Finite Group Databasebeta

← #6986646

Subgroups — #6986646

Order 512.

1479 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36480C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#32838C212yesmaximal, complemented
4#32284C214yesmaximal
5#32052C214yesmaximal
6#30338C212yesmaximal, complemented
7#30063C212yesmaximal, complemented
8#29691C211yesmaximal, characteristic
9#29208C212yesmaximal, complemented
10#28531C212yesmaximal, complemented
11#27815C212yesmaximal, complemented
12#25813C211yesmaximal, characteristic
13#25455C212yesmaximal, complemented
14#24682C212yesmaximal, complemented
15#24302C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic
17#22639C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#500C2212yes—
21C2×#477C2212yes—
22C2×#477C2212yes—
23C2×C4×Q16C2212yes—
24#2868C2212yes—
25#2863C2212yes—
26#2862C2212yes—
27#2857C2211yescharacteristic
28#2660—22no—
29#2605—22no—
30#2567C2212yes—
31#2567C2211yescharacteristic
32#2567C2212yes—
33#2551C2211yescharacteristic
34#2546—21no—
35#2546—21no—
36#2509—21no—
37#2502C2214yes—
38#2497C2212yes—
39#2492C2212yes—
40#2465—22no—
41#2439—21no—
42#2434—22no—
43#2369C2211yescharacteristic
44#2365—21no—
45#2355C2211yescharacteristic
46#2347—21no—
47#2317—21no—
48#2317—21no—
49C4×C4⋊Q8C2212yes—
50C4×C22⋊Q8C2211yescharacteristic
51C4×#117C2211yescharacteristic
52#2100C2214yes—
53#2095C2214yes—
54#2095C2214yes—
55#2083C2214yes—
56#2072C2214yes—
57#2068C2214yes—
58#2057C2211yescharacteristic
59#2057C2212yes—
60#2057C2212yes—
61#2057C2212yes—
62#2057C2212yes—
63#2056C2214yes—
64#2043C2211yescharacteristic
65#2043C2212yes—
66#2036C2212yes—
67#2036C2214yes—
68#2024C2211yescharacteristic
69#2024C2214yes—
70#2024C2212yes—
71#2016C2211yescharacteristic
72#2001C2212yes—
73#1990C2214yes—
74#1990C2212yes—
75#1987C2212yes—
76#1969C2214yes—
77#1947C2214yes—
78#1947C2214yes—
79#1939C2212yes—
80#1933C2214yes—
81#1928C2214yes—
82#1928C2214yes—
83#1923C2214yes—
84#1872C2212yes—
85#1852C2212yes—
86#1852C2214yes—
87#1845C2214yes—
88#1839C2212yes—
89#1803C2212yes—
90#1797C2211yescharacteristic
91C8×C4⋊C4C2212yes—
92C4×C4⋊C8C2212yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C22×Q16C2312yes—
97#533—44no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8—22no—
102C2×C4⋊Q8—22no—
103C2×C4⋊Q8—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2312yes—
109C2×C4⋊Q8C2311yescharacteristic
110C2×#125—21no—
111C2×C22⋊Q8—21no—
112C2×C22⋊Q8—21no—
113C2×C22⋊Q8C2311yescharacteristic
114C2×C22⋊Q8C2312yes—
115C2×C4×Q8—21no—
116C2×C4×Q8—22no—
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2312yes—
119C2×#117—21no—
120C2×#117D411yescharacteristic
121C2×#117D411yescharacteristic
122C22×C4⋊C4—21no—
123C22×C4⋊C4C2312yes—
124C22×C42C2311yescharacteristic
125#500—44no—
126#477—44no—
127#477—44no—
128C4×Q16—44no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#107C2314yes—
132C2×#107C2314yes—
133C2×#107C2312yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2312yes—
137C2×#103C2312yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C4×C8C2312yes—
146C2×C4×C8C2314yes—
147#400—22no—
148#400—22no—
149#398—24no—
150#398—24no—
151#396—22no—
152#395—22no—
153#395—24no—
154#395—24no—
155#395—22no—
156#393—22no—
157#391—22no—
158#391—21no—
159#391—22no—
160#391—22no—
161#391—22no—
162#391—22no—
163#391C2314yes—
164#391C2314yes—
165#391C2312yes—
166#391C2312yes—
167#391C2311yescharacteristic
168#389—22no—
169#389—22no—
170#389—22no—
171#389—22no—
172#389—21no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—22no—
177#389—21no—
178#389C2311yescharacteristic
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#388—22no—
183#388—22no—
184#387—21no—
185#387—22no—
186#387—22no—
187#387—22no—
188#387—22no—
189#387—22no—
190#385—21no—
191#385—24no—
192#385—24no—
193#384—22no—
194#384—22no—
195#384C2312yes—
196#384C2314yes—
197#384C2314yes—
198#384C2311yescharacteristic
199#383—22no—
200#382—21no—
201#382—22no—
202#382—22no—
203#380—21no—
204#380C2312yes—
205#380C2311yescharacteristic
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4D411yescharacteristic
212C4×C4⋊C4C2312yes—
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2311yescharacteristic
216C4×C4⋊C4C2311yescharacteristic
217C4×C22⋊C4—21no—
218C4×C22⋊C4D412yes—
219C4×C22⋊C4C2312yes—
220#376D412yes—
221C43C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2314yes—
228#340C2312yes—
229#336C2312yes—
230#336C2311yescharacteristic
231#336C2311yescharacteristic
232#336C2314yes—
233#336C2314yes—
234C22×Q8—21no—
235C22×Q8—22no—
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—22no—
239C22×Q8—44no—
240C22×Q8—82no—
241C22×Q8—82no—
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C23×C4—22no—
245C23×C4C2×D411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4C2411yescharacteristic
248C2×Q16—24no—
249C2×Q16—42no—
250C2×Q16—42no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—24no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—24no—
258C4⋊Q8—44no—
259C4⋊Q8—42no—
260C4⋊Q8—42no—
261C4⋊Q8—42no—
262C4⋊Q8—44no—
263C4⋊Q8—44no—
264C4⋊Q8—44no—
265C4⋊Q8—84no—
266C4⋊Q8—84no—
267#125—44no—
268C22⋊Q8—22no—
269C22⋊Q8—41no—
270C22⋊Q8—42no—
271C22⋊Q8—42no—
272C22⋊Q8—44no—
273C22⋊Q8—82no—
274C4×Q8—24no—
275C4×Q8—24no—
276C4×Q8—24no—
277C4×Q8—44no—
278C4×Q8—84no—
279#117—24no—
280#117—44no—
281#117Q1618yescomplemented
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—44no—
307C2×C4⋊C4—44no—
308C2×C4⋊C4C2×D411yescharacteristic
309C2×C4⋊C4C2×D411yescharacteristic
310C2×C4⋊C4C2×D411yescharacteristic
311C2×C4⋊C4C2×D411yescharacteristic
312C2×C4⋊C4C2414yes—
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C4⋊C4C2412yes—
316C2×C4⋊C4C2412yes—
317C2×C4⋊C4C2411yescharacteristic
318C2×C4⋊C4C2411yescharacteristic
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4C2×D412yes—
324C2×C22⋊C4C2×D412yes—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—21no—
328C2×C42—21no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D412yes—
336C2×C42C2×D412yes—
337C2×C42C2411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341#107—42no—
342#107—44no—
343#107—44no—
344#107—44no—
345#107—44no—
346C4⋊C8—28no—
347C4⋊C8—24no—
348#103—24no—
349#103—24no—
350#103—44no—
351#103—42no—
352#103—42no—
353#103—44no—
354#103—44no—
355C22⋊C8—24no—
356C22⋊C8—42no—
357C4×C8—24no—
358C4×C8—44no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—21no—
366#95—22no—
367#95—21no—
368#95—22no—
369#95—22no—
370#95—24no—
371#95—24no—
372#95—24no—
373#95—24no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381#95C2×D412yes—
382C24C22×D411yescharacteristic
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8—22no—
388C2×Q8—21no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—21no—
393C2×Q8—44no—
394C2×Q8—42no—
395C2×Q8—44no—
396C2×Q8—42no—
397C2×Q8—44no—
398C2×Q8—44no—
399C2×Q8—42no—
400C2×Q8—44no—
401C2×Q8—42no—
402C2×Q8—44no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×Q8—82no—
406C2×Q8—81no—
407C2×Q8—81no—
408C2×Q8—816no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—41no—
427C22×C4—44no—
428C22×C4—82no—
429C22×C4—82no—
430C22×C4C2×Q1614yes—
431C22×C4C22×D412yes—
432C22×C4C22×D411yescharacteristic
433C22×C4C22×D411yescharacteristic
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D411yescharacteristic
437C22×C4C22×D412yes—
438C22×C4C22×D411yescharacteristic
439C22×C4C22×D411yescharacteristic
440C22×C4C2511yescharacteristic
441Q16—48no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—24no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—24no—
449C2×C8—24no—
450C2×C82−1+414yes—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—24no—
455C4⋊C4—24no—
456C4⋊C4—22no—
457C4⋊C4—24no—
458C4⋊C4—22no—
459C4⋊C4—22no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—44no—
469C4⋊C4—42no—
470C4⋊C4—44no—
471C4⋊C4—42no—
472C4⋊C4—44no—
473C4⋊C4—42no—
474C4⋊C4—44no—
475C4⋊C4—42no—
476C4⋊C4—42no—
477C4⋊C4—44no—
478C4⋊C4—82no—
479C4⋊C4—82no—
480C4⋊C4—82no—
481C4⋊C4—81no—
482C4⋊C4—84no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—84no—
486C4⋊C4C2×Q1614yes—
487C4⋊C4C2×Q1614yes—
488C4⋊C4#13714yes—
489C4⋊C42−1+412yes—
490C22⋊C4—24no—
491C22⋊C4—42no—
492C22⋊C4—44no—
493C22⋊C4—81no—
494C22⋊C4—82no—
495C22⋊C4C2×Q1618yes—
496C42—22no—
497C42—22no—
498C42—24no—
499C42—24no—
500C42—22no—
501C42—24no—
502C42—22no—
503C42—22no—
504C42—42no—
505C42—44no—
506C42—42no—
507C42—42no—
508C42—82no—
509C42—82no—
510C42C2×Q1618yes—
511C42H(5,𝔽2)14yes—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23C22×Q1612yes—
519C23C23×D411yescharacteristic
520Q8—42no—
521Q8—42no—
522Q8—42no—
523Q8—42no—
524Q8—42no—
525Q8—42no—
526Q8—88no—
527Q8—84no—
528Q8—88no—
529Q8—88no—
530Q8—162no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—28no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—44no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—88no—
581C2×C4#53512yes—
582C2×C4#53512yes—
583C2×C4C22×Q1612yes—
584C2×C4C22×Q1614yes—
585C2×C4C22×Q1614yes—
586C2×C4C22×Q1612yes—
587C2×C4C2×#13714yes—
588C2×C4C2×#13712yes—
589C2×C4#57612yes—
590C2×C4C2×H(5,𝔽2)12yes—
591C2×C4C2×2−1+412yes—
592C2×C4C2×2−1+411yescharacteristic
593C2×C4#58511yescharacteristic
594C8—24no—
595C8—44no—
596C8—42no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22C2×#53511yescharacteristic
605C22#348911yescharacteristic
606C22#349911yescharacteristic
607C22C23×Q1611yescharacteristic
608C22C22×#13711yescharacteristic
609C22C2×#57611yescharacteristic
610C22#358511yescharacteristic
611C4—24no—
612C4—22no—
613C4—22no—
614C4—22no—
615C4—24no—
616C4—24no—
617C4—42no—
618C4—41no—
619C4—42no—
620C4—81no—
621C4—81no—
622C4—84no—
623C4—82no—
624C4—82no—
625C4—82no—
626C4—81no—
627C4—82no—
628C4#320914yes—
629C4#354112yes—
630C4#355012yes—
631C2—42no—
632C2#4291111yescharacteristic
633C2C2×#320911yescharacteristic
634C2#6081111yescharacteristic
635C2#6210011yescharacteristic
636C2#6212711yescharacteristic
637C2#6247211yescharacteristic
638C2#6251011yescharacteristic
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. #32838C2semidirect
6. #30338C2semidirect
7. #30063C2semidirect
9. #29208C2semidirect
10. #28531C2semidirect
11. #27815C2semidirect
13. #25455C2semidirect
14. #24682C2semidirect
281. #117Q16semidirect

Maximal subgroups

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

subgroupindex
#364802
#352812
#328382
#322842
#320522
#303382
#300632
#296912
#292082
#285312
#278152
#258132
#254552
#246822
#243022
#235712
#226392