Finite Group Databasebeta

← #6544582

Subgroups — #6544582

Order 512.

1423 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36450C211yesmaximal, characteristic
2#34214C211yesmaximal, characteristic
3#30131C212yesmaximal, complemented
4#29688C211yesmaximal, characteristic
5#29519C212yesmaximal, complemented
6#29278C212yesmaximal, complemented
7#27961C211yesmaximal, characteristic
8#27681C211yesmaximal, characteristic
9#27052C214yesmaximal
10#26740C214yesmaximal
11#26023C212yesmaximal, complemented
12#24683C212yesmaximal, complemented
13#23825C212yesmaximal, complemented
14#23571C211yesmaximal, characteristic
15#22970C212yesmaximal, complemented
16#22763C212yesmaximal, complemented
17#22636C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#475C2212yes—
20C2×C4×Q16C2212yes—
21#2863C2212yes—
22#2857C2211yescharacteristic
23#2577—22no—
24#2568C2211yescharacteristic
25#2552C2211yescharacteristic
26#2528—22no—
27#2514—21no—
28#2499—21no—
29#2494C2212yes—
30#2489C2212yes—
31#2365C2211yescharacteristic
32#2362—21no—
33#2360—22no—
34#2360C2214yes—
35#2355—21no—
36#2355—21no—
37#2355C2211yescharacteristic
38#2355C2212yes—
39Q8×C4⋊C4—21no—
40Q8×C4⋊C4C2212yes—
41#2330—22no—
42#2317—21no—
43C4×C4⋊Q8C2212yes—
44C4×C4⋊Q8C2212yes—
45C4×#125C2212yes—
46C4×C22⋊Q8C2211yescharacteristic
47C2×#384C2211yescharacteristic
48#2274—21no—
49C4×#117C2211yescharacteristic
50#2100C2212yes—
51#2100C2212yes—
52#2100C2212yes—
53#2095C2212yes—
54#2074C2214yes—
55#2068C2214yes—
56#2058C2212yes—
57#2057C2211yescharacteristic
58#2056C2214yes—
59#2043C2211yescharacteristic
60#2043C2212yes—
61#2036C2212yes—
62#2035C2214yes—
63#2026C2214yes—
64#2024C2211yescharacteristic
65#2016C2211yescharacteristic
66#2001C2212yes—
67#1990C2212yes—
68#1989C2214yes—
69#1949C2214yes—
70#1947C2212yes—
71#1944C2214yes—
72#1939C2212yes—
73#1935C2214yes—
74#1927C2214yes—
75#1924C2214yes—
76#1919C2214yes—
77#1918C2214yes—
78#1918C2214yes—
79#1878C2214yes—
80#1872C2212yes—
81#1863C2214yes—
82#1852C2212yes—
83#1834C2212yes—
84#1831C2212yes—
85#1824C2214yes—
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⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2311yescharacteristic
100C2×#125—21no—
101C2×#125C2312yes—
102C2×C22⋊Q8—21no—
103C2×C22⋊Q8—21no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2311yescharacteristic
106C2×C4×Q8—21no—
107C2×C4×Q8—21no—
108C2×C4×Q8—22no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2314yes—
112C2×#117—21no—
113C2×#117D411yescharacteristic
114C2×#117D411yescharacteristic
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4C2312yes—
118C22×C42C2311yescharacteristic
119#475—28no—
120C4×Q16—28no—
121C2×#107C2312yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2312yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2312yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×#97C2314yes—
136C2×C4×C8C2312yes—
137#401—22no—
138#400—22no—
139#400—22no—
140#398—22no—
141#397—22no—
142#397—22no—
143#391—21no—
144#391—21no—
145#391—21no—
146#391—22no—
147#391—21no—
148#391C2314yes—
149#391C2312yes—
150#391C2311yescharacteristic
151#391C2312yes—
152#389—21no—
153#389—22no—
154#389—22no—
155#389—22no—
156#389—21no—
157#389—24no—
158#389—24no—
159#389C2311yescharacteristic
160#389C2312yes—
161#389C2314yes—
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—21no—
172#387—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#387—22no—
177#385—21no—
178#385—24no—
179#384—22no—
180#384—21no—
181#384—21no—
182#384—21no—
183#384C2312yes—
184#384C2311yescharacteristic
185#384C2312yes—
186#383—22no—
187#382—22no—
188#382—21no—
189#382—22no—
190#382C2312yes—
191#380—21no—
192#380C2312yes—
193#380C2311yescharacteristic
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
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⋊C4C2312yes—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2314yes—
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#340C2312yes—
219#340C2314yes—
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—21no—
233C22×Q8C2411yescharacteristic
234C22×Q8C2412yes—
235C23×C4—22no—
236C23×C4C2×D411yescharacteristic
237C23×C4C2×D411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×Q16—24no—
240C2×Q16—42no—
241C2×Q16—42no—
242C22×C8C2412yes—
243C22×C8C2412yes—
244C22×C8C2414yes—
245C4⋊Q8—24no—
246C4⋊Q8—24no—
247C4⋊Q8—24no—
248#125—28no—
249#125—24no—
250#125—42no—
251C22⋊Q8—28no—
252C22⋊Q8—24no—
253C22⋊Q8—24no—
254C22⋊Q8—22no—
255C22⋊Q8—41no—
256C22⋊Q8—44no—
257C4×Q8—28no—
258C4×Q8—24no—
259C4×Q8—24no—
260C4×Q8—44no—
261C4×Q8—44no—
262C4×Q8—44no—
263C4×Q8—84no—
264C4×Q8#4118yes—
265#117—28no—
266#117—24no—
267#117—42no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—82no—
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2×D411yescharacteristic
297C2×C4⋊C4C2×D411yescharacteristic
298C2×C4⋊C4C2414yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2412yes—
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2411yescharacteristic
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—21no—
314C2×C42—22no—
315C2×C42—21no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42C2×D412yes—
320C2×C42C2×D411yescharacteristic
321C2×C42C2×D411yescharacteristic
322C2×C42C2×D412yes—
323C2×C42C2×D412yes—
324C2×C42C2×D412yes—
325C2×C42C2412yes—
326C2×C42C2412yes—
327C2×C42C2411yescharacteristic
328C2×C42C2411yescharacteristic
329#107—24no—
330#107—44no—
331#106—28no—
332C4⋊C8—24no—
333C4⋊C8—44no—
334#103—24no—
335#103—28no—
336#103—24no—
337#103—24no—
338#103—44no—
339#103—42no—
340#103—44no—
341C22⋊C8—42no—
342C22⋊C8—42no—
343#97—44no—
344C4×C8—24no—
345#95—21no—
346#95—24no—
347#95—24no—
348#95—21no—
349#95—22no—
350#95—21no—
351#95—24no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—21no—
356#95—24no—
357#95—22no—
358#95—21no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95C2×D412yes—
367#95C2×D412yes—
368#95C2×D412yes—
369#95C2×D412yes—
370C24C22×D411yescharacteristic
371C2×Q8—24no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8—22no—
375C2×Q8—21no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—21no—
381C2×Q8—42no—
382C2×Q8—42no—
383C2×Q8—42no—
384C2×Q8—42no—
385C2×Q8—42no—
386C2×Q8C2×#4114yes—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—41no—
405C22×C4—44no—
406C22×C4—82no—
407C22×C4—82no—
408C22×C4#13714yes—
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C2511yescharacteristic
419Q16—48no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—22no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—28no—
432C4⋊C4—24no—
433C4⋊C4—28no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—24no—
447C4⋊C4—42no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—82no—
461C4⋊C4—82no—
462C4⋊C4—82no—
463C4⋊C4—84no—
464C4⋊C4—84no—
465C4⋊C4C2×#4114yes—
466C4⋊C42−1+412yes—
467C22⋊C4—28no—
468C22⋊C4—24no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—44no—
472C22⋊C4—81no—
473C42—22no—
474C42—22no—
475C42—28no—
476C42—24no—
477C42—24no—
478C42—22no—
479C42—22no—
480C42—24no—
481C42—22no—
482C42—42no—
483C42—42no—
484C42—44no—
485C42—42no—
486C42—42no—
487C42—82no—
488C42—84no—
489C42—84no—
490C42C2×#4114yes—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23C2×#13712yes—
498C23C23×D411yescharacteristic
499Q8—24no—
500Q8—24no—
501Q8—48no—
502Q8—42no—
503Q8—42no—
504Q8—42no—
505Q8—42no—
506Q8—84no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—28no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—44no—
543C2×C4—42no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—41no—
559C2×C4—164no—
560C2×C4#52512yes—
561C2×C4#52512yes—
562C2×C4C2×#13712yes—
563C2×C4C2×#13712yes—
564C2×C4#57512yes—
565C2×C4#57812yes—
566C2×C4C22×#4112yes—
567C2×C4C2×2−1+411yescharacteristic
568C2×C4#58511yescharacteristic
569C2×C4#58512yes—
570C8—44no—
571C8—42no—
572C8—42no—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22C2×#52511yescharacteristic
581C22#348911yescharacteristic
582C22#350011yescharacteristic
583C22C22×#13711yescharacteristic
584C22C2×#57511yescharacteristic
585C22C2×#57811yescharacteristic
586C22#358511yescharacteristic
587C4—22no—
588C4—22no—
589C4—22no—
590C4—42no—
591C4—42no—
592C4—42no—
593C4—44no—
594C4—41no—
595C4—44no—
596C4—42no—
597C4—42no—
598C4—82no—
599C4—81no—
600C4—84no—
601C4—82no—
602C4—81no—
603C4—81no—
604C4#311714yes—
605C4#349312yes—
606C4#350712yes—
607C2—42no—
608C2#3575911yescharacteristic
609C2C2×#311711yescharacteristic
610C2#6058911yescharacteristic
611C2#6211211yescharacteristic
612C2#6213311yescharacteristic
613C2#6253611yescharacteristic
614C2#6259011yescharacteristic
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. #30131C2semidirect
5. #29519C2semidirect
6. #29278C2semidirect
11. #26023C2semidirect
12. #24683C2semidirect
13. #23825C2semidirect
15. #22970C2semidirect
16. #22763C2semidirect

Maximal subgroups

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

subgroupindex
#364502
#342142
#301312
#296882
#295192
#292782
#279612
#276812
#270522
#267402
#260232
#246832
#238252
#235712
#229702
#227632
#226362