Finite Group Databasebeta

← #203033

Subgroups — #203033

Order 512.

676 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21856C211yesmaximal, characteristic
2#20549C211yesmaximal, characteristic, complemented
3#17822C211yesmaximal, characteristic, complemented
4#17737C211yesmaximal, characteristic
5#17488C211yesmaximal, 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#8079C211yesmaximal, characteristic, complemented
15#7870C211yesmaximal, characteristic, complemented
16#3444C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1994C2211yescharacteristic
24#1994C2211yescharacteristic
25#1962C2211yescharacteristic
26#1962C2211yescharacteristic
27#1871—21no—
28#1871—21no—
29#1806—21no—
30#1806—21no—
31#1771—21no—
32#1771—21no—
33#1767C2211yescharacteristic
34#1767C2211yescharacteristic
35C4×C4⋊C8C2211yescharacteristic
36C4×C4⋊C8C2211yescharacteristic
37#1753—21no—
38#1753—21no—
39#1573—22no—
40#1571C2211yescharacteristic
41#1571C2211yescharacteristic
42#1570C2211yescharacteristic
43#1570C2211yescharacteristic
44#1569—22no—
45#1561C412yescomplemented
46#1560C412yescomplemented
47#1559C412yescomplemented
48#1557C412yescomplemented
49#1542C2211yescharacteristic
50#1542C2211yescharacteristic
51#1540C2211yescharacteristic
52#1540C2211yescharacteristic
53#1494—22no—
54#1488—22no—
55#1481C2211yescharacteristic
56#1481C2211yescharacteristic
57#1480C2211yescharacteristic
58#1480C2211yescharacteristic
59#1301C2211yescharacteristic
60#1301C2211yescharacteristic
61#1301C2211yescharacteristic
62#1301C2211yescharacteristic
63#1276C2211yescharacteristic
64#1276C2211yescharacteristic
65#1276C2211yescharacteristic
66#1276C2211yescharacteristic
67C22×M16—21no—
68C22×M16—21no—
69#533C2×C411yescharacteristic
70#533C2×C411yescharacteristic
71#533C2311yescharacteristic
72#532C2×C411yescharacteristic
73#532C2×C411yescharacteristic
74#532C2311yescharacteristic
75#529—21no—
76#529—21no—
77#529—21no—
78#529—21no—
79#529—21no—
80#529—21no—
81#525—21no—
82#525—21no—
83#525—21no—
84C4×#41—21no—
85C4×#41—21no—
86C4×#41—21no—
87C2×#117—21no—
88C22×C42C2311yescharacteristic
89#432—21no—
90#432—21no—
91#432—21no—
92#432C2×C412yes—
93#432D412yes—
94#432D412yes—
95#431—21no—
96#431—21no—
97#431—21no—
98#431—21no—
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103#423—21no—
104#423—21no—
105#423—21no—
106#423—21no—
107#423—21no—
108#423C2×C412yes—
109#423C2×C412yes—
110#423C2×C412yes—
111#423C2311yescharacteristic
112#423C2311yescharacteristic
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×#104—21no—
122C2×#104—21no—
123C2×#104—21no—
124C2×#104—21no—
125#407—21no—
126C2×C22⋊C8—21no—
127C2×C22⋊C8—21no—
128C4×M16—21no—
129C2×#97—21no—
130C2×#97—21no—
131C2×#97—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4D411yescharacteristic
137C4×C4⋊C4D411yescharacteristic
138C4×C22⋊C4—21no—
139C4×C22⋊C4D411yescharacteristic
140C4×C22⋊C4D411yescharacteristic
141#376—21no—
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—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—41no—
180C2×M16—81no—
181C2×M16—81no—
182C2×M16—81no—
183C2×M16—81no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—41no—
189C22×C8—41no—
190C4⋊Q8C22×C411yescharacteristic
191C4⋊Q8C22×C411yescharacteristic
192C4⋊Q8C22×C411yescharacteristic
193#127C22×C411yescharacteristic
194#126—41no—
195#126—41no—
196#125—21no—
197#125—21no—
198#125—21no—
199#124—21no—
200#124—21no—
201#124—21no—
202#123—41no—
203#123—41no—
204#123—41no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8—21no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211#121—21no—
212#121—21no—
213#121—21no—
214#121—21no—
215#121—21no—
216#121—21no—
217#120—41no—
218C4×Q8—21no—
219C4×Q8—21no—
220C4×Q8—21no—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4—41no—
231C4×D4—41no—
232C4×D4—41no—
233C4×D4—41no—
234C4×D4—41no—
235C4×D4—41no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—44no—
240#117—41no—
241#117—41no—
242#117—41no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—41no—
252C2×C42C22×C411yescharacteristic
253C2×C42C22×C411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4111yescharacteristic
262C2×C42#4111yescharacteristic
263C2×C42#4111yescharacteristic
264C2×C42#4111yescharacteristic
265C2×C42#4111yescharacteristic
266C2×C42#4111yescharacteristic
267C2×C42C2411yescharacteristic
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—41no—
282C4⋊C8—41no—
283C4⋊C8—41no—
284C4⋊C8C22×C412yes—
285C4⋊C8C22×C412yes—
286C4⋊C8C22×C412yes—
287C4⋊C8C22×C412yes—
288C4⋊C8C2×D412yes—
289C4⋊C8C2×D412yes—
290#104—81no—
291#104—81no—
292#104—81no—
293#104—81no—
294C22⋊C8—22no—
295C22⋊C8—22no—
296C22⋊C8—22no—
297C22⋊C8—21no—
298C22⋊C8—21no—
299C22⋊C8—21no—
300C22⋊C8—21no—
301C22⋊C8—41no—
302#97—22no—
303#97—22no—
304#97—22no—
305#97—81no—
306#97—81no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—41no—
312C4×C8—81no—
313C4×C8—81no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318C24#12311yescharacteristic
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—82no—
324#41—82no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—81no—
347C22×C4#11711yescharacteristic
348C22×C4#11711yescharacteristic
349C22×C4C4×D411yescharacteristic
350C22×C4C4×D411yescharacteristic
351C22×C4C4×D411yescharacteristic
352C22×C4C4×D411yescharacteristic
353C22×C4#12111yescharacteristic
354C22×C4#12111yescharacteristic
355C22×C4#12311yescharacteristic
356C22×C4#12311yescharacteristic
357C22×C4#12311yescharacteristic
358C22×C4#12411yescharacteristic
359C22×C4#12411yescharacteristic
360C22×C4#12611yescharacteristic
361C22×C4#12611yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—82no—
370M16—82no—
371M16—82no—
372M16—82no—
373C2×C8—22no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
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—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—82no—
392C2×C8—82no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—41no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C42—24no—
419C42—21no—
420C42—21no—
421C42—21no—
422C42—21no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—42no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—81no—
432C42—81no—
433C42#11711yescharacteristic
434C42#11711yescharacteristic
435C42C4×D411yescharacteristic
436C42C4×D411yescharacteristic
437C42C4×D411yescharacteristic
438C42C4×D411yescharacteristic
439C42#12111yescharacteristic
440C42#12111yescharacteristic
441C42#12411yescharacteristic
442C42#12411yescharacteristic
443C42#12611yescharacteristic
444C42#12611yescharacteristic
445C42C23×C411yescharacteristic
446C42C22×D411yescharacteristic
447C42C2×#4111yescharacteristic
448C42C2×#4111yescharacteristic
449C23—21no—
450C23—21no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23#38811yescharacteristic
455C23#38811yescharacteristic
456C23C2×#12311yescharacteristic
457Q8—41no—
458Q8—41no—
459Q8—42no—
460Q8—81no—
461D4—41no—
462D4—41no—
463D4—42no—
464D4—81no—
465D4—82no—
466D4—82no—
467D4—81no—
468D4—81no—
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—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
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—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4#38811yescharacteristic
511C2×C4#38811yescharacteristic
512C2×C4#38811yescharacteristic
513C2×C4#38811yescharacteristic
514C2×C4#38811yescharacteristic
515C2×C4#38811yescharacteristic
516C2×C4C2×#11711yescharacteristic
517C2×C4C2×C4×D411yescharacteristic
518C2×C4C2×C4×D411yescharacteristic
519C2×C4C2×#12111yescharacteristic
520C2×C4C2×#12411yescharacteristic
521C2×C4C2×#12611yescharacteristic
522C2×C4#57611yescharacteristic
523C2×C4#57711yescharacteristic
524C2×C4#57711yescharacteristic
525C2×C4#57811yescharacteristic
526C8—42no—
527C8—42no—
528C8—42no—
529C8—42no—
530C8—42no—
531C8—42no—
532C8—81no—
533C8—81no—
534C8—81no—
535C8—81no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22C2×#38811yescharacteristic
550C22#309811yescharacteristic
551C22#309911yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—42no—
562C4—42no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4#294611yescharacteristic
569C4#294711yescharacteristic
570C4#295612yes—
571C4#304511yescharacteristic
572C4#304611yescharacteristic
573C2—21no—
574C2—21no—
575C2—41no—
576C2—41no—
577C2—161no—
578C2#2106211yescharacteristic
579C2#2108011yescharacteristic
580C2#2109911yescharacteristic
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. #20549C2semidirect
3. #17822C2semidirect
6. #9584C2semidirect
7. #9583C2semidirect
8. #9582C2semidirect
9. #9581C2semidirect
10. #9580C2semidirect
11. #9579C2semidirect
12. #9578C2semidirect
13. #9577C2semidirect
14. #8079C2semidirect
15. #7870C2semidirect
45. #1561C4semidirect
46. #1560C4semidirect
47. #1559C4semidirect
48. #1557C4semidirect

Maximal subgroups

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

subgroupindex
#218562
#205492
#178222
#177372
#174882
#95842
#95832
#95822
#95812
#95802
#95792
#95782
#95772
#80792
#78702