Finite Group Databasebeta

← #6489379

Subgroups — #6489379

Order 512.

1660 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35337C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#32965C212yesmaximal, complemented
4#32963C212yesmaximal, complemented
5#30088C214yesmaximal, complemented
6#28723C211yesmaximal, characteristic, complemented
7#28722C211yesmaximal, characteristic, complemented
8#28522C211yesmaximal, characteristic, complemented
9#28521C211yesmaximal, characteristic, complemented
10#28212C212yesmaximal, complemented
11#28211C212yesmaximal, complemented
12#27953C212yesmaximal, complemented
13#27593C214yesmaximal, complemented
14#25862C212yesmaximal
15#22226C212yesmaximal, complemented
16#22225C212yesmaximal, complemented
17#20783C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520C2212yes—
20C2×#491C2212yes—
21C2×#490C2212yes—
22C2×#477C2212yes—
23C2×#475C2212yes—
24C2×#476C2212yes—
25C2×#474C2212yes—
26#2862C2211yescharacteristic
27#2859C2211yescharacteristic
28#2851C2212yes—
29#2647C2212yes—
30#2628C2211yescharacteristic
31#2628C2211yescharacteristic
32#2589—22no—
33#2589—22no—
34#2588—22no—
35#2588—22no—
36#2361—22no—
37#2361—22no—
38#2354—22no—
39D4×C4⋊C4—22no—
40#2331C2211yescharacteristic
41#2331C2211yescharacteristic
42#2326C2212yes—
43#2326C2212yes—
44#2314C2212yes—
45#2311C2212yes—
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#127C2211yescharacteristic
49C4×#127C2211yescharacteristic
50C4×#125C2212yes—
51C2×#384C2211yescharacteristic
52C4×#117C2211yescharacteristic
53#2097C2214yes—
54#2092C2214yes—
55#2084C2214yes—
56#2083C2214yes—
57#2072C2214yes—
58#2071C2214yes—
59#2042C2214yes—
60#2042C2214yes—
61#1992C2212yes—
62#1992C2212yes—
63#1961C2212yes—
64#1960C2212yes—
65#1928C2214yes—
66#1927C2214yes—
67#1926C2214yes—
68#1925C2214yes—
69#1865C2214yes—
70#1864C2214yes—
71#1863C2214yes—
72#1862C2214yes—
73#1853C2214yes—
74#1852C2214yes—
75#1847C2212yes—
76#1847C2212yes—
77#1840C2212yes—
78#1832C2212yes—
79#1818C2212yes—
80#1805C2212yes—
81#1805C2212yes—
82#1804C2212yes—
83#1804C2212yes—
84#1798C2212yes—
85#1788C2214yes—
86#1788C2214yes—
87#1767C2211yescharacteristic
88#1767C2211yescharacteristic
89#1749C2212yes—
90#1745C2212yes—
91C22×#41C2312yes—
92#532—24no—
93#532—24no—
94C2×C4⋊Q8C2311yescharacteristic
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#125C2312yes—
99C2×#124C2312yes—
100C2×#121—22no—
101#520—44no—
102#520—44no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—22no—
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119#491—28no—
120#490—28no—
121#477—44no—
122#476—44no—
123#475—44no—
124#474—44no—
125C2×#107C2314yes—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#102C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×#97C2312yes—
142#400—21no—
143#400—21no—
144#400—22no—
145#400—22no—
146#399—21no—
147#399—21no—
148#398—22no—
149#398—22no—
150#395—21no—
151#395—21no—
152#393—21no—
153#393—21no—
154#391—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391C2312yes—
159#391C2312yes—
160#390—22no—
161#390—22no—
162#390C2312yes—
163#390C2312yes—
164#389—22no—
165#389—22no—
166#389—22no—
167#389—22no—
168#389C2312yes—
169#389C2312yes—
170#389C2312yes—
171#389C2312yes—
172#388—22no—
173#388—22no—
174#388—24no—
175#388—24no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#387—22no—
184#386—24no—
185#385—22no—
186#385—22no—
187#385—24no—
188#385—24no—
189#384—22no—
190#384C2311yescharacteristic
191#384C2311yescharacteristic
192#384C2312yes—
193#384C2312yes—
194#382C2312yes—
195#382C2312yes—
196#382C2311yescharacteristic
197#382C2311yescharacteristic
198#380—24no—
199#380C2312yes—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4D412yes—
206C4×C4⋊C4D412yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4D412yes—
211#376C2311yescharacteristic
212#376C2311yescharacteristic
213C43C2311yescharacteristic
214C43C2311yescharacteristic
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2314yes—
220#336C2312yes—
221#336C2312yes—
222#336C2314yes—
223#336C2314yes—
224C2×#41—24no—
225C2×#41—24no—
226C2×#41—24no—
227C2×#41—24no—
228C2×#41—42no—
229C2×#41—42no—
230C22×Q8—21no—
231C22×Q8—21no—
232C22×Q8C2412yes—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4C2412yes—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—22no—
240C23×C4C2×D412yes—
241C23×C4C2411yescharacteristic
242C22×C8C2414yes—
243C22×C8C2414yes—
244C4⋊Q8—22no—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8#4114yes—
248#127—22no—
249#127#4114yes—
250#125—24no—
251#125—24no—
252#124—24no—
253#124—24no—
254#121—216no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—42no—
258C4×Q8—42no—
259C4×D4—216no—
260C4×D4—24no—
261C4×D4—24no—
262C4×D4—42no—
263C4×D4—42no—
264C4×D4—48no—
265C4×D4—48no—
266C4×D4—84no—
267C4×D4—84no—
268#117—24no—
269#117—24no—
270#117—48no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—48no—
291C2×C4⋊C4—84no—
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2412yes—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4C2×D411yescharacteristic
309C2×C22⋊C4C2×D411yescharacteristic
310C2×C22⋊C4C2×D412yes—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2×D412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C2×C42C2411yescharacteristic
325C2×C42C2411yescharacteristic
326C2×C42C2411yescharacteristic
327C2×C42C2411yescharacteristic
328#107—44no—
329#107—44no—
330#106—44no—
331#106—44no—
332C4⋊C8—24no—
333C4⋊C8—24no—
334#103—28no—
335#103—28no—
336#103—44no—
337#103—44no—
338#102—28no—
339#102—28no—
340#102—44no—
341#102—44no—
342C22⋊C8—28no—
343#97—24no—
344#97—24no—
345#95—24no—
346#95—24no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—21no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367C24—22no—
368C24C22×D411yescharacteristic
369#41—48no—
370#41—48no—
371#41—48no—
372#41—48no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—82no—
381C2×Q8—82no—
382C2×Q8C2×#4114yes—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—22no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4C2×#4114yes—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—44no—
413C22×C4—44no—
414C22×C4—84no—
415C22×C4—84no—
416C22×C4—84no—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C42−1+414yes—
427C22×C4C2511yescharacteristic
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C4⋊C4—28no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C22⋊C4—216no—
465C22⋊C4—24no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—84no—
471C22⋊C4—84no—
472C22⋊C4—82no—
473C22⋊C4—82no—
474C42—24no—
475C42—21no—
476C42—21no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—42no—
481C42—42no—
482C42—42no—
483C42—42no—
484C42—44no—
485C42—44no—
486C42—82no—
487C42—82no—
488C42C2×#4112yes—
489C42C2×#4112yes—
490C42H(5,𝔽2)14yes—
491C23—22no—
492C23—22no—
493C23—21no—
494C23—21no—
495C23—24no—
496C23—44no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23#53612yes—
501C23C23×D411yescharacteristic
502Q8—24no—
503Q8—24no—
504Q8—42no—
505Q8—42no—
506Q8—162no—
507Q8—162no—
508D4—24no—
509D4—24no—
510D4—416no—
511D4—416no—
512D4—42no—
513D4—42no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—48no—
537C2×C4—416no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—164no—
557C2×C4—164no—
558C2×C4—164no—
559C2×C4#52514yes—
560C2×C4#53512yes—
561C2×C4#53512yes—
562C2×C4#53612yes—
563C2×C4C22×#4111yescharacteristic
564C2×C4C22×#4111yescharacteristic
565C2×C4C2×H(5,𝔽2)12yes—
566C2×C4C2×2−1+412yes—
567C8—44no—
568C8—44no—
569C22—22no—
570C22—24no—
571C22—42no—
572C22—41no—
573C22—41no—
574C22—48no—
575C22—44no—
576C22—82no—
577C22—82no—
578C22—84no—
579C22C2×#52511yescharacteristic
580C22C2×#53511yescharacteristic
581C22C2×#53611yescharacteristic
582C22#358512yes—
583C22#358612yes—
584C4—24no—
585C4—22no—
586C4—22no—
587C4—22no—
588C4—42no—
589C4—41no—
590C4—41no—
591C4—44no—
592C4—42no—
593C4—42no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4#352012yes—
601C4#352012yes—
602C2—42no—
603C2—84no—
604C2#3524011yescharacteristic
605C2#6056512yes—
606C2#6082711yescharacteristic
607C2#6082811yescharacteristic
608C2#6088612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35337C2semidirect
3. #32965C2semidirect
4. #32963C2semidirect
5. #30088C2semidirect
6. #28723C2semidirect
7. #28722C2semidirect
8. #28522C2semidirect
9. #28521C2semidirect
10. #28212C2semidirect
11. #28211C2semidirect
12. #27953C2semidirect
13. #27593C2semidirect
15. #22226C2semidirect
16. #22225C2semidirect
17. #20783C2semidirect

Maximal subgroups

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

subgroupindex
#353372
#342072
#329652
#329632
#300882
#287232
#287222
#285222
#285212
#282122
#282112
#279532
#275932
#258622
#222262
#222252
#207832