
138#407—21no—
139#407—21no—
140#407—22no—
141C2×C22⋊C8—22no—
142C2×C22⋊C8C2×C412yes—
143C2×C22⋊C8D412yes—
144#405—22no—
145#405—22no—
146C4×M16—21no—
147C4×M16—21no—
148C4×M16—21no—
149C4×M16C2×C412yes—
150C4×M16C2×C412yes—
151C4×M16C2312yes—
152C2×#97—22no—
153C2×#97C2311yescharacteristic
154C2×#97C2311yescharacteristic
155C2×C4×C8—21no—
156C2×C4×C8—21no—
157C2×C4×C8—22no—
158C2×C4×C8C2312yes—
159#383C2×C412yes—
160#382—21no—
161#381—21no—
162#380C2×C412yes—
163C4×C4⋊C4—21no—
164C4×C22⋊C4—21no—
165#376C2312yes—
166#336—22no—
167#336—22no—
168#330C2×C414yes—
169#330C2×C414yes—
170#329C2×C414yes—
171#329C2×C414yes—
172#328C2×C414yes—
173#328C2×C414yes—
174#327C2×C414yes—
175#327C2×C414yes—
176#322—22no—
177#322—22no—
178#322—22no—
179C82—22no—
180C22×Q8C4211yescharacteristic
181C22×D4—21no—
182C22×D4C4211yescharacteristic
183C23×C4—21no—
184C23×C4C22×C411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4111yescharacteristic
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—42no—
195C2×M16—42no—
196C22×C8—21no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—22no—
200C22×C8—22no—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203#126—22no—
204#126—22no—
205#126—22no—
206#126—22no—
207#125—21no—
208#125—21no—
209#125C4212yes—
210#125C22×C412yes—
211#124—21no—
212#124—21no—
213#124C4212yes—
214#124C22×C412yes—
215#123—22no—
216#123—22no—
217#123—22no—
218#123—22no—
219C22⋊Q8—21no—
220C22⋊Q8—21no—
221C22⋊Q8C4212yes—
222C22⋊Q8C22×C412yes—
223#121—21no—
224#121—21no—
225#121C4212yes—
226#121C22×C412yes—
227C4×Q8—21no—
228C4×Q8—21no—
229C4×Q8C4212yes—
230C4×Q8C22×C412yes—
231C4×D4—22no—
232C4×D4—22no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—21no—
236C4×D4—21no—
237C4×D4C4212yes—
238C4×D4C22×C412yes—
239#117—22no—
240#117—22no—
241#117—22no—
242#117—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4C4211yescharacteristic
249C2×C4⋊C4C4211yescharacteristic
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—22no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C22×C411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42#4111yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42C2411yescharacteristic
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8—22no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8C22⋊C412yes—
282C4⋊C8C22⋊C412yes—
283C4⋊C8C22⋊C411yescharacteristic
284C4⋊C8C22⋊C411yescharacteristic
285C4⋊C8C22⋊C411yescharacteristic
286C4⋊C8C22⋊C411yescharacteristic
287C4⋊C8C22×C411yescharacteristic
288C4⋊C8C22×C411yescharacteristic
289C4⋊C8C22×C411yescharacteristic
290C4⋊C8C22×C411yescharacteristic
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8C22⋊C412yes—
303C22⋊C8C22⋊C412yes—
304C22⋊C8C22⋊C412yes—
305C22⋊C8C22⋊C412yes—
306C22⋊C8C22×C412yes—
307C22⋊C8C22×C412yes—
308#97—22no—
309#97—22no—
310#97—21no—
311#97—21no—
312#97—21no—
313#97—21no—
314#97—44no—
315#97C22×C412yes—
316#97C22×C412yes—
317#97C22×C414yes—
318C4×C8—22no—
319C4×C8—22no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—22no—
323C4×C8—22no—
324C4×C8—22no—
325C4×C8—21no—
326C4×C8—22no—
327C4×C8—21no—
328C4×C8—44no—
329C4×C8C22×C412yes—
330C4×C8C22×C412yes—
331C4×C8C2×D412yes—
332C4×C8C2×D412yes—
333#95—22no—
334#95—22no—
335#95—22no—
336#95C22×C414yes—
337C24—21no—
338C24C4×D411yescharacteristic
339C2×Q8—21no—
340C2×Q8—42no—
341C2×Q8C2×C4212yes—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—21no—
345C2×D4—42no—
346C2×D4—82no—
347C2×D4C2×C4212yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4C2×C4211yescharacteristic
361C22×C4C2×C4211yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4#11711yescharacteristic
365C22×C4#11711yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C23×C411yescharacteristic
376C22×C4C22×D411yescharacteristic
377C22×C4C2×#4111yescharacteristic
378M16—44no—
379M16—44no—
380M16—44no—
381M16—82no—
382M16—82no—
383M16—81no—
384M16—81no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—24no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—44no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—82no—
402C2×C8—82no—
403C2×C8C2×C22⋊C412yes—
404C2×C8C2×C22⋊C412yes—
405C2×C8C2×C22⋊C412yes—
406C2×C8C2×C22⋊C412yes—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—44no—
418C4⋊C4C2×C4212yes—
419C4⋊C4C2×C4212yes—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—82no—
431C42—22no—
432C42—22no—
433C42—42no—
434C42—42no—
435C42C2×C4211yescharacteristic
436C42C2×C4211yescharacteristic
437C42C2×C22⋊C411yescharacteristic
438C42C2×C22⋊C411yescharacteristic
439C42#11711yescharacteristic
440C42#11711yescharacteristic
441C42C4×D411yescharacteristic
442C42C4×D411yescharacteristic
443C42C4×D411yescharacteristic
444C42C4×D411yescharacteristic
445C42C4×D411yescharacteristic
446C42C4×D411yescharacteristic
447C42C23×C411yescharacteristic
448C42C23×C411yescharacteristic
449C42C22×D411yescharacteristic
450C42C2×#4111yescharacteristic
451C23—22no—
452C23—22no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—81no—
457C23—81no—
458C23C4×C22⋊C411yescharacteristic
459C23C4×C22⋊C411yescharacteristic
460C23C2×C4×D411yescharacteristic
461Q8—42no—
462Q8—42no—
463D4—42no—
464D4—44no—
465D4—42no—
466D4—82no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4C4×C22⋊C411yescharacteristic
501C2×C4C4×C22⋊C411yescharacteristic
502C2×C4C4×C22⋊C411yescharacteristic
503C2×C4C4×C22⋊C411yescharacteristic
504C2×C4C4×C22⋊C411yescharacteristic
505C2×C4C4×C22⋊C411yescharacteristic
506C2×C4C22×C4211yescharacteristic
507C2×C4C22×C22⋊C411yescharacteristic
508C2×C4C2×#11711yescharacteristic
509C2×C4C2×C4×D411yescharacteristic
510C2×C4C2×C4×D411yescharacteristic
511C2×C4C2×C4×D411yescharacteristic
512C8—24no—
513C8—22no—
514C8—22no—
515C8—22no—
516C8—22no—
517C8—42no—
518C8—42no—
519C8—42no—
520C8—44no—
521C8—41no—
522C8—41no—
523C8—41no—
524C8—41no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—82no—
537C22—162no—
538C22C2×C4×C22⋊C411yescharacteristic
539C22#295712yes—
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—42no—
551C4—42no—
552C4—42no—
553C4—84no—
554C4#175712yes—
555C4#290111yescharacteristic
556C4#290111yescharacteristic
557C2—22no—
558C2—41no—
559C2—41no—
560C2—82no—
561C2C2×#175711yescharacteristic
562C2#2040511yescharacteristic
563C2#2042211yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC2×C42C24
socleC22C2×C4×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20482C2semidirect
5. #17326C2semidirect
6. #8250C2semidirect
7. #8249C2semidirect
8. #8248C2semidirect
9. #8247C2semidirect
10. #8084C2semidirect
11. #8070C2semidirect
15. #2993C4semidirect
16. #2993C4semidirect
17. #2983C4semidirect
18. #2982C4semidirect
19. #2975C4semidirect
21. #2972C4semidirect
36. #1746C22semidirect
37. #1726C22semidirect
38. #1529C4semidirect
39. #1528C4semidirect
40. #1527C4semidirect
50. #1451C22semidirect
52. C8×M16C22semidirect
53. #1302C22semidirect
54. #1302C22semidirect
55. #1302C22semidirect
56. #1302C22semidirect
57. #1300C22semidirect
58. #1300C22semidirect

Maximal subgroups

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

subgroupindex
#602902
#204822
#176422
#175672
#173262
#82502
#82492
#82482
#82472
#80842
#80702