Finite Group Databasebeta

← #6555797

Subgroups — #6555797

Order 512.

1547 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36598C211yesmaximal, characteristic, complemented
2#34322C211yesmaximal, characteristic
3#30046C212yesmaximal
4#29798C211yesmaximal, characteristic
5#29559C212yesmaximal, complemented
6#29370C212yesmaximal, complemented
7#29259C211yesmaximal, characteristic, complemented
8#28440C211yesmaximal, characteristic, complemented
9#26610C214yesmaximal
10#26319C212yesmaximal, complemented
11#26183C214yesmaximal, complemented
12#25632C212yesmaximal
13#23807C212yesmaximal
14#23775C211yesmaximal, characteristic
15#23187C212yesmaximal, complemented
16#23002C212yesmaximal
17#22832C211yesmaximal, characteristic, complemented
18C2×#556C2211yescharacteristic
19C2×#494C2212yes—
20C2×#462C2214yes—
21C2×#461C2212yes—
22C2×#441C2214yes—
23C2×#440C2214yes—
24#2868C2211yescharacteristic
25#2862C2212yes—
26#2582—21no—
27#2571—21no—
28#2570—22no—
29#2568C2212yes—
30#2511—22no—
31#2446C2214yes—
32#2373—22no—
33#2367C2212yes—
34#2365C2211yescharacteristic
35#2365C2212yes—
36#2360—21no—
37#2355—21no—
38#2355—21no—
39#2355C2211yescharacteristic
40#2351C2212yes—
41D4×C4⋊C4C2211yescharacteristic
42#2346—21no—
43#2334—22no—
44#2323—21no—
45C4×C4⋊Q8C2211yescharacteristic
46C4×C4⋊Q8C2211yescharacteristic
47C4×C4⋊Q8C2212yes—
48C4×#124C2212yes—
49#2295C2212yes—
50#2293C2211yescharacteristic
51#2274—21no—
52C42×Q8C2211yescharacteristic
53#2100C2211yescharacteristic
54#2097C2214yes—
55#2095C2212yes—
56#2094C2211yescharacteristic
57#2088C2212yes—
58#2074C2214yes—
59#2065C2212yes—
60#2058C2211yescharacteristic
61#2043C2212yes—
62#2043C2212yes—
63#2024C2212yes—
64#2023C2211yescharacteristic
65#2016C2212yes—
66#2016C2212yes—
67#2015C2214yes—
68#2015C2214yes—
69#2013C2212yes—
70#2013C2212yes—
71#1984C2212yes—
72#1981C2212yes—
73#1981C2212yes—
74#1971C2214yes—
75#1956C2212yes—
76#1947C2212yes—
77#1935C2214yes—
78#1928C2214yes—
79#1927C2214yes—
80#1917C2214yes—
81#1878C2214yes—
82#1878C2214yes—
83#1878C2214yes—
84#1878C2214yes—
85#1877C2214yes—
86#1863C2214yes—
87#1863C2214yes—
88#1839C2212yes—
89#1831C2211yescharacteristic
90C4×#106C2212yes—
91#1770C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#102C2212yes—
94#1745C2212yes—
95C22×Q16C2314yes—
96C22×Q16C2314yes—
97C22×SD16C2314yes—
98#556—24no—
99#556—24no—
100C2×C4⋊Q8C2312yes—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2311yescharacteristic
105C2×#126—22no—
106C2×#124C2312yes—
107C2×C22⋊Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8D411yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2312yes—
113C2×C4×Q8C2314yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2312yes—
118C2×#117—21no—
119C2×#117—21no—
120C22×C4⋊C4—21no—
121#494—28no—
122#462—48no—
123#461—28no—
124#441—48no—
125#440—48no—
126C2×#107C2314yes—
127C2×#106C2312yes—
128C2×#106C2312yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#102C2312yes—
139C2×#102C2312yes—
140C2×#97C2314yes—
141C2×C4×C8C2312yes—
142#400—24no—
143#400—24no—
144#398—24no—
145#391—22no—
146#391—21no—
147#391—21no—
148#391—22no—
149#391—22no—
150#391C2314yes—
151#391C2312yes—
152#391C2314yes—
153#391C2314yes—
154#391C2312yes—
155#390C2311yescharacteristic
156#390C2311yescharacteristic
157#389—22no—
158#389—21no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2311yescharacteristic
168#389C2311yescharacteristic
169#389C2312yes—
170#388—22no—
171#388—22no—
172#388—21no—
173#388—22no—
174#388—21no—
175#388—22no—
176#387—22no—
177#387—24no—
178#387—22no—
179#387—22no—
180#387—24no—
181#387—24no—
182#387—22no—
183#385—22no—
184#385—22no—
185#384—21no—
186#384—21no—
187#384—21no—
188#384—21no—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#382—22no—
192#382—22no—
193#382—22no—
194#382—22no—
195#382C2312yes—
196#380—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—22no—
206C4×C4⋊C4—22no—
207C4×C4⋊C4D412yes—
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D412yes—
211C4×C4⋊C4C2312yes—
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2312yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—21no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—22no—
220#376—22no—
221C43C2311yescharacteristic
222C43C2312yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#336C2311yescharacteristic
229#336C2314yes—
230#336C2312yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2314yes—
234#336C2314yes—
235C22×Q8—21no—
236C22×Q8C2×D411yescharacteristic
237C22×Q8C2414yes—
238C22×Q8C2412yes—
239C22×D4C2411yescharacteristic
240C23×C4—21no—
241C23×C4—22no—
242C2×Q16—28no—
243C2×Q16—28no—
244C2×Q16—28no—
245C2×Q16—44no—
246C2×Q16—44no—
247C2×Q16—44no—
248C2×SD16—28no—
249C2×SD16—44no—
250C2×SD16—44no—
251C22×C8C2414yes—
252C22×C8C2412yes—
253C22×C8C2412yes—
254C4⋊Q8—22no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—42no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8#4114yes—
261#126—216no—
262#124—24no—
263#124—24no—
264C22⋊Q8—28no—
265C4×Q8—24no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—22no—
269C4×Q8—22no—
270C4×Q8—42no—
271C4×Q8—44no—
272C4×Q8—44no—
273C4×Q8—44no—
274C4×Q8—44no—
275C4×Q8—44no—
276C4×D4—22no—
277C4×D4—44no—
278C4×D4#4114yes—
279#117—28no—
280#117—44no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4C2×D411yescharacteristic
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D412yes—
307C2×C4⋊C4C2×D412yes—
308C2×C4⋊C4C2×D412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2411yescharacteristic
311C2×C4⋊C4C2412yes—
312C2×C4⋊C4C2412yes—
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—21no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—21no—
319C2×C22⋊C4—21no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C42—22no—
324C2×C42—21no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—24no—
329C2×C42—22no—
330C2×C42C2×D411yescharacteristic
331C2×C42C2×D412yes—
332C2×C42C2×D412yes—
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D412yes—
335C2×C42C2×D412yes—
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2412yes—
339C2×C42C2412yes—
340C2×C42C2411yescharacteristic
341C2×C42C2411yescharacteristic
342C2×C42C2411yescharacteristic
343#107—44no—
344#106—24no—
345#106—42no—
346C4⋊C8—24no—
347C4⋊C8—24no—
348#103—28no—
349#103—28no—
350#103—44no—
351#103—44no—
352#103—44no—
353#103—44no—
354#103—44no—
355#102—24no—
356#102—42no—
357#97—28no—
358C4×C8—24no—
359#95—21no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—21no—
367#95—22no—
368#95—24no—
369#95—24no—
370#95—22no—
371#95—24no—
372#95—22no—
373#95—21no—
374#95—22no—
375#95—24no—
376#95—24no—
377#95—24no—
378#95C2×D412yes—
379#95C2×D412yes—
380#95C2×D412yes—
381C24—21no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—24no—
391C2×Q8—24no—
392C2×Q8—24no—
393C2×Q8—24no—
394C2×Q8—42no—
395C2×Q8—42no—
396C2×Q8—42no—
397C2×Q8C2×#4114yes—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—22no—
401C2×D4C2×#4112yes—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—44no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—82no—
415C22×C4—82no—
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C2511yescharacteristic
427Q16—416no—
428Q16—88no—
429SD16—416no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—24no—
438C2×C8H(5,𝔽2)14yes—
439C4⋊C4—28no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—22no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—24no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—42no—
456C4⋊C4—42no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—44no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—44no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—82no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C4⋊C4—82no—
474C4⋊C4C2×#4112yes—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—24no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—81no—
482C22⋊C4—82no—
483C42—28no—
484C42—24no—
485C42—22no—
486C42—22no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—24no—
491C42—22no—
492C42—22no—
493C42—41no—
494C42—42no—
495C42—42no—
496C42—42no—
497C42—44no—
498C42—82no—
499C42—82no—
500C42—84no—
501C42—82no—
502C42#13714yes—
503C42C2×#4112yes—
504C42C2×#4112yes—
505C23—22no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C23×D411yescharacteristic
513Q8—28no—
514Q8—44no—
515Q8—44no—
516Q8—44no—
517Q8—44no—
518Q8—44no—
519Q8—44no—
520Q8—84no—
521D4—24no—
522D4—42no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—164no—
572C2×C4#52512yes—
573C2×C4#52512yes—
574C2×C4C2×#13712yes—
575C2×C4C2×#13712yes—
576C2×C4C2×#13712yes—
577C2×C4#57512yes—
578C2×C4#57812yes—
579C2×C4C22×#4111yescharacteristic
580C2×C4C22×#4111yescharacteristic
581C2×C4C2×H(5,𝔽2)12yes—
582C2×C4#58512yes—
583C8—24no—
584C8—44no—
585C8—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22C2×#52511yescharacteristic
594C22#348711yescharacteristic
595C22#349811yescharacteristic
596C22C22×#13711yescharacteristic
597C22C2×#57511yescharacteristic
598C22C2×#57811yescharacteristic
599C22#358611yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—84no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—82no—
616C4—82no—
617C4#311614yes—
618C4#350612yes—
619C4#351312yes—
620C2—82no—
621C2#3584411yescharacteristic
622C2C2×#311611yescharacteristic
623C2#6059411yescharacteristic
624C2#6189311yescharacteristic
625C2#6190511yescharacteristic
626C2#6242411yescharacteristic
627C2#6245611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36598C2semidirect
5. #29559C2semidirect
6. #29370C2semidirect
7. #29259C2semidirect
8. #28440C2semidirect
10. #26319C2semidirect
11. #26183C2semidirect
15. #23187C2semidirect
17. #22832C2semidirect

Maximal subgroups

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

subgroupindex
#365982
#343222
#300462
#297982
#295592
#293702
#292592
#284402
#266102
#263192
#261832
#256322
#238072
#237752
#231872
#230022
#228322