Finite Group Databasebeta

← #182932

Subgroups — #182932

Order 512.

943 conjugacy classes of subgroups.

The subgroup lattice has 569 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2966C211yesmaximal, characteristic
2#20476C211yesmaximal, characteristic, complemented
3#17659C211yesmaximal, characteristic
4#17554C211yesmaximal, characteristic
5#17323C211yesmaximal, characteristic, complemented
6#8250C214yesmaximal, complemented
7#8249C214yesmaximal, complemented
8#8085C211yesmaximal, characteristic, complemented
9#8069C211yesmaximal, characteristic, complemented
10C2×#529C2211yescharacteristic
11#2966—21no—
12#2966—21no—
13#2966—21no—
14#2966—21no—
15#2966C412yescomplemented
16#2966C412yescomplemented
17#2966C2212yes—
18#2966C2212yes—
19C2×#446C411yescharacteristic, complemented
20C2×C8×Q8C411yescharacteristic, complemented
21C2×#436C411yescharacteristic, complemented
22C2×#435—21no—
23C2×C8×D4—21no—
24C2×C8×D4C411yescharacteristic, complemented
25#2920C2211yescharacteristic
26#2920C2211yescharacteristic
27C2×#433—21no—
28C2×#432C2211yescharacteristic
29C2×#432C2211yescharacteristic
30C2×#405—21no—
31C22×C4×C8C2211yescharacteristic
32#2283C2211yescharacteristic
33#1770—21no—
34C8×C4⋊C4—21no—
35#1752—21no—
36C8×C22⋊C4—21no—
37#1746C2212yescomplemented
38#1726C2212yescomplemented
39#1535C414yescomplemented
40#1534C412yescomplemented
41#1533C412yescomplemented
42#1524C2214yes—
43#1512C414yes—
44#1511C412yes—
45#1510C412yes—
46#1500C2214yes—
47#1452—21no—
48#1451—21no—
49#1451C2212yescomplemented
50#1450—21no—
51C8×M16—21no—
52C8×M16C2212yescomplemented
53#1302C2214yescomplemented
54#1300C2214yescomplemented
55C22×M16—21no—
56C23×C8—21no—
57C23×C8D411yescharacteristic
58C23×C8D411yescharacteristic
59#529—21no—
60#529—21no—
61#529—21no—
62#529—21no—
63#529C2×C412yes—
64#529C2×C412yes—
65#529C2×C412yes—
66#529C2312yes—
67C2×#126—21no—
68C2×#125C2×C411yescharacteristic
69C2×#124C2×C411yescharacteristic
70C2×#123—21no—
71C2×C22⋊Q8C2×C411yescharacteristic
72C2×#121C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×#117—21no—
77C22×C42C2311yescharacteristic
78#446—21no—
79#446—21no—
80#446C2×C412yes—
81#446C2×C412yes—
82C8×Q8—21no—
83C8×Q8—21no—
84C8×Q8C2×C412yes—
85C8×Q8C2×C412yes—
86#436—21no—
87#436—21no—
88#436C2×C412yes—
89#436C2×C412yes—
90#435—22no—
91#435—22no—
92#435—22no—
93#435—22no—
94C8×D4—22no—
95C8×D4—22no—
96C8×D4—22no—
97C8×D4—22no—
98C8×D4—21no—
99C8×D4—21no—
100C8×D4C2×C412yes—
101C8×D4C2×C412yes—
102#433—22no—
103#433—22no—
104#433—22no—
105#433—22no—
106#432—21no—
107#432—21no—
108#432—21no—
109#432—21no—
110#432—21no—
111#432C2×C412yes—
112#432C2×C412yes—
113#432C2×C412yes—
114#432C2×C412yes—
115#432C2×C412yes—
116#432C2×C412yes—
117#432C2312yes—
118#431—21no—
119#431—21no—
120#431—21no—
121#431C2312yes—
122#423—21no—
123#423—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8C2×C411yescharacteristic
128C2×C4⋊C8C2×C411yescharacteristic
129#407—21no—
130#407C2×C412yes—
131C2×C22⋊C8—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135#405—22no—
136#405—22no—
137#405—22no—
138#405—22no—
139C4×M16—21no—
140C4×M16C2312yes—
141C2×#97—21no—
142C2×#97—21no—
143C2×#97—22no—
144C2×#97C2312yes—
145C2×C4×C8—21no—
146C2×C4×C8—22no—
147C2×C4×C8C2×C411yescharacteristic
148C2×C4×C8C2×C411yescharacteristic
149C2×C4×C8C2×C411yescharacteristic
150C2×C4×C8C2×C411yescharacteristic
151C2×C4×C8D411yescharacteristic
152C2×C4×C8D411yescharacteristic
153C2×C4×C8D411yescharacteristic
154C2×C4×C8D411yescharacteristic
155C2×C4×C8D411yescharacteristic
156C2×C4×C8D411yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#383—21no—
160#382C2×C412yes—
161#381C2×C412yes—
162#380—21no—
163C4×C4⋊C4—21no—
164C4×C22⋊C4—21no—
165#376C2312yes—
166#336—22no—
167#336—22no—
168#330C2×C418yes—
169#329C2×C418yes—
170#328C2×C418yes—
171#327C2×C418yes—
172#322—22no—
173#322—22no—
174#322—22no—
175C82—22no—
176C22×Q8C4211yescharacteristic
177C22×D4—21no—
178C22×D4C4211yescharacteristic
179C23×C4—21no—
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D411yescharacteristic
182C23×C4#4111yescharacteristic
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—44no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—22no—
191C22×C8—22no—
192C22×C8—22no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—44no—
196C22×C8C22⋊C411yescharacteristic
197C22×C8C22⋊C411yescharacteristic
198C22×C8C22⋊C411yescharacteristic
199C22×C8C22⋊C411yescharacteristic
200C22×C8C22⋊C411yescharacteristic
201C22×C8C22⋊C411yescharacteristic
202C22×C8C22⋊C411yescharacteristic
203C22×C8C22⋊C411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C22×C8C2×D411yescharacteristic
207C22×C8C2×D411yescharacteristic
208#126—22no—
209#126—22no—
210#126—22no—
211#126—22no—
212#125—21no—
213#125—21no—
214#125C4212yes—
215#125C22×C412yes—
216#124—21no—
217#124—21no—
218#124C4212yes—
219#124C22×C412yes—
220#123—22no—
221#123—22no—
222#123—22no—
223#123—22no—
224C22⋊Q8—21no—
225C22⋊Q8—21no—
226C22⋊Q8C4212yes—
227C22⋊Q8C22×C412yes—
228#121—21no—
229#121—21no—
230#121C4212yes—
231#121C22×C412yes—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8C4212yes—
235C4×Q8C22×C412yes—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—22no—
239C4×D4—22no—
240C4×D4—21no—
241C4×D4—21no—
242C4×D4C4212yes—
243C4×D4C22×C412yes—
244#117—22no—
245#117—22no—
246#117—22no—
247#117—22no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4C4211yescharacteristic
254C2×C4⋊C4C4211yescharacteristic
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C42—21no—
262C2×C42—22no—
263C2×C42C22×C411yescharacteristic
264C2×C42C22×C411yescharacteristic
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42#4111yescharacteristic
272C2×C42#4111yescharacteristic
273C2×C42#4111yescharacteristic
274C2×C42C2411yescharacteristic
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8—22no—
286C4⋊C8—22no—
287C4⋊C8C22×C412yes—
288C4⋊C8C22×C412yes—
289C22⋊C8—22no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—22no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8C22×C414yes—
301#97—22no—
302#97—22no—
303#97—22no—
304#97—22no—
305#97—22no—
306#97—22no—
307#97—44no—
308#97C22×C414yes—
309#97C22×C414yes—
310C4×C8—22no—
311C4×C8—22no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—44no—
315C4×C8C22⋊C411yescharacteristic
316C4×C8C22⋊C411yescharacteristic
317C4×C8C22⋊C411yescharacteristic
318C4×C8C22⋊C411yescharacteristic
319C4×C8C22⋊C411yescharacteristic
320C4×C8C22⋊C411yescharacteristic
321C4×C8C22⋊C411yescharacteristic
322C4×C8C22⋊C411yescharacteristic
323C4×C8C22×C411yescharacteristic
324C4×C8C22×C411yescharacteristic
325C4×C8C22×C411yescharacteristic
326C4×C8C22×C411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331#95—22no—
332#95—22no—
333#95—22no—
334#95C22×C414yes—
335C24—21no—
336C24C4×D411yescharacteristic
337C2×Q8—21no—
338C2×Q8—42no—
339C2×Q8C2×C4212yes—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—42no—
344C2×D4—44no—
345C2×D4C2×C4212yes—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4C2×C4211yescharacteristic
359C22×C4C2×C4211yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4#11711yescharacteristic
363C22×C4#11711yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4C4×D411yescharacteristic
369C22×C4C4×D411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C23×C411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C2×#4111yescharacteristic
376M16—44no—
377M16—44no—
378M16—44no—
379M16—44no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—44no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—44no—
403C2×C8C2×C22⋊C411yescharacteristic
404C2×C8C2×C22⋊C411yescharacteristic
405C2×C8C2×C22⋊C411yescharacteristic
406C2×C8C2×C22⋊C411yescharacteristic
407C2×C8C2×C22⋊C411yescharacteristic
408C2×C8C2×C22⋊C411yescharacteristic
409C2×C8C2×C22⋊C411yescharacteristic
410C2×C8C2×C22⋊C411yescharacteristic
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—44no—
422C4⋊C4C2×C4212yes—
423C4⋊C4C2×C4212yes—
424C22⋊C4—22no—
425C22⋊C4—22no—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—44no—
435C42—22no—
436C42—22no—
437C42—42no—
438C42—42no—
439C42C2×C4211yescharacteristic
440C42C2×C4211yescharacteristic
441C42C2×C22⋊C411yescharacteristic
442C42C2×C22⋊C411yescharacteristic
443C42#11711yescharacteristic
444C42#11711yescharacteristic
445C42C4×D411yescharacteristic
446C42C4×D411yescharacteristic
447C42C4×D411yescharacteristic
448C42C4×D411yescharacteristic
449C42C4×D411yescharacteristic
450C42C4×D411yescharacteristic
451C42C23×C411yescharacteristic
452C42C23×C411yescharacteristic
453C42C22×D411yescharacteristic
454C42C2×#4111yescharacteristic
455C23—22no—
456C23—22no—
457C23—22no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23C4×C22⋊C411yescharacteristic
463C23C4×C22⋊C411yescharacteristic
464C23C2×C4×D411yescharacteristic
465Q8—42no—
466Q8—42no—
467D4—42no—
468D4—42no—
469D4—44no—
470D4—44no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
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—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4C4×C22⋊C411yescharacteristic
505C2×C4C4×C22⋊C411yescharacteristic
506C2×C4C4×C22⋊C411yescharacteristic
507C2×C4C4×C22⋊C411yescharacteristic
508C2×C4C4×C22⋊C411yescharacteristic
509C2×C4C4×C22⋊C411yescharacteristic
510C2×C4C22×C4211yescharacteristic
511C2×C4C22×C22⋊C411yescharacteristic
512C2×C4C2×#11711yescharacteristic
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C2×C4×D411yescharacteristic
515C2×C4C2×C4×D411yescharacteristic
516C8—21no—
517C8—21no—
518C8—21no—
519C8—21no—
520C8—21no—
521C8—21no—
522C8—44no—
523C8—44no—
524C8—44no—
525C8—42no—
526C8—42no—
527C8#42112yes—
528C8#42112yes—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—44no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—84no—
541C22#175712yes—
542C22C2×C4×C22⋊C411yescharacteristic
543C22#295512yes—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—84no—
559C4C2×#42111yescharacteristic
560C4C2×#42111yescharacteristic
561C2—22no—
562C2—22no—
563C2—41no—
564C2—82no—
565C2C2×#175711yescharacteristic
566C2C4×#42111yescharacteristic
567C2#2042011yescharacteristic
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 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20476C2semidirect
5. #17323C2semidirect
6. #8250C2semidirect
7. #8249C2semidirect
8. #8085C2semidirect
9. #8069C2semidirect
15. #2966C4semidirect
16. #2966C4semidirect
19. C2×#446C4semidirect
20. C2×C8×Q8C4semidirect
21. C2×#436C4semidirect
24. C2×C8×D4C4semidirect
37. #1746C22semidirect
38. #1726C22semidirect
39. #1535C4semidirect
40. #1534C4semidirect
41. #1533C4semidirect
49. #1451C22semidirect
52. C8×M16C22semidirect
53. #1302C22semidirect
54. #1300C22semidirect

Maximal subgroups

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

subgroupindex
C2×#29662
#204762
#176592
#175542
#173232
#82502
#82492
#80852
#80692