Finite Group Databasebeta

← #6544581

Subgroups — #6544581

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#36480C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#30132C212yesmaximal, complemented
4#29691C211yesmaximal, characteristic
5#29520C212yesmaximal, complemented
6#29272C212yesmaximal, complemented
7#27960C211yesmaximal, characteristic
8#27680C211yesmaximal, characteristic
9#27053C214yesmaximal
10#26743C214yesmaximal
11#26022C212yesmaximal, complemented
12#24682C212yesmaximal, complemented
13#23823C212yesmaximal, complemented
14#23571C211yesmaximal, characteristic
15#22971C212yesmaximal, complemented
16#22763C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#477C2212yes—
20C2×C4×Q16C2212yes—
21#2863C2212yes—
22#2857C2211yescharacteristic
23#2605—22no—
24#2567C2211yescharacteristic
25#2551C2211yescharacteristic
26#2509—21no—
27#2497C2212yes—
28#2492C2212yes—
29#2465—22no—
30#2439—21no—
31#2369C2211yescharacteristic
32#2365—21no—
33#2360—22no—
34#2360C2214yes—
35#2355—21no—
36#2355C2212yes—
37#2355C2211yescharacteristic
38Q8×C4⋊C4—21no—
39Q8×C4⋊C4C2212yes—
40#2347—21no—
41#2330—22no—
42#2317—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×C4⋊Q8C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2274—21no—
49C4×#117C2211yescharacteristic
50#2099C2212yes—
51#2099C2212yes—
52#2099C2212yes—
53#2096C2212yes—
54#2072C2214yes—
55#2066C2214yes—
56#2057C2211yescharacteristic
57#2057C2212yes—
58#2056C2214yes—
59#2043C2211yescharacteristic
60#2043C2212yes—
61#2036C2212yes—
62#2036C2214yes—
63#2024C2214yes—
64#2024C2211yescharacteristic
65#2016C2211yescharacteristic
66#2001C2212yes—
67#1990C2212yes—
68#1988C2214yes—
69#1948C2212yes—
70#1947C2214yes—
71#1943C2214yes—
72#1939C2212yes—
73#1935C2214yes—
74#1928C2214yes—
75#1924C2214yes—
76#1919C2214yes—
77#1918C2214yes—
78#1918C2214yes—
79#1878C2214yes—
80#1872C2212yes—
81#1865C2214yes—
82#1853C2212yes—
83#1834C2212yes—
84#1831C2212yes—
85#1823C2214yes—
86#1803C2212yes—
87#1803C2212yes—
88#1797C2211yescharacteristic
89#1770C2212yes—
90#1763C2214yes—
91#1763C2214yes—
92C4×#103C2212yes—
93C4×C22⋊C8C2211yescharacteristic
94#1745C2212yes—
95C22×Q16C2312yes—
96#533—28no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2312yes—
102C2×#125—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2311yescharacteristic
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8—22no—
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2314yes—
113C2×#117—21no—
114C2×#117D411yescharacteristic
115C2×#117D411yescharacteristic
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120#477—28no—
121C4×Q16—28no—
122C2×#107C2314yes—
123C2×#107C2312yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2314yes—
137C2×C4×C8C2312yes—
138#400—22no—
139#400—22no—
140#396—22no—
141#395—22no—
142#395—22no—
143#393—22no—
144#391—22no—
145#391—22no—
146#391—21no—
147#391—22no—
148#391—22no—
149#391C2314yes—
150#391C2312yes—
151#391C2311yescharacteristic
152#391C2312yes—
153#389—21no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—21no—
162#389—24no—
163#389—24no—
164#389C2311yescharacteristic
165#389C2312yes—
166#389C2314yes—
167#389C2312yes—
168#389C2312yes—
169#388—22no—
170#388—22no—
171#387—21no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—24no—
176#387—24no—
177#387—22no—
178#387—22no—
179#385—21no—
180#385—24no—
181#384—22no—
182#384—21no—
183#384—21no—
184#384C2312yes—
185#384C2312yes—
186#384C2311yescharacteristic
187#384C2312yes—
188#383—22no—
189#382—22no—
190#382—21no—
191#380—21no—
192#380C2312yes—
193#380C2311yescharacteristic
194C4×C4⋊C4—21no—
195C4×C4⋊C4—21no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2314yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C22⋊C4—21no—
212C4×C22⋊C4—24no—
213C4×C22⋊C4D412yes—
214C4×C22⋊C4C2312yes—
215#376D412yes—
216C43C2312yes—
217#340C2314yes—
218#340C2314yes—
219#340C2312yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#336C2312yes—
227#336C2311yescharacteristic
228#336C2314yes—
229#336C2314yes—
230#336C2311yescharacteristic
231C22×Q8—21no—
232C22×Q8—22no—
233C22×Q8—21no—
234C22×Q8C2411yescharacteristic
235C22×Q8C2412yes—
236C23×C4—22no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2411yescharacteristic
240C2×Q16—24no—
241C2×Q16—42no—
242C2×Q16—42no—
243C22×C8C2412yes—
244C22×C8C2412yes—
245C22×C8C2414yes—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248C4⋊Q8—24no—
249C4⋊Q8—24no—
250C4⋊Q8—42no—
251C4⋊Q8—44no—
252#125—28no—
253C22⋊Q8—28no—
254C22⋊Q8—24no—
255C22⋊Q8—24no—
256C22⋊Q8—22no—
257C22⋊Q8—41no—
258C22⋊Q8—82no—
259C4×Q8—28no—
260C4×Q8—24no—
261C4×Q8—24no—
262C4×Q8—44no—
263C4×Q8—44no—
264C4×Q8—84no—
265C4×Q8—84no—
266C4×Q8#4118yes—
267#117—28no—
268#117—24no—
269#117—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—82no—
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2×D411yescharacteristic
299C2×C4⋊C4C2×D411yescharacteristic
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2414yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2411yescharacteristic
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C22⋊C4—22no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4C2×D412yes—
313C2×C22⋊C4C2×D412yes—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D411yescharacteristic
327C2×C42C2×D412yes—
328C2×C42C2412yes—
329C2×C42C2411yescharacteristic
330C2×C42C2412yes—
331C2×C42C2411yescharacteristic
332#107—24no—
333#107—28no—
334#106—44no—
335C4⋊C8—24no—
336C4⋊C8—44no—
337#103—24no—
338#103—28no—
339#103—24no—
340#103—24no—
341#103—44no—
342#103—42no—
343#103—44no—
344C22⋊C8—42no—
345C22⋊C8—42no—
346#97—44no—
347C4×C8—24no—
348#95—24no—
349#95—24no—
350#95—21no—
351#95—21no—
352#95—22no—
353#95—21no—
354#95—24no—
355#95—22no—
356#95—22no—
357#95—21no—
358#95—24no—
359#95—22no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370#95C2×D412yes—
371C24C22×D411yescharacteristic
372C2×Q8—24no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—22no—
376C2×Q8—21no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—21no—
382C2×Q8—44no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—44no—
386C2×Q8—42no—
387C2×Q8—82no—
388C2×Q8—81no—
389C2×Q8—81no—
390C2×Q8C2×#4114yes—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—41no—
409C22×C4—44no—
410C22×C4—82no—
411C22×C4—82no—
412C22×C4#13714yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C2511yescharacteristic
423Q16—48no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—24no—
431C2×C8—22no—
432C2×C8—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—28no—
436C4⋊C4—24no—
437C4⋊C4—28no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—44no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—84no—
466C4⋊C4—82no—
467C4⋊C4—84no—
468C4⋊C4—81no—
469C4⋊C4#13714yes—
470C4⋊C4C2×#4114yes—
471C4⋊C42−1+412yes—
472C22⋊C4—28no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—42no—
476C22⋊C4—81no—
477C22⋊C4—82no—
478C42—22no—
479C42—22no—
480C42—28no—
481C42—24no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—24no—
487C42—22no—
488C42—42no—
489C42—42no—
490C42—42no—
491C42—84no—
492C42—84no—
493C42—82no—
494C42—82no—
495C42C2×#4114yes—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C2×#13712yes—
503C23C23×D411yescharacteristic
504Q8—24no—
505Q8—24no—
506Q8—48no—
507Q8—42no—
508Q8—42no—
509Q8—42no—
510Q8—42no—
511Q8—88no—
512Q8—162no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—28no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—164no—
565C2×C4#52512yes—
566C2×C4#52512yes—
567C2×C4C2×#13712yes—
568C2×C4C2×#13714yes—
569C2×C4C2×#13712yes—
570C2×C4C2×#13712yes—
571C2×C4#57612yes—
572C2×C4C22×#4112yes—
573C2×C4C2×2−1+411yescharacteristic
574C2×C4#58511yescharacteristic
575C2×C4#58512yes—
576C8—44no—
577C8—42no—
578C8—42no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22C2×#52511yescharacteristic
587C22#348911yescharacteristic
588C22#350011yescharacteristic
589C22C22×#13711yescharacteristic
590C22C22×#13711yescharacteristic
591C22C2×#57611yescharacteristic
592C22#358611yescharacteristic
593C4—22no—
594C4—22no—
595C4—22no—
596C4—24no—
597C4—42no—
598C4—42no—
599C4—42no—
600C4—44no—
601C4—41no—
602C4—44no—
603C4—42no—
604C4—84no—
605C4—81no—
606C4—81no—
607C4—82no—
608C4—81no—
609C4—82no—
610C4#311314yes—
611C4#349312yes—
612C4#350712yes—
613C2—42no—
614C2#3575911yescharacteristic
615C2C2×#311311yescharacteristic
616C2#6058411yescharacteristic
617C2#6211311yescharacteristic
618C2#6212711yescharacteristic
619C2#6253711yescharacteristic
620C2#6258911yescharacteristic
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. #30132C2semidirect
5. #29520C2semidirect
6. #29272C2semidirect
11. #26022C2semidirect
12. #24682C2semidirect
13. #23823C2semidirect
15. #22971C2semidirect
16. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#364802
#342142
#301322
#296912
#295202
#292722
#279602
#276802
#270532
#267432
#260222
#246822
#238232
#235712
#229712
#227632
#226362