Finite Group Databasebeta

← #6993614

Subgroups — #6993614

Order 512.

1425 conjugacy classes of subgroups.

The subgroup lattice has 633 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36287C211yesmaximal, characteristic
2#35320C211yesmaximal, characteristic
3#32820C212yesmaximal, complemented
4#32296C214yesmaximal
5#31951C214yesmaximal
6#30310C212yesmaximal, complemented
7#29985C212yesmaximal, complemented
8#29691C211yesmaximal, characteristic
9#29559C212yesmaximal, complemented
10#28542C212yesmaximal, complemented
11#27737C212yesmaximal, complemented
12#25811C211yesmaximal, characteristic
13#25454C212yesmaximal, complemented
14#24818C212yesmaximal, complemented
15#24301C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic
17#22636C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#498C2212yes—
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×C4×Q16C2212yes—
24#2862C2212yes—
25#2862C2212yes—
26#2859C2212yes—
27#2857C2211yescharacteristic
28#2642—22no—
29#2612—22no—
30#2568—21no—
31#2568—21no—
32#2568C2212yes—
33#2568C2212yes—
34#2567C2211yescharacteristic
35#2556—22no—
36#2552C2211yescharacteristic
37#2506—21no—
38#2503—22no—
39#2503C2214yes—
40#2497C2212yes—
41#2441—21no—
42#2423C2212yes—
43#2370—21no—
44#2369C2211yescharacteristic
45Q8×C4⋊C4C2211yescharacteristic
46#2346—21no—
47#2310—21no—
48#2310—21no—
49C4×C22⋊Q8C2211yescharacteristic
50#2295C2212yes—
51C4×#117C2211yescharacteristic
52#2101C2214yes—
53#2097C2214yes—
54#2096C2214yes—
55#2085C2214yes—
56#2085C2214yes—
57#2072C2214yes—
58#2066C2214yes—
59#2066C2214yes—
60#2058C2212yes—
61#2058C2212yes—
62#2058C2212yes—
63#2057C2211yescharacteristic
64#2057C2212yes—
65#2043C2211yescharacteristic
66#2043C2212yes—
67#2036C2212yes—
68#2036C2214yes—
69#2024C2211yescharacteristic
70#2024C2212yes—
71#2016C2211yescharacteristic
72#2001C2212yes—
73#1990C2212yes—
74#1988C2212yes—
75#1971C2214yes—
76#1949C2214yes—
77#1939C2212yes—
78#1935C2214yes—
79#1927C2214yes—
80#1924C2214yes—
81#1878C2214yes—
82#1872C2212yes—
83#1853C2212yes—
84#1853C2214yes—
85#1847C2214yes—
86#1836C2214yes—
87#1831C2212yes—
88#1805C2214yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91#1770C2212yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C23×Q8—21no—
96C22×Q16C2312yes—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—44no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×#125—21no—
107C2×#125—21no—
108C2×#125—22no—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8—21no—
111C2×C22⋊Q8C2311yescharacteristic
112C2×C22⋊Q8C2312yes—
113C2×C4×Q8—21no—
114C2×C4×Q8C2311yescharacteristic
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×#117—21no—
118C2×#117D411yescharacteristic
119C2×#117D411yescharacteristic
120C22×C4⋊C4—21no—
121C22×C4⋊C4C2312yes—
122C22×C42C2311yescharacteristic
123#498—44no—
124#477—44no—
125#475—44no—
126C4×Q16—44no—
127C2×#107C2314yes—
128C2×#107C2312yes—
129C2×#106C2314yes—
130C2×#106C2314yes—
131C2×#106C2314yes—
132C2×C4⋊C8C2312yes—
133C2×C4⋊C8C2314yes—
134C2×#103C2312yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2312yes—
139C2×#103C2312yes—
140C2×#103C2312yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×#97C2314yes—
144C2×C4×C8C2312yes—
145#400—24no—
146#400—24no—
147#400—24no—
148#400—24no—
149#400—22no—
150#399—22no—
151#398—22no—
152#398—22no—
153#396—22no—
154#395—22no—
155#391—22no—
156#391—22no—
157#391—22no—
158#391—22no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391—22no—
164#391—22no—
165#391—21no—
166#391—22no—
167#391C2312yes—
168#391C2314yes—
169#391C2314yes—
170#391C2311yescharacteristic
171#389—21no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—22no—
178#389—22no—
179#389C2312yes—
180#389C2311yescharacteristic
181#389C2312yes—
182#389C2312yes—
183#389C2312yes—
184#388—22no—
185#388—22no—
186#387—21no—
187#387—22no—
188#387—22no—
189#387—22no—
190#387—22no—
191#385—21no—
192#384—21no—
193#384C2312yes—
194#384C2311yescharacteristic
195#384C2312yes—
196#382—22no—
197#382—22no—
198#382—22no—
199#382C2314yes—
200#382C2314yes—
201#381—24no—
202#381—24no—
203#380—21no—
204#380C2312yes—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—21no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2311yescharacteristic
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C22⋊C4—21no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4C2312yes—
219#376—22no—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—44no—
239C22×Q8—44no—
240C22×Q8—44no—
241C22×Q8C2412yes—
242C22×Q8C2411yescharacteristic
243C23×C4—22no—
244C23×C4C2×D411yescharacteristic
245C23×C4C2×D411yescharacteristic
246C23×C4C2411yescharacteristic
247C2×Q16—24no—
248C2×Q16—24no—
249C2×Q16—42no—
250C22×C8C2412yes—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—44no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—44no—
261#125—44no—
262#125—44no—
263#125—84no—
264C22⋊Q8—22no—
265C22⋊Q8—41no—
266C22⋊Q8—42no—
267C22⋊Q8—42no—
268C22⋊Q8—44no—
269C22⋊Q8—82no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—24no—
274C4×Q8—42no—
275C4×Q8—44no—
276#117—24no—
277#117—42no—
278#117—44no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2414yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2412yes—
334C2×C42C2411yescharacteristic
335C2×C42C2412yes—
336C2×C42C2411yescharacteristic
337#107—42no—
338#107—44no—
339#106—44no—
340#106—44no—
341#106—44no—
342C4⋊C8—24no—
343C4⋊C8—44no—
344#103—24no—
345#103—44no—
346#103—42no—
347#103—42no—
348#103—42no—
349#103—44no—
350#103—44no—
351C22⋊C8—24no—
352C22⋊C8—42no—
353#97—28no—
354C4×C8—24no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—24no—
375#95—24no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24C22×D411yescharacteristic
381C2×Q8—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—42no—
391C2×Q8—44no—
392C2×Q8—42no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—42no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—81no—
401C2×Q8—81no—
402C2×Q8—816no—
403C2×Q82−1+412yes—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—41no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4#13714yes—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D411yescharacteristic
435C22×C4C2511yescharacteristic
436Q16—48no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—24no—
441C2×C8—24no—
442C2×C8—22no—
443C2×C8—24no—
444C2×C8—22no—
445C2×C82−1+414yes—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—24no—
452C4⋊C4—24no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—42no—
469C4⋊C4—44no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—42no—
473C4⋊C4—44no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—44no—
477C4⋊C4—42no—
478C4⋊C4—82no—
479C4⋊C4—81no—
480C4⋊C4—82no—
481C4⋊C4—82no—
482C4⋊C4—82no—
483C4⋊C4—82no—
484C22⋊C4—24no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—44no—
488C22⋊C4—81no—
489C22⋊C4—82no—
490C42—22no—
491C42—22no—
492C42—24no—
493C42—24no—
494C42—22no—
495C42—24no—
496C42—22no—
497C42—22no—
498C42—42no—
499C42—42no—
500C42—44no—
501C42—44no—
502C42—42no—
503C42—42no—
504C42—82no—
505C42H(5,𝔽2)14yes—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23C2×#13712yes—
513C23C23×D411yescharacteristic
514Q8—24no—
515Q8—42no—
516Q8—42no—
517Q8—42no—
518Q8—42no—
519Q8—42no—
520Q8—84no—
521Q8—88no—
522Q8—88no—
523Q8—162no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—88no—
577C2×C4#53512yes—
578C2×C4#53512yes—
579C2×C4C2×#13712yes—
580C2×C4C2×#13712yes—
581C2×C4#57512yes—
582C2×C4#57712yes—
583C2×C4C2×H(5,𝔽2)12yes—
584C2×C4C2×2−1+411yescharacteristic
585C2×C4C2×2−1+412yes—
586C2×C4#58511yescharacteristic
587C8—24no—
588C8—42no—
589C8—44no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#53511yescharacteristic
598C22#348911yescharacteristic
599C22#349911yescharacteristic
600C22C22×#13711yescharacteristic
601C22C2×#57511yescharacteristic
602C22C2×#57711yescharacteristic
603C22#358611yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—81no—
614C4—84no—
615C4—81no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—81no—
620C4—82no—
621C4#321514yes—
622C4#353812yes—
623C4#354712yes—
624C2—42no—
625C2#4299411yescharacteristic
626C2C2×#321511yescharacteristic
627C2#6081311yescharacteristic
628C2#6211311yescharacteristic
629C2#6213511yescharacteristic
630C2#6247811yescharacteristic
631C2#6251111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #32820C2semidirect
6. #30310C2semidirect
7. #29985C2semidirect
9. #29559C2semidirect
10. #28542C2semidirect
11. #27737C2semidirect
13. #25454C2semidirect
14. #24818C2semidirect

Maximal subgroups

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

subgroupindex
#362872
#353202
#328202
#322962
#319512
#303102
#299852
#296912
#295592
#285422
#277372
#258112
#254542
#248182
#243012
#235712
#226362