Finite Group Databasebeta

← #6563363

Subgroups — #6563363

Order 512.

1487 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36301C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#30093C212yesmaximal, complemented
4#29939C211yesmaximal, characteristic, complemented
5#29388C212yesmaximal, complemented
6#29317C212yesmaximal, complemented
7#27476C214yesmaximal, complemented
8#27178C214yesmaximal
9#26511C212yesmaximal, complemented
10#25813C211yesmaximal, characteristic, complemented
11#25450C212yesmaximal, complemented
12#24299C211yesmaximal, characteristic
13#23780C212yesmaximal, complemented
14#23755C211yesmaximal, characteristic
15#23162C212yesmaximal, complemented
16#22969C212yesmaximal, complemented
17#22643C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C2×#500C2212yes—
20C2×#476C2212yes—
21#2864C2212yes—
22#2857C2211yescharacteristic
23#2571C2211yescharacteristic
24#2563C2211yescharacteristic
25#2527—22no—
26#2494—21no—
27#2494C2212yes—
28#2485—21no—
29#2477—22no—
30#2420C2212yes—
31#2369C2212yes—
32#2369C2211yescharacteristic
33#2369C2212yes—
34#2365—21no—
35#2365—21no—
36#2352—21no—
37D4×C4⋊C4C2211yescharacteristic
38#2345C2214yes—
39#2344—21no—
40Q8×C22⋊C4—22no—
41#2331—22no—
42#2316—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#121C2211yescharacteristic
46#2293C2212yes—
47#2285C2211yescharacteristic
48#2274—21no—
49C4×#117C2211yescharacteristic
50#2101C2214yes—
51#2100C2211yescharacteristic
52#2100C2212yes—
53#2094C2211yescharacteristic
54#2088C2214yes—
55#2085C2214yes—
56#2084C2211yescharacteristic
57#2083C2212yes—
58#2077C2212yes—
59#2076C2214yes—
60#2071C2211yescharacteristic
61#2067C2214yes—
62#2058C2212yes—
63#2057C2212yes—
64#2057C2212yes—
65#2034C2212yes—
66#2023C2212yes—
67#1990C2212yes—
68#1987C2212yes—
69#1987C2212yes—
70#1949C2214yes—
71#1949C2214yes—
72#1947C2212yes—
73#1944C2214yes—
74#1943C2214yes—
75#1939C2212yes—
76#1925C2214yes—
77#1924C2214yes—
78#1920C2214yes—
79#1918C2214yes—
80#1918C2214yes—
81#1918C2214yes—
82#1874C2212yes—
83#1862C2214yes—
84#1853C2212yes—
85#1852C2214yes—
86#1839C2212yes—
87#1797C2211yescharacteristic
88#1776C2214yes—
89C4×#106C2212yes—
90#1770C2212yes—
91#1767C2212yes—
92#1762C2214yes—
93C4×#102C2212yes—
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×#121C2311yescharacteristic
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×D4—21no—
107C2×C4×D4C2311yescharacteristic
108C2×C4×D4C2312yes—
109C2×#117—21no—
110C2×#117—22no—
111C2×#117D411yescharacteristic
112C2×#117D411yescharacteristic
113C22×C4⋊C4—21no—
114C22×C4⋊C4—21no—
115C22×C4⋊C4C2312yes—
116C22×C42C2311yescharacteristic
117#500—28no—
118#476—28no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#106C2314yes—
123C2×#106C2312yes—
124C2×#106C2312yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×#102C2312yes—
128C2×#102C2314yes—
129C2×#102C2312yes—
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×#97C2314yes—
133C2×C4×C8C2312yes—
134#401—22no—
135#400—22no—
136#397—22no—
137#397—22no—
138#394—22no—
139#393—22no—
140#391—22no—
141#391—22no—
142#391—21no—
143#391—24no—
144#391—24no—
145#391—22no—
146#390C2312yes—
147#390C2311yescharacteristic
148#389—21no—
149#389—22no—
150#389—22no—
151#389—21no—
152#389—22no—
153#389—22no—
154#389—21no—
155#389C2314yes—
156#389C2314yes—
157#389C2312yes—
158#389C2311yescharacteristic
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—21no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—24no—
171#387—24no—
172#387—22no—
173#386—21no—
174#386—22no—
175#385—22no—
176#384C2311yescharacteristic
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#382—21no—
180#382—21no—
181#382C2312yes—
182#382C2312yes—
183#381—22no—
184#380—21no—
185#380—21no—
186#380—21no—
187#380—21no—
188#380—21no—
189#380—24no—
190#380C2312yes—
191#380C2312yes—
192#380C2311yescharacteristic
193C4×C4⋊C4—21no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4D411yescharacteristic
200C4×C4⋊C4D411yescharacteristic
201C4×C4⋊C4C2314yes—
202C4×C4⋊C4C2314yes—
203C4×C4⋊C4C2311yescharacteristic
204C4×C4⋊C4C2312yes—
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2311yescharacteristic
208C4×C22⋊C4—24no—
209C4×C22⋊C4—21no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4D412yes—
212C4×C22⋊C4C2312yes—
213#376D412yes—
214C43C2312yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2312yes—
223#340C2312yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2311yescharacteristic
229#336C2314yes—
230#336C2314yes—
231C22×Q8—28no—
232C22×Q8—22no—
233C22×Q8—22no—
234C22×Q8—21no—
235C22×Q8—82no—
236C22×Q8—82no—
237C22×D4—21no—
238C22×D4C2411yescharacteristic
239C23×C4—22no—
240C23×C4—21no—
241C23×C4—22no—
242C23×C4C2×D411yescharacteristic
243C23×C4C2×D411yescharacteristic
244C23×C4C2411yescharacteristic
245C22×C8C2412yes—
246C22×C8C2414yes—
247C22×C8C2412yes—
248C4⋊Q8—28no—
249C4⋊Q8—28no—
250C4⋊Q8—28no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8#4118yes—
254C22⋊Q8—82no—
255#121—22no—
256#121—41no—
257C4×Q8—84no—
258C4×Q8—84no—
259C4×D4—24no—
260C4×D4—24no—
261C4×D4—42no—
262C4×D4—44no—
263#117—24no—
264#117—42no—
265#117—44no—
266#117—84no—
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—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D411yescharacteristic
288C2×C4⋊C4C2×D411yescharacteristic
289C2×C4⋊C4C2×D411yescharacteristic
290C2×C4⋊C4C2×D411yescharacteristic
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2411yescharacteristic
293C2×C4⋊C4C2411yescharacteristic
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C4⋊C4C2411yescharacteristic
297C2×C4⋊C4C2412yes—
298C2×C4⋊C4C2414yes—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C42—24no—
312C2×C42—22no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42C2×D412yes—
318C2×C42C2×D412yes—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D412yes—
322C2×C42C2×D412yes—
323C2×C42C2412yes—
324C2×C42C2411yescharacteristic
325C2×C42C2412yes—
326C2×C42C2411yescharacteristic
327#107—28no—
328#107—28no—
329#107—44no—
330#106—24no—
331#106—28no—
332#106—42no—
333C4⋊C8—28no—
334C4⋊C8—24no—
335#102—24no—
336#102—24no—
337#102—28no—
338C22⋊C8—24no—
339C22⋊C8—42no—
340#97—44no—
341C4×C8—24no—
342#95—24no—
343#95—24no—
344#95—22no—
345#95—21no—
346#95—22no—
347#95—21no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—24no—
352#95—24no—
353#95—21no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95C2×D412yes—
362#95C2×D412yes—
363#95C2×D412yes—
364#95C2×D412yes—
365C24—21no—
366C24C22×D411yescharacteristic
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—44no—
370C2×Q8—42no—
371C2×Q8—44no—
372C2×Q8—82no—
373C2×Q8—82no—
374C2×Q8—81no—
375C2×Q8—81no—
376C2×Q8—816no—
377C2×D4—22no—
378C2×D4—22no—
379C2×D4—21no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—81no—
383C2×D42−1+412yes—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—41no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—82no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4#13714yes—
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C2511yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C82−1+414yes—
430C4⋊C4—28no—
431C4⋊C4—28no—
432C4⋊C4—28no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—81no—
463C4⋊C4C2×#4118yes—
464C22⋊C4—24no—
465C22⋊C4—42no—
466C22⋊C4—44no—
467C22⋊C4—42no—
468C22⋊C4—44no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C42—22no—
477C42—22no—
478C42—24no—
479C42—24no—
480C42—22no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—42no—
485C42—42no—
486C42—42no—
487C42—44no—
488C42—82no—
489C42—84no—
490C42—84no—
491C42C2×#4114yes—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—22no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23C2×#13712yes—
506C23C23×D411yescharacteristic
507Q8—416no—
508Q8—88no—
509Q8—162no—
510D4—24no—
511D4—42no—
512D4—84no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—28no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—28no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—88no—
563C2×C4—88no—
564C2×C4—84no—
565C2×C4#52512yes—
566C2×C4#52512yes—
567C2×C4C2×#13712yes—
568C2×C4C2×#13712yes—
569C2×C4#57512yes—
570C2×C4#57612yes—
571C2×C4C22×#4112yes—
572C2×C4C2×2−1+412yes—
573C2×C4C2×2−1+411yescharacteristic
574C2×C4#58511yescharacteristic
575C8—24no—
576C8—44no—
577C8—42no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—81no—
592C22C2×#52511yescharacteristic
593C22#348911yescharacteristic
594C22#349911yescharacteristic
595C22C22×#13711yescharacteristic
596C22C2×#57511yescharacteristic
597C22C2×#57611yescharacteristic
598C22#358511yescharacteristic
599C4—22no—
600C4—22no—
601C4—22no—
602C4—42no—
603C4—42no—
604C4—42no—
605C4—41no—
606C4—48no—
607C4—42no—
608C4—81no—
609C4—82no—
610C4—81no—
611C4—81no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4#311714yes—
616C4#351612yes—
617C4#352312yes—
618C2—42no—
619C2—82no—
620C2#3599711yescharacteristic
621C2C2×#311711yescharacteristic
622C2#6057811yescharacteristic
623C2#6210811yescharacteristic
624C2#6214311yescharacteristic
625C2#6248011yescharacteristic
626C2#6250811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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
2. #34216C2semidirect
3. #30093C2semidirect
4. #29939C2semidirect
5. #29388C2semidirect
6. #29317C2semidirect
7. #27476C2semidirect
9. #26511C2semidirect
10. #25813C2semidirect
11. #25450C2semidirect
13. #23780C2semidirect
15. #23162C2semidirect
16. #22969C2semidirect
17. #22643C2semidirect

Maximal subgroups

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

subgroupindex
#363012
#342162
#300932
#299392
#293882
#293172
#274762
#271782
#265112
#258132
#254502
#242992
#237802
#237552
#231622
#229692
#226432