Finite Group Databasebeta

← #6594219

Subgroups — #6594219

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36191C211yesmaximal, characteristic
2#36181C211yesmaximal, characteristic
3#33296C211yesmaximal, characteristic, complemented
4#33059C211yesmaximal, characteristic, complemented
5#33059C211yesmaximal, characteristic, complemented
6#29227C211yesmaximal, characteristic, complemented
7#29147C211yesmaximal, characteristic, complemented
8#27983C211yesmaximal, characteristic, complemented
9#27980C211yesmaximal, characteristic, complemented
10#27967C211yesmaximal, characteristic, complemented
11#27965C211yesmaximal, characteristic, complemented
12#27732C214yesmaximal, complemented
13#27706C214yesmaximal, complemented
14#27705C214yesmaximal, complemented
15#22499C214yesmaximal
16C4×#491C211yesmaximal, characteristic, complemented
17#20765C211yesmaximal, characteristic, complemented
18#20764C211yesmaximal, characteristic, complemented
19#20494C211yesmaximal, characteristic
20C2×#552C2212yes—
21C2×#491C2211yescharacteristic
22C2×#436C2212yes—
23C2×C4×M16C2211yescharacteristic
24C2×C4×M16C2211yescharacteristic
25#2862C2211yescharacteristic
26#2859C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2851C2211yescharacteristic
30#2851C2211yescharacteristic
31#2574—22no—
32#2570C2212yes—
33#2565C2212yes—
34#2520—22no—
35#2506—22no—
36#2503C2212yes—
37#2499C2212yes—
38#2496—22no—
39#2486—22no—
40#2450—22no—
41#2440C2212yes—
42#2439C2212yes—
43#2368—22no—
44#2352C2212yes—
45#2344—22no—
46C4×C4⋊Q8C2211yescharacteristic
47C4×#125C2211yescharacteristic
48#2285C2211yescharacteristic
49#2285C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C42×D4C2211yescharacteristic
53#2101C2214yes—
54#2101C2214yes—
55#2099C2214yes—
56#2074C2214yes—
57#2072C2214yes—
58#2024C2214yes—
59#1993C2214yes—
60#1992C2211yescharacteristic
61#1992C2211yescharacteristic
62#1992C2211yescharacteristic
63#1991C2214yes—
64#1989C2214yes—
65#1960C2211yescharacteristic
66#1960C2211yescharacteristic
67#1960C2211yescharacteristic
68#1935C2214yes—
69#1933C2214yes—
70#1933C2214yes—
71#1894C2214yes—
72#1894C2214yes—
73#1873C2214yes—
74#1872C2214yes—
75#1836C2211yescharacteristic
76#1836C2211yescharacteristic
77#1836C2211yescharacteristic
78#1836C2211yescharacteristic
79#1836C2211yescharacteristic
80#1836C2211yescharacteristic
81#1836C2211yescharacteristic
82#1836C2211yescharacteristic
83#1836C2211yescharacteristic
84#1835C2214yes—
85#1834C2214yes—
86#1825C2214yes—
87#1824C2214yes—
88#1805C2214yes—
89C8×C4⋊C4C2212yes—
90#1767C2211yescharacteristic
91C4×#103C2214yes—
92#1753C2212yes—
93#1753C2212yes—
94#1749C2211yescharacteristic
95#1749C2211yescharacteristic
96#1736C2214yes—
97#1736C2214yes—
98C4×#97C2211yescharacteristic
99C22×M16C2312yes—
100C22×M16C2312yes—
101#552—216no—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2312yes—
104C2×#126—22no—
105C2×#125—22no—
106C2×#125C2311yescharacteristic
107C2×#125C2312yes—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×D4D412yes—
114C2×C4×D4C2311yescharacteristic
115C2×#117—22no—
116C2×#117C2312yes—
117C2×#117C2312yes—
118C22×C4⋊C4—22no—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#491—24no—
124#436—28no—
125C2×C4⋊C8C2312yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C4×M16—24no—
133C4×M16—24no—
134C2×#97C2311yescharacteristic
135C2×#97C2311yescharacteristic
136C2×#97C2311yescharacteristic
137C2×#97C2312yes—
138C2×#97C2311yescharacteristic
139C2×C4×C8C2312yes—
140C2×C4×C8C2312yes—
141C2×C4×C8C2312yes—
142#400—22no—
143#400—24no—
144#398—24no—
145#398—22no—
146#398—22no—
147#397—24no—
148#396—22no—
149#395—22no—
150#393—22no—
151#391—22no—
152#391—22no—
153#391—22no—
154#391C2312yes—
155#391C2312yes—
156#391C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#388—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—24no—
171#387—22no—
172#387—22no—
173#387—24no—
174#387—22no—
175#386—22no—
176#385—22no—
177#385—22no—
178#385—22no—
179#384—22no—
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#382—22no—
186#382C2311yescharacteristic
187#382C2312yes—
188#382C2312yes—
189#382C2312yes—
190#382C2311yescharacteristic
191#382C2311yescharacteristic
192#382C2312yes—
193#381—24no—
194#380—22no—
195#380—22no—
196#380—22no—
197#380—22no—
198#380C2312yes—
199#380C2312yes—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D412yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C22⋊C4—22no—
208C4×C22⋊C4D412yes—
209C4×C22⋊C4D412yes—
210C4×C22⋊C4C2311yescharacteristic
211C4×C22⋊C4C2311yescharacteristic
212C2×#95—22no—
213C43C2311yescharacteristic
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#336C2312yes—
227#336C2312yes—
228#336C2312yes—
229C22×Q8—22no—
230C22×Q8C2412yes—
231C22×D4C2×D411yescharacteristic
232C23×C4—22no—
233C23×C4—22no—
234C23×C4C2×D412yes—
235C23×C4C2×D412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C2×M16—24no—
239C2×M16—24no—
240C2×M16—24no—
241C2×M16—24no—
242C2×M16—42no—
243C2×M16—42no—
244C22×C8C2412yes—
245C22×C8C2412yes—
246C22×C8C2412yes—
247C22×C8C2412yes—
248C4⋊Q8—22no—
249C4⋊Q8—22no—
250C4⋊Q8—44no—
251#126—48no—
252#125—28no—
253#125—22no—
254#125—22no—
255#125—48no—
256#123—48no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—48no—
260C4×Q8—28no—
261C4×D4—44no—
262C4×D4#4118yes—
263#117—28no—
264#117—48no—
265#117—44no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
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—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2412yes—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4C2×D412yes—
304C2×C22⋊C4C2×D412yes—
305C2×C22⋊C4C2×D411yescharacteristic
306C2×C22⋊C4C2×D411yescharacteristic
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324C4⋊C8—24no—
325#103—28no—
326#103—44no—
327#103—44no—
328#103—44no—
329C22⋊C8—24no—
330C22⋊C8—24no—
331#97—22no—
332#97—22no—
333#97—22no—
334#97—42no—
335#97#4114yes—
336C4×C8—24no—
337C4×C8—24no—
338C4×C8—24no—
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—24no—
348#95—22no—
349#95—22no—
350#95—24no—
351#95—24no—
352#95—22no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95C2×D412yes—
358#95C2×D412yes—
359#95C2×D412yes—
360C24C22×D411yescharacteristic
361C24C22×D411yescharacteristic
362C2×Q8—22no—
363C2×Q8—24no—
364C2×Q8—22no—
365C2×Q8—24no—
366C2×Q8—44no—
367C2×Q8—82no—
368C2×Q8—82no—
369C2×D4—42no—
370C2×D4#12414yes—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—82no—
395C22×C4—82no—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D411yescharacteristic
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C2×#4114yes—
405C22×C4C2×#4114yes—
406C22×C4C2511yescharacteristic
407M16—48no—
408M16—48no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
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×C8C2×#4114yes—
421C4⋊C4—24no—
422C4⋊C4—22no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—22no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—44no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4#12414yes—
450C22⋊C4—24no—
451C22⋊C4—42no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—44no—
455C22⋊C4—44no—
456C22⋊C4—82no—
457C22⋊C4—82no—
458C22⋊C4#12414yes—
459C22⋊C4#12414yes—
460C42—24no—
461C42—24no—
462C42—21no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—22no—
467C42—42no—
468C42—42no—
469C42—44no—
470C42—44no—
471C42—82no—
472C42—82no—
473C42C2×#4114yes—
474C42C2×#4112yes—
475C422−1+412yes—
476C23—22no—
477C23—22no—
478C23—21no—
479C23—21no—
480C23—21no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23C2×#12412yes—
485C23C2×#12412yes—
486C23C23×D411yescharacteristic
487Q8—44no—
488Q8—44no—
489Q8—164no—
490D4—44no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—48no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—88no—
535C2×C4—164no—
536C2×C4C2×#12412yes—
537C2×C4C2×#12412yes—
538C2×C4C2×#12412yes—
539C2×C4C2×#12412yes—
540C2×C4#54714yes—
541C2×C4#57514yes—
542C2×C4C22×#4111yescharacteristic
543C2×C4C22×#4112yes—
544C2×C4C2×2−1+411yescharacteristic
545C2×C4#58512yes—
546C8—24no—
547C8—42no—
548C8—42no—
549C8—42no—
550C22—22no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22C22×#12411yescharacteristic
561C22C2×#54711yescharacteristic
562C22#350011yescharacteristic
563C22C2×#57512yes—
564C22#358612yes—
565C4—22no—
566C4—22no—
567C4—24no—
568C4—24no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—42no—
573C4—82no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—82no—
581C4#347512yes—
582C4#351512yes—
583C2—42no—
584C2—42no—
585C2#3616411yescharacteristic
586C2#6061811yescharacteristic
587C2#6062711yescharacteristic
588C2#6125312yes—
589C2#6259312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33296C2semidirect
4. #33059C2semidirect
5. #33059C2semidirect
6. #29227C2semidirect
7. #29147C2semidirect
8. #27983C2semidirect
9. #27980C2semidirect
10. #27967C2semidirect
11. #27965C2semidirect
12. #27732C2semidirect
13. #27706C2semidirect
14. #27705C2semidirect
16. C4×#491C2semidirect
17. #20765C2semidirect
18. #20764C2semidirect

Maximal subgroups

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

subgroupindex
#361912
#361812
#332962
#330592
#292272
#291472
#279832
#279802
#279672
#279652
#277322
#277062
#277052
#224992
C4×#4912
#207652
#207642
#204942