Finite Group Databasebeta

← #6850719

Subgroups — #6850719

Order 512.

1535 conjugacy classes of subgroups.

The subgroup lattice has 601 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36209C211yesmaximal, characteristic, complemented
2#36164C211yesmaximal, characteristic
3#33296C211yesmaximal, characteristic, complemented
4#32060C214yesmaximal, complemented
5#32058C214yesmaximal, complemented
6#29815C212yesmaximal, complemented
7#29146C211yesmaximal, characteristic, complemented
8#28522C211yesmaximal, characteristic, complemented
9#27983C212yesmaximal, complemented
10#27894C212yesmaximal, complemented
11#24699C212yesmaximal, complemented
12#24695C212yesmaximal, complemented
13#24134C212yesmaximal, complemented
14#24132C212yesmaximal
15#21986C212yesmaximal, complemented
16#21702C211yesmaximal, characteristic, complemented
17#21271C211yesmaximal, characteristic, complemented
18C2×#554C2211yescharacteristic
19C2×#547C2211yescharacteristic
20C2×#497C2211yescharacteristic
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×#441C2212yes—
24C2×#440C2212yes—
25C2×#432C2212yes—
26#2862C2211yescharacteristic
27#2859C2211yescharacteristic
28#2857C2212yes—
29#2850C2212yes—
30#2568C2211yescharacteristic
31#2563C2211yescharacteristic
32#2536C2212yes—
33#2508—22no—
34#2504C2212yes—
35#2503C2212yes—
36#2495—22no—
37#2489—22no—
38#2489C2214yes—
39#2488—22no—
40#2488—22no—
41#2485—22no—
42#2356C2212yes—
43#2346—22no—
44#2339—22no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×#124C2211yescharacteristic
47#2285C2212yes—
48C2×#382C2212yes—
49C42×D4C2211yescharacteristic
50#2097C2214yes—
51#2074C2214yes—
52#2074C2214yes—
53#2056C2214yes—
54#2055C2214yes—
55#2035C2214yes—
56#2033C2214yes—
57#2026C2212yes—
58#2025C2212yes—
59#2001C2214yes—
60#2000C2214yes—
61#1993C2212yes—
62#1993C2212yes—
63#1991C2214yes—
64#1969C2212yes—
65#1968C2212yes—
66#1960C2211yescharacteristic
67#1960C2212yes—
68#1959C2212yes—
69#1959C2211yescharacteristic
70#1955C2212yes—
71#1949C2214yes—
72#1949C2214yes—
73#1940C2212yes—
74#1928C2214yes—
75#1926C2214yes—
76#1923C2214yes—
77#1872C2214yes—
78#1872C2214yes—
79#1853C2214yes—
80#1852C2214yes—
81#1847C2212yes—
82#1847C2211yescharacteristic
83#1836C2212yes—
84#1836C2212yes—
85#1836C2212yes—
86#1805C2212yes—
87#1804C2212yes—
88#1767C2211yescharacteristic
89#1749C2212yes—
90C4×#97C2211yescharacteristic
91C22×#41—21no—
92C22×Q16C2312yes—
93C22×SD16C2312yes—
94#554—44no—
95#547—44no—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×#125—21no—
100C2×#125C2312yes—
101C2×#124C2311yescharacteristic
102C2×#123—22no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—22no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×#121C2312yes—
108C2×C4×Q8C2312yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×D4—21no—
111C2×C4×D4D412yes—
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×#117—22no—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C42C2312yes—
118#497—24no—
119#477—44no—
120#475—44no—
121#441—44no—
122#440—44no—
123#432—28no—
124C2×#107C2314yes—
125C2×#107C2312yes—
126C2×#106C2314yes—
127C2×#106C2312yes—
128C2×C4⋊C8C2312yes—
129C2×C4⋊C8C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×#103C2312yes—
135C2×#103C2312yes—
136C2×#102C2314yes—
137C2×#102C2312yes—
138C2×C22⋊C8C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2311yescharacteristic
141C2×#97C2312yes—
142C2×#97C2312yes—
143C2×#97C2311yescharacteristic
144C2×C4×C8C2312yes—
145#400—21no—
146#400—21no—
147#400—22no—
148#400—22no—
149#400—24no—
150#398—22no—
151#397—24no—
152#397—22no—
153#397—22no—
154#394—22no—
155#393—21no—
156#393—21no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391C2312yes—
161#391C2314yes—
162#391C2312yes—
163#391C2311yescharacteristic
164#390C2312yes—
165#390C2311yescharacteristic
166#389—22no—
167#389—22no—
168#389—22no—
169#389—22no—
170#389—22no—
171#388—24no—
172#388—24no—
173#388—22no—
174#388—22no—
175#388—22no—
176#387—24no—
177#386—22no—
178#386—22no—
179#386—22no—
180#386—22no—
181#384C2311yescharacteristic
182#384C2312yes—
183#383—22no—
184#382—22no—
185#382—22no—
186#382—22no—
187#382—24no—
188#382C2312yes—
189#382C2312yes—
190#382C2312yes—
191#382C2312yes—
192#382C2311yescharacteristic
193#382C2312yes—
194#381—22no—
195#381—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380C2314yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D412yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C22⋊C4—22no—
207C4×C22⋊C4D412yes—
208C4×C22⋊C4D412yes—
209C4×C22⋊C4C2312yes—
210#376—22no—
211C2×#95—22no—
212C2×#95—22no—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#336C2314yes—
221#336C2312yes—
222C2×#41—44no—
223C2×#41—48no—
224C22×Q8—21no—
225C22×Q8—21no—
226C22×Q8C2411yescharacteristic
227C22×Q8C2412yes—
228C22×D4—22no—
229C22×D4C2×D411yescharacteristic
230C22×D4C2411yescharacteristic
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—21no—
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2412yes—
237C2×Q16—24no—
238C2×Q16—42no—
239C2×Q16—42no—
240C2×SD16—24no—
241C2×SD16—42no—
242C2×SD16—42no—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C4⋊Q8—22no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—22no—
250#125—44no—
251#125—44no—
252#124—22no—
253#124—22no—
254#123—48no—
255C22⋊Q8—48no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C22⋊Q8—44no—
259C22⋊Q8—44no—
260#121—44no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×Q8—24no—
264C4×D4—24no—
265C4×D4—24no—
266C4×D4—44no—
267C4×D4—44no—
268#117—48no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D412yes—
286C2×C4⋊C4C2411yescharacteristic
287C2×C4⋊C4C2411yescharacteristic
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2414yes—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42C2×D412yes—
310C2×C42C2×D412yes—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2411yescharacteristic
314C2×C42C2412yes—
315C2×C42C2411yescharacteristic
316C2×C42C2411yescharacteristic
317#107—42no—
318#107—44no—
319#106—42no—
320#106—44no—
321C4⋊C8—28no—
322C4⋊C8—24no—
323#103—44no—
324#103—42no—
325#103—42no—
326#103—42no—
327#103—44no—
328#103—44no—
329#102—44no—
330#102—42no—
331C22⋊C8—28no—
332C22⋊C8—44no—
333#97—24no—
334#97—24no—
335#97—22no—
336#97#4114yescomplemented
337C4×C8—42no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95C2×D411yescharacteristic
358#95C2×D411yescharacteristic
359#95C2×D412yes—
360#95C2×D412yes—
361C24—21no—
362C24C22×D412yes—
363#41—816no—
364C2×Q8—21no—
365C2×Q8—22no—
366C2×Q8—22no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8—22no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—44no—
375C2×Q8—42no—
376C2×Q8—42no—
377C2×Q8—82no—
378C2×D4—24no—
379C2×D4—22no—
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—42no—
411C22×C4—82no—
412C22×C4—82no—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C2511yescharacteristic
421Q16—48no—
422SD16—48no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8C2×#4114yes—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—42no—
442C4⋊C4—44no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—84no—
454C4⋊C4—84no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—42no—
458C22⋊C4—44no—
459C22⋊C4—44no—
460C22⋊C4—48no—
461C22⋊C4—42no—
462C22⋊C4—44no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—84no—
467C42—24no—
468C42—24no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—42no—
473C42—42no—
474C42—48no—
475C42—42no—
476C42—44no—
477C42C2×#4112yes—
478C42H(5,𝔽2)14yes—
479C422−1+412yes—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—42no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23#57514yes—
492C23C23×D411yescharacteristic
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498Q8—42no—
499Q8—84no—
500Q8—162no—
501D4—28no—
502D4—42no—
503D4—42no—
504D4—88no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—48no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—41no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—84no—
546C2×C4—84no—
547C2×C4—88no—
548C2×C4—164no—
549C2×C4#53512yes—
550C2×C4#53512yes—
551C2×C4#54714yes—
552C2×C4C22×#4111yescharacteristic
553C2×C4C2×H(5,𝔽2)12yes—
554C2×C4C2×2−1+411yescharacteristic
555C2×C4#58512yes—
556C8—24no—
557C8—44no—
558C8—42no—
559C22—24no—
560C22—24no—
561C22—24no—
562C22—42no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—81no—
567C22—82no—
568C22—82no—
569C22—81no—
570C22—81no—
571C22C2×#53511yescharacteristic
572C22C2×#54711yescharacteristic
573C22#350011yescharacteristic
574C22C2×#57512yes—
575C22#358612yes—
576C4—24no—
577C4—22no—
578C4—22no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—81no—
584C4—82no—
585C4—84no—
586C4—82no—
587C4—81no—
588C4—82no—
589C4—81no—
590C4—82no—
591C4#350612yes—
592C4#354712yes—
593C2—44no—
594C2—161no—
595C2#3949111yescharacteristic
596C2#6079911yescharacteristic
597C2#6082611yescharacteristic
598C2#6125312yes—
599C2#6257912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#547C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36209C2semidirect
3. #33296C2semidirect
4. #32060C2semidirect
5. #32058C2semidirect
6. #29815C2semidirect
7. #29146C2semidirect
8. #28522C2semidirect
9. #27983C2semidirect
10. #27894C2semidirect
11. #24699C2semidirect
12. #24695C2semidirect
13. #24134C2semidirect
15. #21986C2semidirect
16. #21702C2semidirect
17. #21271C2semidirect
336. #97#41semidirect

Maximal subgroups

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

subgroupindex
#362092
#361642
#332962
#320602
#320582
#298152
#291462
#285222
#279832
#278942
#246992
#246952
#241342
#241322
#219862
#217022
#212712