Finite Group Databasebeta

← #203001

Subgroups — #203001

Order 512.

678 conjugacy classes of subgroups.

The subgroup lattice has 577 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21418C211yesmaximal, characteristic
2#20470C211yesmaximal, characteristic, complemented
3#17803C211yesmaximal, characteristic, complemented
4#17713C211yesmaximal, characteristic
5#17458C211yesmaximal, characteristic
6#9584C211yesmaximal, characteristic, complemented
7#9583C211yesmaximal, characteristic, complemented
8#9582C211yesmaximal, characteristic, complemented
9#9581C211yesmaximal, characteristic, complemented
10#9580C211yesmaximal, characteristic, complemented
11#9579C211yesmaximal, characteristic, complemented
12#9578C211yesmaximal, characteristic, complemented
13#9577C211yesmaximal, characteristic, complemented
14#8080C211yesmaximal, characteristic, complemented
15#7866C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1994C2211yescharacteristic
24#1994C2211yescharacteristic
25#1962C2211yescharacteristic
26#1962C2211yescharacteristic
27#1897—21no—
28#1897—21no—
29#1767C2211yescharacteristic
30#1767C2211yescharacteristic
31C4×C4⋊C8C2211yescharacteristic
32C4×C4⋊C8C2211yescharacteristic
33#1759—21no—
34#1759—21no—
35#1750—21no—
36#1750—21no—
37#1746—21no—
38#1726—21no—
39#1572—22no—
40#1571C2211yescharacteristic
41#1571C2211yescharacteristic
42#1570C2211yescharacteristic
43#1570C2211yescharacteristic
44#1564—22no—
45#1554C412yescomplemented
46#1553C412yescomplemented
47#1552C412yescomplemented
48#1551C412yescomplemented
49#1542C2211yescharacteristic
50#1542C2211yescharacteristic
51#1540C2211yescharacteristic
52#1540C2211yescharacteristic
53#1482—22no—
54#1481C2211yescharacteristic
55#1481C2211yescharacteristic
56#1480C2211yescharacteristic
57#1480C2211yescharacteristic
58#1476—22no—
59#1301C2211yescharacteristic
60#1301C2211yescharacteristic
61#1301C2211yescharacteristic
62#1301C2211yescharacteristic
63#1276C2211yescharacteristic
64#1276C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67#533C2×C411yescharacteristic
68#533C2×C411yescharacteristic
69#533C2311yescharacteristic
70#532C2×C411yescharacteristic
71#532C2×C411yescharacteristic
72#532C2311yescharacteristic
73#529—21no—
74#529—21no—
75#529—21no—
76#529—21no—
77#529—21no—
78#529—21no—
79#525—21no—
80#525—21no—
81#525—21no—
82C4×#41—21no—
83C4×#41—21no—
84C4×#41—21no—
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C22×C42C2311yescharacteristic
88#432—21no—
89#432—21no—
90#432—21no—
91#432—21no—
92#432C2×C412yes—
93#432C2×C412yes—
94#431—21no—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431—21no—
100#431—21no—
101#431—21no—
102#431—21no—
103#431C2×C412yes—
104#431D412yes—
105#431D412yes—
106#431C2311yescharacteristic
107#431C2311yescharacteristic
108#431C2311yescharacteristic
109#431C2311yescharacteristic
110#423C2311yescharacteristic
111#423C2311yescharacteristic
112C2×C4⋊C8—21no—
113C2×C4⋊C8C2×C412yes—
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×#104—21no—
119C2×#104—21no—
120C2×#104—21no—
121C2×#104—21no—
122#407—21no—
123C2×C22⋊C8—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8—21no—
127C4×M16—21no—
128C2×#97—21no—
129C2×#97—21no—
130C2×#97—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140#376D411yescharacteristic
141#376D411yescharacteristic
142C43C2311yescharacteristic
143C43C2311yescharacteristic
144#339—21no—
145#339—21no—
146#339—21no—
147#339—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#335—22no—
153#335—22no—
154#334—22no—
155#334—22no—
156#333C2×C412yes—
157#333C2×C412yes—
158#332C2×C412yes—
159#332C2×C412yes—
160#332C2×C412yes—
161#332C2×C412yes—
162#331C2×C412yes—
163#331C2×C412yes—
164Q8⋊C8—22no—
165Q8⋊C8—22no—
166D4⋊C8—22no—
167D4⋊C8—22no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—21no—
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173C23×C4#4111yescharacteristic
174C2×M16—21no—
175C2×M16—21no—
176C22×C8—22no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—41no—
182C4⋊Q8C22×C411yescharacteristic
183C4⋊Q8C22×C411yescharacteristic
184C4⋊Q8C22×C411yescharacteristic
185#127C22×C411yescharacteristic
186#126—41no—
187#126—41no—
188#125—21no—
189#125—21no—
190#125—21no—
191#124—21no—
192#124—21no—
193#124—21no—
194#123—41no—
195#123—41no—
196#123—41no—
197C22⋊Q8—21no—
198C22⋊Q8—21no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203#121—21no—
204#121—21no—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209#120—41no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×D4—21no—
217C4×D4—21no—
218C4×D4—21no—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—41no—
223C4×D4—41no—
224C4×D4—41no—
225C4×D4—41no—
226C4×D4—41no—
227C4×D4—41no—
228#117—22no—
229#117—22no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—41no—
234#117—41no—
235#117—41no—
236#117—41no—
237#117—41no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—41no—
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D411yescharacteristic
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42#4111yescharacteristic
257C2×C42#4111yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42#4111yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42C2411yescharacteristic
263C4⋊C8—22no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—41no—
275C4⋊C8C22×C412yes—
276C4⋊C8C22×C412yes—
277C4⋊C8C22×C412yes—
278C4⋊C8C22×C412yes—
279C4⋊C8C2×D412yes—
280C4⋊C8C2×D412yes—
281#104—42no—
282#104—42no—
283#104—42no—
284#104—42no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—21no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296C22⋊C8—41no—
297#97—22no—
298#97—22no—
299#97—42no—
300#97—41no—
301#97—42no—
302C4×C8—22no—
303C4×C8—22no—
304C4×C8—22no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—42no—
310C4×C8—41no—
311C4×C8—41no—
312C4×C8—42no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—21no—
317C24#12311yescharacteristic
318#41—44no—
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—82no—
324C2×Q8—21no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—42no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—81no—
346C22×C4#11711yescharacteristic
347C22×C4#11711yescharacteristic
348C22×C4C4×D411yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4#12111yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12311yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4#12311yescharacteristic
357C22×C4#12411yescharacteristic
358C22×C4#12411yescharacteristic
359C22×C4#12611yescharacteristic
360C22×C4#12611yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—44no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—82no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—42no—
420C42—42no—
421C42—41no—
422C42—41no—
423C42—41no—
424C42—81no—
425C42—81no—
426C42#11711yescharacteristic
427C42#11711yescharacteristic
428C42C4×D411yescharacteristic
429C42C4×D411yescharacteristic
430C42C4×D411yescharacteristic
431C42C4×D411yescharacteristic
432C42#12111yescharacteristic
433C42#12111yescharacteristic
434C42#12411yescharacteristic
435C42#12411yescharacteristic
436C42#12611yescharacteristic
437C42#12611yescharacteristic
438C42#13512yes—
439C42#13512yes—
440C42C23×C411yescharacteristic
441C42C22×D411yescharacteristic
442C42C2×#4111yescharacteristic
443C42C2×#4111yescharacteristic
444C23—21no—
445C23—21no—
446C23—41no—
447C23—41no—
448C23—41no—
449C23#38811yescharacteristic
450C23#38811yescharacteristic
451C23C2×#12311yescharacteristic
452Q8—41no—
453Q8—41no—
454Q8—42no—
455Q8—81no—
456D4—41no—
457D4—41no—
458D4—42no—
459D4—81no—
460D4—82no—
461D4—82no—
462D4—81no—
463D4—81no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—41no—
480C2×C4—41no—
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—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4#38811yescharacteristic
506C2×C4#38811yescharacteristic
507C2×C4#38811yescharacteristic
508C2×C4#38811yescharacteristic
509C2×C4#38811yescharacteristic
510C2×C4#38811yescharacteristic
511C2×C4C2×#11711yescharacteristic
512C2×C4C2×C4×D411yescharacteristic
513C2×C4C2×C4×D411yescharacteristic
514C2×C4C2×#12111yescharacteristic
515C2×C4C2×#12411yescharacteristic
516C2×C4C2×#12611yescharacteristic
517C2×C4C2×#13511yescharacteristic
518C2×C4C2×#13511yescharacteristic
519C2×C4#57511yescharacteristic
520C2×C4#57511yescharacteristic
521C8—42no—
522C8—42no—
523C8—42no—
524C8—42no—
525C8—42no—
526C8—42no—
527C8—81no—
528C8—81no—
529C8—81no—
530C8—81no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22C2×#38811yescharacteristic
545C22#309111yescharacteristic
546C22#309111yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—42no—
557C4—42no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C4C4×#13511yescharacteristic
564C4#294311yescharacteristic
565C4#295512yes—
566C4#303511yescharacteristic
567C4#303511yescharacteristic
568C2—21no—
569C2—21no—
570C2—41no—
571C2—41no—
572C2—161no—
573C2#2105811yescharacteristic
574C2#2108011yescharacteristic
575C2#2110011yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#388
prime radical#3444C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20470C2semidirect
3. #17803C2semidirect
6. #9584C2semidirect
7. #9583C2semidirect
8. #9582C2semidirect
9. #9581C2semidirect
10. #9580C2semidirect
11. #9579C2semidirect
12. #9578C2semidirect
13. #9577C2semidirect
14. #8080C2semidirect
15. #7866C2semidirect
45. #1554C4semidirect
46. #1553C4semidirect
47. #1552C4semidirect
48. #1551C4semidirect

Maximal subgroups

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

subgroupindex
#214182
#204702
#178032
#177132
#174582
#95842
#95832
#95822
#95812
#95802
#95792
#95782
#95772
#80802
#78662