Finite Group Databasebeta

← #205704

Subgroups — #205704

Order 512.

856 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3231C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic
3#17751C211yesmaximal, characteristic
4#17747C211yesmaximal, characteristic
5#17718C211yesmaximal, characteristic
6#17708C211yesmaximal, characteristic
7#17326C211yesmaximal, characteristic
8#9778C211yesmaximal, characteristic, complemented
9#9776C211yesmaximal, characteristic, complemented
10#9775C211yesmaximal, characteristic, complemented
11#9775C211yesmaximal, characteristic, complemented
12#9771C211yesmaximal, characteristic, complemented
13#9770C211yesmaximal, characteristic, complemented
14#9770C211yesmaximal, characteristic, complemented
15#9767C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#533C2211yescharacteristic
18#3231—21no—
19#3231—21no—
20#3231—21no—
21#3231—21no—
22#3231C414yes—
23#3231C414yes—
24C2×#500C2211yescharacteristic
25C2×#498C2211yescharacteristic
26C2×#487C2211yescharacteristic
27C2×#487C2211yescharacteristic
28C2×#485—22no—
29C2×#479—22no—
30#2920C2211yescharacteristic
31#2920C2211yescharacteristic
32C2×#431C2211yescharacteristic
33C2×#431C2211yescharacteristic
34C2×#423C2211yescharacteristic
35#1950—22no—
36#1855—22no—
37#1848C2211yescharacteristic
38#1848C2211yescharacteristic
39#1837C2211yescharacteristic
40#1837C2211yescharacteristic
41#1561—22no—
42#1559—22no—
43#1554—22no—
44#1554—22no—
45#1552—22no—
46#1550—22no—
47#1549C2211yescharacteristic
48#1549C2211yescharacteristic
49#1549C2211yescharacteristic
50#1549C2211yescharacteristic
51#1549C2211yescharacteristic
52#1549C2211yescharacteristic
53#1548C2211yescharacteristic
54#1548C2211yescharacteristic
55#1548C2211yescharacteristic
56#1548C2211yescharacteristic
57#1548C2211yescharacteristic
58#1548C2211yescharacteristic
59#1546C2211yescharacteristic
60#1546C2211yescharacteristic
61#1546C2211yescharacteristic
62#1546C2211yescharacteristic
63#1542—22no—
64#1542—22no—
65#1454—21no—
66#1454—21no—
67#1452C2211yescharacteristic
68#1452C2211yescharacteristic
69#1451—21no—
70#1450C2211yescharacteristic
71#1450C2211yescharacteristic
72C8×M16—21no—
73C22×M16—21no—
74#533—21no—
75#533—21no—
76#533—21no—
77#533—21no—
78#533—21no—
79#533—21no—
80#533—21no—
81#533—21no—
82#533—21no—
83#533—21no—
84#533—21no—
85#533—21no—
86#533C2×C412yes—
87#533C2×C412yes—
88#533C2×C412yes—
89#533C2×C412yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×C4⋊Q8C2311yescharacteristic
94C2×#125—21no—
95C2×#125—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8—21no—
103C2×C4×Q8—21no—
104C2×#117—21no—
105C2×#117—21no—
106C22×C42C2311yescharacteristic
107#500—21no—
108#500—21no—
109#500C2×C414yes—
110#498—21no—
111#498—21no—
112#498C2×C414yes—
113#487—21no—
114#487—21no—
115#487—21no—
116#487—21no—
117#487C2×C414yes—
118#487C2×C414yes—
119#485—24no—
120#485—24no—
121#485—24no—
122#485—24no—
123#479—24no—
124#479—24no—
125#479—24no—
126#479—24no—
127#432—21no—
128#432—21no—
129#432—21no—
130#432D412yes—
131#431—21no—
132#431—21no—
133#431—21no—
134#431—21no—
135#431—21no—
136#431C2×C412yes—
137#431C2×C412yes—
138#431D412yes—
139#431D411yescharacteristic
140#431D411yescharacteristic
141#431D411yescharacteristic
142#431D411yescharacteristic
143#431C2311yescharacteristic
144#431C2311yescharacteristic
145#431C2311yescharacteristic
146#431C2311yescharacteristic
147C2×#108—22no—
148C2×#108—22no—
149C2×#107—22no—
150C2×#106—22no—
151#423—21no—
152#423—21no—
153#423—21no—
154#423D412yes—
155#423C2311yescharacteristic
156#423C2311yescharacteristic
157#423C2311yescharacteristic
158#423C2311yescharacteristic
159C2×C4⋊C8—21no—
160C2×C4⋊C8—21no—
161C2×C4⋊C8D412yes—
162C2×C4⋊C8D412yes—
163C2×#103—22no—
164C2×#103—22no—
165#407—21no—
166#407—21no—
167C2×C22⋊C8—21no—
168C2×C22⋊C8D412yes—
169C4×M16—21no—
170C4×M16—21no—
171C2×C4×C8—21no—
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174#341—22no—
175#341—22no—
176#333—22no—
177#333—22no—
178#333—22no—
179#333—22no—
180#333—22no—
181#333—22no—
182#332—22no—
183#332—22no—
184#332—22no—
185#332—22no—
186#332—22no—
187#332—22no—
188#332—22no—
189#332—22no—
190#332—22no—
191#332—22no—
192#322—21no—
193#322—21no—
194#322—21no—
195#322—21no—
196#322—21no—
197#322—21no—
198C82—21no—
199C82—21no—
200C22×Q8—21no—
201C22×Q8—21no—
202C22×Q8—21no—
203C22×Q8—21no—
204C23×C4C2×D411yescharacteristic
205C23×C4#4111yescharacteristic
206C23×C4#4111yescharacteristic
207C2×M16—22no—
208C2×M16—21no—
209C2×M16—21no—
210C2×M16—21no—
211C2×M16—21no—
212C2×M16—41no—
213C2×M16—82no—
214C22×C8—22no—
215C22×C8—41no—
216C22×C8C2×D412yes—
217C22×C8C2×D412yes—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—21no—
221C4⋊Q8—21no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8C22×C412yes—
231C4⋊Q8C22×C412yes—
232C4⋊Q8C22×C412yes—
233C4⋊Q8C22×C412yes—
234#125—22no—
235#125—22no—
236#125—22no—
237#125—22no—
238#125—41no—
239#125—41no—
240#125—41no—
241#125—41no—
242C22⋊Q8—22no—
243C22⋊Q8—22no—
244C22⋊Q8—22no—
245C22⋊Q8—22no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8—22no—
250C22⋊Q8—41no—
251C22⋊Q8—41no—
252C22⋊Q8—41no—
253C22⋊Q8—41no—
254C22⋊Q8—41no—
255C22⋊Q8—41no—
256C22⋊Q8—41no—
257C22⋊Q8—41no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—22no—
265C4×Q8—22no—
266C4×Q8—41no—
267C4×Q8—41no—
268C4×Q8—41no—
269C4×Q8—41no—
270C4×Q8—41no—
271C4×Q8—41no—
272C4×Q8—41no—
273C4×Q8—41no—
274#117—22no—
275#117—22no—
276#117—22no—
277#117—22no—
278#117—41no—
279#117—41no—
280#117—41no—
281#117—41no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—41no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—41no—
293C2×C4⋊C4—41no—
294C2×C22⋊C4—41no—
295C2×C22⋊C4—41no—
296C2×C42—41no—
297C2×C42—41no—
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42#4111yescharacteristic
308C2×C42#4111yescharacteristic
309C2×C42C2411yescharacteristic
310#108—82no—
311#108—82no—
312#107—24no—
313#107—24no—
314#106—24no—
315#106—24no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—21no—
319C4⋊C8—21no—
320C4⋊C8—21no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—41no—
327C4⋊C8—41no—
328C4⋊C8C22⋊C412yes—
329C4⋊C8C22⋊C412yes—
330C4⋊C8C22⋊C412yes—
331C4⋊C8C22⋊C412yes—
332C4⋊C8C2×D412yes—
333C4⋊C8C2×D412yes—
334C4⋊C8C2×D412yes—
335C4⋊C8C2×D412yes—
336#103—24no—
337#103—24no—
338#103—24no—
339#103—24no—
340C22⋊C8—22no—
341C22⋊C8—22no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344C22⋊C8—21no—
345C22⋊C8—21no—
346C22⋊C8—41no—
347C22⋊C8—41no—
348C22⋊C8C22⋊C412yes—
349C22⋊C8C22⋊C412yes—
350C22⋊C8C22⋊C412yes—
351C22⋊C8C22⋊C412yes—
352#97—22no—
353#97—41no—
354C4×C8—22no—
355C4×C8—21no—
356C4×C8—21no—
357C4×C8—21no—
358C4×C8—21no—
359C4×C8—41no—
360C4×C8C22×C412yes—
361C4×C8C22×C412yes—
362C4×C8C2×D411yescharacteristic
363C4×C8C2×D411yescharacteristic
364C4×C8C2×D411yescharacteristic
365C4×C8C2×D411yescharacteristic
366C24#12411yescharacteristic
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—22no—
370C2×Q8—22no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—41no—
374C2×Q8—41no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—82no—
378C2×Q8—82no—
379C2×Q8—82no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C4×D411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4#12111yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12111yescharacteristic
397C22×C4#12111yescharacteristic
398C22×C4#12411yescharacteristic
399C22×C4#12411yescharacteristic
400C22×C4#12411yescharacteristic
401C22×C4#12711yescharacteristic
402C22×C4#12711yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C2×#4111yescharacteristic
405C22×C4C2×#4111yescharacteristic
406M16—44no—
407M16—42no—
408M16—42no—
409M16—42no—
410M16—42no—
411M16—82no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—82no—
425C2×C8—82no—
426C2×C8C2×C22⋊C412yes—
427C2×C8C2×C22⋊C412yes—
428C2×C8C2×C22⋊C412yes—
429C2×C8C2×C22⋊C412yes—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C42—42no—
459C42—42no—
460C42—42no—
461C42—42no—
462C42C2×C22⋊C411yescharacteristic
463C42C2×C22⋊C411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42C4×D411yescharacteristic
467C42C4×D411yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42#12111yescharacteristic
471C42#12111yescharacteristic
472C42#12711yescharacteristic
473C42#12711yescharacteristic
474C42C23×C411yescharacteristic
475C42C22×D411yescharacteristic
476C42C22×D411yescharacteristic
477C42C22×D411yescharacteristic
478C23—22no—
479C23—41no—
480C23—41no—
481C23#39011yescharacteristic
482C23#39011yescharacteristic
483C23C2×#12411yescharacteristic
484Q8—44no—
485Q8—82no—
486Q8—82no—
487Q8—82no—
488Q8—81no—
489Q8—81no—
490Q8—82no—
491Q8—82no—
492Q8—82no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#39011yescharacteristic
521C2×C4#39011yescharacteristic
522C2×C4#39011yescharacteristic
523C2×C4#39011yescharacteristic
524C2×C4#39011yescharacteristic
525C2×C4#39011yescharacteristic
526C2×C4C22×C22⋊C411yescharacteristic
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×#12111yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12711yescharacteristic
532C2×C4#57712yes—
533C2×C4#57812yes—
534C8—22no—
535C8—22no—
536C8—22no—
537C8—22no—
538C8—42no—
539C8—42no—
540C8—42no—
541C8—42no—
542C8—81no—
543C8—81no—
544C8—81no—
545C8—81no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—82no—
555C22C2×#39011yescharacteristic
556C22#313712yes—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4#289812yes—
570C4#306712yes—
571C4#306912yes—
572C4#315411yescharacteristic
573C4#315611yescharacteristic
574C2—22no—
575C2—41no—
576C2—41no—
577C2#2120511yescharacteristic
578C2#2124811yescharacteristic
579C2#2125011yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#124
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #9778C2semidirect
9. #9776C2semidirect
10. #9775C2semidirect
11. #9775C2semidirect
12. #9771C2semidirect
13. #9770C2semidirect
14. #9770C2semidirect
15. #9767C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32312
#219932
#177512
#177472
#177182
#177082
#173262
#97782
#97762
#97752
#97712
#97702
#97672