Finite Group Databasebeta

← #6563375

Subgroups — #6563375

Order 512.

1545 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36314C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic
3#30096C212yesmaximal, complemented
4#29936C211yesmaximal, characteristic
5#29394C212yesmaximal, complemented
6#29317C212yesmaximal, complemented
7#27470C214yesmaximal
8#27176C214yesmaximal
9#26509C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic
11#25456C212yesmaximal, complemented
12#24301C211yesmaximal, characteristic
13#23783C212yesmaximal, complemented
14#23756C211yesmaximal, characteristic
15#23161C212yesmaximal, complemented
16#22970C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#519—21no—
20C2×#500C2212yes—
21C2×#475C2212yes—
22#2864C2212yes—
23#2857C2211yescharacteristic
24#2571C2211yescharacteristic
25#2568C2211yescharacteristic
26#2533—22no—
27#2530—22no—
28#2511—21no—
29#2499—21no—
30#2495C2212yes—
31#2421C2212yes—
32#2365—21no—
33#2365—21no—
34#2365C2212yes—
35#2365C2211yescharacteristic
36#2365C2212yes—
37#2362—21no—
38#2356—21no—
39Q8×C4⋊C4C2211yescharacteristic
40#2344—22no—
41#2344C2214yes—
42#2327—22no—
43#2310—21no—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47#2295C2212yes—
48#2285C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2099C2211yescharacteristic
52#2099C2212yes—
53#2096C2211yescharacteristic
54#2088C2214yes—
55#2085C2214yes—
56#2084C2211yescharacteristic
57#2083C2212yes—
58#2077C2212yes—
59#2075C2214yes—
60#2072C2211yescharacteristic
61#2068C2214yes—
62#2058C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2035C2212yes—
66#2024C2212yes—
67#1990C2212yes—
68#1988C2212yes—
69#1988C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1947C2212yes—
73#1944C2214yes—
74#1943C2214yes—
75#1939C2212yes—
76#1928C2214yes—
77#1924C2214yes—
78#1921C2214yes—
79#1921C2214yes—
80#1921C2214yes—
81#1919C2214yes—
82#1872C2212yes—
83#1863C2214yes—
84#1853C2212yes—
85#1852C2214yes—
86#1839C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92#1763C2214yes—
93C4×#103C2212yes—
94C4×C22⋊C8C2211yescharacteristic
95C23×Q8—21no—
96C2×C4⋊Q8—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8C2312yes—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8C2311yescharacteristic
104#519—48no—
105C2×C4×Q8—21no—
106C2×C4×Q8—22no—
107C2×C4×Q8—22no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×#117—21no—
113C2×#117—24no—
114C2×#117D411yescharacteristic
115C2×#117D411yescharacteristic
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119#500—28no—
120#475—28no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×#106C2312yes—
126C2×#106C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2314yes—
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×C22⋊C8C2312yes—
133C2×C22⋊C8C2312yes—
134C2×#97C2314yes—
135C2×C4×C8C2312yes—
136#400—22no—
137#400—22no—
138#400—22no—
139#400—22no—
140#397—22no—
141#397—22no—
142#391—22no—
143#391—22no—
144#391—21no—
145#391—21no—
146#391—21no—
147#391—22no—
148#391—22no—
149#391—21no—
150#391C2312yes—
151#391C2311yescharacteristic
152#389—21no—
153#389—22no—
154#389—22no—
155#389—21no—
156#389—22no—
157#389C2314yes—
158#389C2314yes—
159#389C2312yes—
160#389C2311yescharacteristic
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#388—22no—
165#388—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—21no—
173#387—24no—
174#387—24no—
175#387—24no—
176#387—24no—
177#387—22no—
178#387—22no—
179#385—21no—
180#384—21no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#383—22no—
185#382—21no—
186#382—21no—
187#382C2312yes—
188#382C2312yes—
189#381—24no—
190#380—21no—
191#380—21no—
192#380C2312yes—
193#380C2312yes—
194#380C2311yescharacteristic
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4D411yescharacteristic
202C4×C4⋊C4D411yescharacteristic
203C4×C4⋊C4C2314yes—
204C4×C4⋊C4C2314yes—
205C4×C4⋊C4C2311yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C22⋊C4—21no—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213#376—22no—
214#376D412yes—
215C43C2312yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2314yes—
226#340C2312yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2314yes—
231#336C2314yes—
232C22×Q8—28no—
233C22×Q8—22no—
234C22×Q8—22no—
235C22×Q8—21no—
236C22×Q8—44no—
237C22×Q8—44no—
238C22×Q8C2411yescharacteristic
239C23×C4—22no—
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2411yescharacteristic
243C22×C8C2412yes—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C4⋊Q8—28no—
247C4⋊Q8—28no—
248C4⋊Q8—28no—
249C4⋊Q8—24no—
250C4⋊Q8—24no—
251C4⋊Q8#4118yes—
252C22⋊Q8—22no—
253C22⋊Q8—41no—
254C22⋊Q8—44no—
255C4×Q8—24no—
256C4×Q8—24no—
257C4×Q8—42no—
258C4×Q8—48no—
259C4×Q8—48no—
260C4×Q8—44no—
261C4×Q8—48no—
262C4×Q8—48no—
263#117—24no—
264#117—416no—
265#117—42no—
266#117—44no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2×D411yescharacteristic
292C2×C4⋊C4C2×D411yescharacteristic
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2411yescharacteristic
295C2×C4⋊C4C2411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2412yes—
300C2×C4⋊C4C2414yes—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4C2×D412yes—
306C2×C22⋊C4C2×D412yes—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—21no—
313C2×C42—22no—
314C2×C42—21no—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D412yes—
319C2×C42C2×D412yes—
320C2×C42C2×D412yes—
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2412yes—
324C2×C42C2411yescharacteristic
325#107—28no—
326#107—28no—
327#107—44no—
328#106—24no—
329#106—28no—
330#106—42no—
331C4⋊C8—28no—
332C4⋊C8—24no—
333#103—24no—
334#103—24no—
335#103—28no—
336C22⋊C8—24no—
337C22⋊C8—42no—
338#97—44no—
339C4×C8—24no—
340#95—21no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—21no—
345#95—21no—
346#95—22no—
347#95—21no—
348#95—24no—
349#95—24no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—24no—
354#95—24no—
355#95—21no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95C2×D412yes—
364#95C2×D412yes—
365#95C2×D412yes—
366#95C2×D412yes—
367C24C22×D411yescharacteristic
368C2×Q8—28no—
369C2×Q8—28no—
370C2×Q8—21no—
371C2×Q8—22no—
372C2×Q8—22no—
373C2×Q8—44no—
374C2×Q8—432no—
375C2×Q8—42no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—42no—
379C2×Q8—44no—
380C2×Q8—42no—
381C2×Q82−1+412yes—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—41no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4AGL(1,ℤ/8)14yes—
404C22×C4C22×D412yes—
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C2511yescharacteristic
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—24no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C82−1+414yes—
423C4⋊C4—28no—
424C4⋊C4—28no—
425C4⋊C4—28no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—44no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—48no—
446C4⋊C4—48no—
447C4⋊C4—44no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4C2×#4118yes—
460C22⋊C4—24no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—44no—
464C22⋊C4—44no—
465C22⋊C4—81no—
466C42—22no—
467C42—22no—
468C42—24no—
469C42—24no—
470C42—22no—
471C42—24no—
472C42—22no—
473C42—22no—
474C42—42no—
475C42—42no—
476C42—44no—
477C42—48no—
478C42—48no—
479C42—44no—
480C42—42no—
481C42—44no—
482C42—42no—
483C42C2×#4114yes—
484C23—21no—
485C23—21no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23C2×AGL(1,ℤ/8)12yes—
491C23C23×D411yescharacteristic
492Q8—24no—
493Q8—416no—
494Q8—416no—
495Q8—42no—
496Q8—84no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—28no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—21no—
507C2×C4—28no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—44no—
522C2×C4—416no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4#52512yes—
551C2×C4#52512yes—
552C2×C4C2×AGL(1,ℤ/8)12yes—
553C2×C4C2×AGL(1,ℤ/8)12yes—
554C2×C4#57512yes—
555C2×C4#57812yes—
556C2×C4C22×#4112yes—
557C2×C4C2×2−1+412yes—
558C2×C4C2×2−1+411yescharacteristic
559C2×C4#58511yescharacteristic
560C8—24no—
561C8—44no—
562C8—42no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—41no—
567C22—41no—
568C22—41no—
569C22—41no—
570C22C2×#52511yescharacteristic
571C22#348911yescharacteristic
572C22#349911yescharacteristic
573C22C22×AGL(1,ℤ/8)11yescharacteristic
574C22C2×#57511yescharacteristic
575C22C2×#57811yescharacteristic
576C22#358611yescharacteristic
577C4—22no—
578C4—22no—
579C4—22no—
580C4—42no—
581C4—42no—
582C4—48no—
583C4—41no—
584C4—42no—
585C4—42no—
586C4—42no—
587C4—82no—
588C4—81no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—81no—
593C4—81no—
594C4#311414yes—
595C4#351612yes—
596C4#352312yes—
597C2—42no—
598C2#3599711yescharacteristic
599C2C2×#311411yescharacteristic
600C2#6058311yescharacteristic
601C2#6211111yescharacteristic
602C2#6214611yescharacteristic
603C2#6247911yescharacteristic
604C2#6250711yescharacteristic
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. #30096C2semidirect
5. #29394C2semidirect
6. #29317C2semidirect
9. #26509C2semidirect
11. #25456C2semidirect
13. #23783C2semidirect
15. #23161C2semidirect
16. #22970C2semidirect

Maximal subgroups

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

subgroupindex
#363142
#342092
#300962
#299362
#293942
#293172
#274702
#271762
#265092
#258132
#254562
#243012
#237832
#237562
#231612
#229702
#226432