Finite Group Databasebeta

← #511574

Subgroups — #511574

Order 512.

745 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#32931C211yesmaximal, characteristic, complemented
2#32930C211yesmaximal, characteristic
3#13494C211yesmaximal, characteristic, complemented
4#13493C211yesmaximal, characteristic, complemented
5#13492C211yesmaximal, characteristic, complemented
6#13383C211yesmaximal, characteristic, complemented
7#13383C211yesmaximal, characteristic, complemented
8#13382C211yesmaximal, characteristic, complemented
9#13381C211yesmaximal, characteristic, complemented
10#13380C211yesmaximal, characteristic, complemented
11#13380C211yesmaximal, characteristic, complemented
12#13305C211yesmaximal, characteristic, complemented
13#13305C211yesmaximal, characteristic, complemented
14#13305C211yesmaximal, characteristic, complemented
15#12679C211yesmaximal, characteristic, complemented
16C24×C8C2211yescharacteristic
17C22×SD32C2211yescharacteristic, complemented
18C2×#467C2211yescharacteristic
19C2×#467C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×#466C2211yescharacteristic
22C2×#466C2211yescharacteristic
23C2×#466C2211yescharacteristic
24#2848—21no—
25#2847—21no—
26C2×#367C2211yescharacteristic, complemented
27C2×#366C2211yescharacteristic, complemented
28C2×#366C2211yescharacteristic, complemented
29C2×#358C2211yescharacteristic, complemented
30C2×#358C2211yescharacteristic, complemented
31C2×#358C2211yescharacteristic, complemented
32C2×#357C2211yescharacteristic, complemented
33C2×#357C2211yescharacteristic, complemented
34C2×#357C2211yescharacteristic, complemented
35C2×C22⋊C16C2211yescharacteristic
36C2×C22⋊C16C2211yescharacteristic
37C2×C22⋊C16C2211yescharacteristic
38C2×C22⋊C16C2211yescharacteristic
39C2×C22⋊C16C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41#1893—21no—
42#1893—21no—
43#1893—21no—
44#1892—21no—
45#1892—21no—
46#1892—21no—
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1808C2211yescharacteristic
50#1808C2211yescharacteristic
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1380C2211yescharacteristic, complemented
54#1380C2211yescharacteristic, complemented
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59C24×C4D411yescharacteristic
60C22×Q16C2311yescharacteristic
61C22×D8C2311yescharacteristic
62C23×C8—41no—
63C23×C8—41no—
64C23×C8—41no—
65C23×C8—41no—
66C23×C8—41no—
67C23×C8—41no—
68C23×C8D411yescharacteristic
69C23×C8D411yescharacteristic
70C23×C8D411yescharacteristic
71C23×C8C2311yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×C22⋊Q8—21no—
75C2×C22⋊Q8—21no—
76C2×C22⋊Q8—21no—
77C2×#121—21no—
78C2×#121—21no—
79C2×#121—21no—
80C2×#120—41no—
81C2×SD32—41no—
82C2×SD32—41no—
83C2×SD32—41no—
84C22×C16C2311yescharacteristic
85C22×C16C2311yescharacteristic
86C22×C16C2311yescharacteristic
87C22×C16C2311yescharacteristic
88#467—41no—
89#467—41no—
90#467—41no—
91#467—41no—
92#467—41no—
93#467—41no—
94#466—41no—
95#466—41no—
96#466—41no—
97#466—41no—
98#466—41no—
99#466—41no—
100C2×#107C2311yescharacteristic
101C2×#107C2311yescharacteristic
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112#386—41no—
113#386—41no—
114#386—41no—
115#385—41no—
116#385—41no—
117#385—41no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124#379—41no—
125#367—41no—
126#366—41no—
127#366—41no—
128#358—41no—
129#358—41no—
130#358—41no—
131#357—41no—
132#357—41no—
133#357—41no—
134C22⋊C16—41no—
135C22⋊C16—41no—
136C22⋊C16—41no—
137C22⋊C16—41no—
138C22⋊C16—41no—
139C22⋊C16—41no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146C25D811yescharacteristic
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4—41no—
150C22×D4—41no—
151C22×D4—41no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4—41no—
158C23×C4D811yescharacteristic
159C23×C4D811yescharacteristic
160C23×C4D811yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×Q16—21no—
165C2×Q16—21no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—21no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—21no—
173C2×D8—21no—
174C2×D8—21no—
175C2×D8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—41no—
195C22×C8—41no—
196C22×C8—41no—
197C22×C8—41no—
198C22×C8—41no—
199C22×C8—41no—
200C22×C8—41no—
201C22×C8—41no—
202C22×C8—41no—
203C22×C8—44no—
204C22×C8—44no—
205C22×C8—44no—
206C22×C8—44no—
207C22×C8C2×D411yescharacteristic
208C22×C8C2×D411yescharacteristic
209C22×C8C2×D411yescharacteristic
210C22×C8C2411yescharacteristic
211C22⋊Q8—42no—
212C22⋊Q8—42no—
213C22⋊Q8—42no—
214C22⋊Q8—42no—
215C22⋊Q8—42no—
216C22⋊Q8—42no—
217#121—42no—
218#121—42no—
219#121—42no—
220#121—42no—
221#121—42no—
222#121—42no—
223#120—84no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C22⋊C4—41no—
238C2×C22⋊C4—41no—
239C2×C22⋊C4—41no—
240C2×C22⋊C4—41no—
241C2×C22⋊C4—41no—
242C2×C22⋊C4—41no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245SD32—44no—
246C2×C16—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251C2×C16—21no—
252C2×C16—21no—
253C2×C16—21no—
254C2×C16—21no—
255C2×C16—21no—
256C2×C16—21no—
257C2×C16—21no—
258#107—21no—
259#107—21no—
260#107—21no—
261#107—21no—
262#107—21no—
263#107—21no—
264#107—21no—
265#107—21no—
266#107—21no—
267#107—21no—
268#107—21no—
269#107—21no—
270#103—42no—
271#103—42no—
272#103—42no—
273#102—42no—
274#102—42no—
275#102—42no—
276#95—41no—
277#95—41no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—41no—
286C24—41no—
287C24—41no—
288C24—41no—
289C24—41no—
290C24C2×D811yescharacteristic
291C24C2×D811yescharacteristic
292C24C2×D811yescharacteristic
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—82no—
306C2×D4—82no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—82no—
310C2×D4—82no—
311C2×D4—82no—
312C2×D4—82no—
313C2×D4—82no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C2×D811yescharacteristic
353C22×C4C2×D811yescharacteristic
354C22×C4C2×D811yescharacteristic
355C22×C4C22×D411yescharacteristic
356Q16—41no—
357Q16—41no—
358Q16—41no—
359Q16—41no—
360D8—41no—
361D8—41no—
362D8—41no—
363D8—41no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—44no—
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—41no—
388C2×C8H(5,𝔽2)11yescharacteristic
389C2×C8H(5,𝔽2)11yescharacteristic
390C2×C8H(5,𝔽2)11yescharacteristic
391C2×C8H(5,𝔽2)11yescharacteristic
392C2×C8H(5,𝔽2)11yescharacteristic
393C2×C8H(5,𝔽2)11yescharacteristic
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—82no—
409C22⋊C4—82no—
410C22⋊C4—82no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C22⋊C4—82no—
416C22⋊C4—82no—
417C22⋊C4—82no—
418C22⋊C4—82no—
419C22⋊C4—82no—
420C22⋊C4—82no—
421C16—41no—
422C16—41no—
423C16—41no—
424C16—41no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—81no—
451C23—81no—
452C23—81no—
453C23—81no—
454C23—81no—
455C23—81no—
456C23—81no—
457C23#50812yes—
458C23#50912yes—
459C23#51012yes—
460C23C22×D811yescharacteristic
461Q8—81no—
462Q8—81no—
463Q8—81no—
464Q8—81no—
465D4—81no—
466D4—81no—
467D4—81no—
468D4—81no—
469D4—164no—
470D4—164no—
471D4—164no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
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—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4#53411yescharacteristic
518C2×C4#53411yescharacteristic
519C2×C4#53411yescharacteristic
520C2×C4#57511yescharacteristic
521C2×C4#57511yescharacteristic
522C2×C4#57511yescharacteristic
523C8—41no—
524C8—41no—
525C8—41no—
526C8#56011yescharacteristic
527C8#56011yescharacteristic
528C8#56011yescharacteristic
529C8H(3,𝔽4)11yescharacteristic
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—416no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—41no—
552C22—161no—
553C22—161no—
554C22—161no—
555C22—161no—
556C22—161no—
557C22—161no—
558C22—161no—
559C22#318611yescharacteristic
560C22#318611yescharacteristic
561C22#318611yescharacteristic
562C22C2×#50811yescharacteristic
563C22C2×#50911yescharacteristic
564C22C2×#51011yescharacteristic
565C22#341411yescharacteristic
566C4—41no—
567C4—41no—
568C4—41no—
569C4—161no—
570C4—161no—
571C4—161no—
572C4—161no—
573C4—161no—
574C4—161no—
575C4—161no—
576C4#281211yescharacteristic
577C4#319711yescharacteristic
578C4#319711yescharacteristic
579C4#319711yescharacteristic
580C2—42no—
581C2—42no—
582C2—42no—
583C2—321no—
584C2#3270611yescharacteristic
585C2#3375411yescharacteristic
586C2#3375611yescharacteristic
587C2#3375711yescharacteristic
588C2#3375911yescharacteristic
589C2#3376011yescharacteristic
590C2#3376211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radical#32930C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #32931C2semidirect
3. #13494C2semidirect
4. #13493C2semidirect
5. #13492C2semidirect
6. #13383C2semidirect
7. #13383C2semidirect
8. #13382C2semidirect
9. #13381C2semidirect
10. #13380C2semidirect
11. #13380C2semidirect
12. #13305C2semidirect
13. #13305C2semidirect
14. #13305C2semidirect
15. #12679C2semidirect
17. C22×SD32C22semidirect
26. C2×#367C22semidirect
27. C2×#366C22semidirect
28. C2×#366C22semidirect
29. C2×#358C22semidirect
30. C2×#358C22semidirect
31. C2×#358C22semidirect
32. C2×#357C22semidirect
33. C2×#357C22semidirect
34. C2×#357C22semidirect
53. #1380C22semidirect
54. #1380C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect

Maximal subgroups

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

subgroupindex
#329312
#329302
#134942
#134932
#134922
#133832
#133822
#133812
#133802
#133052
#126792