Finite Group Databasebeta

← #6565052

Subgroups — #6565052

Order 512.

1447 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34324C211yesmaximal, characteristic
3#30109C212yesmaximal, complemented
4#29984C211yesmaximal, characteristic, complemented
5#29338C212yesmaximal, complemented
6#29218C211yesmaximal, characteristic, complemented
7#28205C211yesmaximal, characteristic, complemented
8#27612C212yesmaximal, complemented
9#27352C214yesmaximal, complemented
10#26750C214yesmaximal, complemented
11#24970C212yesmaximal
12#24123C212yesmaximal
13#23762C212yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23068C212yesmaximal, complemented
16#22839C211yesmaximal, characteristic
17#22763C212yesmaximal, complemented
18C2×#556C2211yescharacteristic
19C2×#475C2212yes—
20C2×C4×Q16C2212yes—
21#2868C2211yescharacteristic
22#2857C2212yes—
23#2571—21no—
24#2563—21no—
25#2538C2212yes—
26#2497—22no—
27#2492C2214yes—
28#2439—22no—
29#2373—22no—
30#2369—21no—
31#2367—21no—
32#2365C2211yescharacteristic
33#2362C2212yes—
34#2355—21no—
35#2355C2212yes—
36#2354—21no—
37#2354C2211yescharacteristic
38Q8×C4⋊C4C2211yescharacteristic
39#2346C2212yes—
40#2334—22no—
41#2326—21no—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C22⋊Q8C2212yes—
45#2296C2212yes—
46#2295C2211yescharacteristic
47#2285C2212yes—
48#2276—21no—
49C42×D4C2211yescharacteristic
50#2100C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2095C2212yes—
54#2084C2212yes—
55#2083C2211yescharacteristic
56#2075C2214yes—
57#2072C2211yescharacteristic
58#2068C2212yes—
59#2057C2212yes—
60#2056C2212yes—
61#2043C2212yes—
62#2043C2211yescharacteristic
63#2036C2214yes—
64#2035C2214yes—
65#2024C2212yes—
66#2016C2211yescharacteristic
67#2001C2214yes—
68#1990C2212yes—
69#1987C2214yes—
70#1954C2212yes—
71#1947C2212yes—
72#1935C2214yes—
73#1935C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1924C2214yes—
77#1920C2214yes—
78#1918C2214yes—
79#1878C2214yes—
80#1873C2214yes—
81#1873C2214yes—
82#1854C2214yes—
83#1834C2212yes—
84#1831C2211yescharacteristic
85#1825C2214yes—
86#1817C2212yes—
87#1803C2212yes—
88#1803C2212yes—
89#1776C2214yes—
90#1770C2212yes—
91C4×C4⋊C8C2211yescharacteristic
92#1763C2214yes—
93C4×#103C2212yes—
94#1749C2212yes—
95C22×Q16C2312yes—
96#556—24no—
97#556—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#126—22no—
104C2×#124—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×C4×Q8—21no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2311yescharacteristic
112C2×C4×Q8C2312yes—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C2×#117C2312yes—
118C2×#117C2312yes—
119C22×C42C2312yes—
120#475—28no—
121C4×Q16—28no—
122C2×#107C2312yes—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2314yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138#400—24no—
139#396—24no—
140#393—24no—
141#391—21no—
142#391—21no—
143#391—24no—
144#391C2312yes—
145#391C2311yescharacteristic
146#391C2311yescharacteristic
147#391C2314yes—
148#390—22no—
149#390—21no—
150#389—22no—
151#389—21no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2311yescharacteristic
159#389C2311yescharacteristic
160#389C2312yes—
161#389C2312yes—
162#388—22no—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—22no—
167#388—24no—
168#387—21no—
169#387—22no—
170#387—21no—
171#387—22no—
172#387—22no—
173#387—24no—
174#385—22no—
175#385—22no—
176#385—22no—
177#384—21no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#384C2312yes—
184#382—22no—
185#382—22no—
186#382—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380C2312yes—
191#380C2314yes—
192C4×C4⋊C4—21no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—22no—
212C4×C22⋊C4D412yes—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376—21no—
216#376—22no—
217C43C2311yescharacteristic
218#340C2312yes—
219#340C2314yes—
220#340C2314yes—
221#340C2312yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2314yes—
226#340C2314yes—
227#336C2312yes—
228#336C2311yescharacteristic
229#336C2311yescharacteristic
230#336C2312yes—
231#336C2314yes—
232C22×Q8—21no—
233C22×Q8—22no—
234C22×Q8C2412yes—
235C22×Q8C2411yescharacteristic
236C22×D4C2×D411yescharacteristic
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2412yes—
240C2×Q16—24no—
241C2×Q16—42no—
242C2×Q16—42no—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C22×C8C2412yes—
246C4⋊Q8—22no—
247C4⋊Q8—44no—
248C4⋊Q8—42no—
249C4⋊Q8—44no—
250C4⋊Q8—82no—
251C4⋊Q8#4114yes—
252#126—216no—
253#124—28no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—44no—
257C22⋊Q8—44no—
258C4×Q8—28no—
259C4×Q8—22no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—44no—
265C4×Q8#4114yes—
266C4×D4—24no—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—42no—
270#117—24no—
271#117—24no—
272#117—44no—
273#117—84no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
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—24no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2414yes—
303C2×C4⋊C4C2411yescharacteristic
304C2×C4⋊C4C2411yescharacteristic
305C2×C4⋊C4C2412yes—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C22⋊C4C2×D412yes—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—21no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—44no—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D411yescharacteristic
328C2×C42C2×D412yes—
329C2×C42C2×D412yes—
330C2×C42C2×D411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334C2×C42C2411yescharacteristic
335#107—24no—
336#107—44no—
337#106—28no—
338C4⋊C8—24no—
339C4⋊C8—42no—
340#103—24no—
341#103—24no—
342#103—28no—
343#103—42no—
344#103—42no—
345#103—44no—
346#103—44no—
347C22⋊C8—44no—
348C22⋊C8—44no—
349#97—44no—
350C4×C8—24no—
351#95—21no—
352#95—22no—
353#95—21no—
354#95—21no—
355#95—22no—
356#95—21no—
357#95—21no—
358#95—22no—
359#95—24no—
360#95—24no—
361#95—22no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—22no—
367#95—22no—
368#95—21no—
369#95—24no—
370#95C2×D412yes—
371#95C2×D412yes—
372#95C2×D412yes—
373C24C22×D412yes—
374C2×Q8—24no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—44no—
385C2×Q8—42no—
386C2×Q8—42no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×Q8C2×#4112yes—
390C2×D4—22no—
391C2×D4—22no—
392C2×D4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2511yescharacteristic
422Q16—48no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—22no—
427C2×C8—24no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—24no—
431C2×C8—22no—
432C4⋊C4—28no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C4⋊C4—82no—
465C4⋊C4—82no—
466C4⋊C4C2×#4114yes—
467C4⋊C4C2×#4112yes—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—24no—
471C22⋊C4—24no—
472C22⋊C4—42no—
473C22⋊C4—82no—
474C22⋊C4—82no—
475C22⋊C4—84no—
476C22⋊C4—84no—
477C42—28no—
478C42—24no—
479C42—22no—
480C42—22no—
481C42—24no—
482C42—22no—
483C42—22no—
484C42—42no—
485C42—42no—
486C42—41no—
487C42—44no—
488C42—44no—
489C42—44no—
490C42—42no—
491C42—82no—
492C42—84no—
493C42—82no—
494C42#13714yes—
495C42C2×#4112yes—
496C42C2×#4112yes—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23C23×D411yescharacteristic
505Q8—24no—
506Q8—48no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—42no—
512Q8—164no—
513D4—44no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—48no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—41no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—84no—
565C2×C4#52512yes—
566C2×C4#52512yes—
567C2×C4C2×#13712yes—
568C2×C4C2×#13712yes—
569C2×C4C2×#13712yes—
570C2×C4#57512yes—
571C2×C4#57812yes—
572C2×C4C22×#4111yescharacteristic
573C2×C4C22×#4111yescharacteristic
574C2×C4#58512yes—
575C2×C4#58512yes—
576C8—42no—
577C8—42no—
578C8—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22C2×#52511yescharacteristic
587C22#348511yescharacteristic
588C22#350011yescharacteristic
589C22C22×#13711yescharacteristic
590C22C2×#57511yescharacteristic
591C22C2×#57811yescharacteristic
592C22#358511yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—22no—
597C4—42no—
598C4—42no—
599C4—41no—
600C4—44no—
601C4—42no—
602C4—42no—
603C4—42no—
604C4—82no—
605C4—81no—
606C4—81no—
607C4—84no—
608C4—82no—
609C4—82no—
610C4#311614yes—
611C4#349312yes—
612C4#352412yes—
613C2—82no—
614C2#3601111yescharacteristic
615C2C2×#311611yescharacteristic
616C2#6060011yescharacteristic
617C2#6171911yescharacteristic
618C2#6173011yescharacteristic
619C2#6253611yescharacteristic
620C2#6258011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #30109C2semidirect
4. #29984C2semidirect
5. #29338C2semidirect
6. #29218C2semidirect
7. #28205C2semidirect
8. #27612C2semidirect
9. #27352C2semidirect
10. #26750C2semidirect
13. #23762C2semidirect
14. #23679C2semidirect
15. #23068C2semidirect
17. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#343242
#301092
#299842
#293382
#292182
#282052
#276122
#273522
#267502
#249702
#241232
#237622
#236792
#230682
#228392
#227632