Finite Group Databasebeta

← #6798777

Subgroups — #6798777

Order 512.

1438 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2D4×#125C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31764C214yesmaximal, complemented
5#31762C214yesmaximal, complemented
6#29852C212yesmaximal, complemented
7#29213C211yesmaximal, characteristic, complemented
8#28725C211yesmaximal, characteristic, complemented
9#27881C212yesmaximal, complemented
10#26027C212yesmaximal, complemented
11#26023C212yesmaximal, complemented
12#25912C212yesmaximal, complemented
13#24971C212yesmaximal
14#24969C212yesmaximal
15#21838C212yesmaximal, complemented
16C4×#491C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic
18C22×#125C2212yes—
19C2×#491C2211yescharacteristic
20C2×#491C2212yes—
21C2×#432C2212yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2851C2212yes—
26#2571—21no—
27#2563—21no—
28#2552C2211yescharacteristic
29#2552C2211yescharacteristic
30#2538C2212yes—
31#2506—21no—
32#2506—21no—
33#2497—22no—
34#2492C2214yes—
35#2473—22no—
36#2473—22no—
37#2442C2212yes—
38#2442C2212yes—
39#2439—22no—
40#2369—21no—
41#2355C2212yes—
42#2354—21no—
43D4×C4⋊C4—21no—
44D4×C4⋊C4—21no—
45D4×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#125C2211yescharacteristic
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2077C2214yes—
52#2076C2214yes—
53#2074C2214yes—
54#2068C2214yes—
55#2066C2214yes—
56#2058C2212yes—
57#2057C2212yes—
58#2036C2214yes—
59#2035C2214yes—
60#2026C2214yes—
61#2026C2214yes—
62#1993C2214yes—
63#1992C2212yes—
64#1992C2212yes—
65#1992C2212yes—
66#1992C2212yes—
67#1992C2211yescharacteristic
68#1989C2214yes—
69#1989C2214yes—
70#1960C2211yescharacteristic
71#1954C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1935C2212yes—
76#1935C2212yes—
77#1924C2214yes—
78#1920C2214yes—
79#1918C2214yes—
80#1872C2214yes—
81#1872C2214yes—
82#1847C2212yes—
83#1836C2211yescharacteristic
84#1834C2212yes—
85#1834C2212yes—
86#1824C2214yes—
87#1823C2214yes—
88#1803C2212yes—
89#1803C2212yes—
90#1767C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2312yes—
99C2×#125—28no—
100C2×#125—22no—
101C2×#125—22no—
102C2×#125—44no—
103C2×#125C2312yes—
104C2×#125C2312yes—
105C2×#125C2312yes—
106C2×#125C2311yescharacteristic
107C2×C22⋊Q8C2314yes—
108C2×C4×Q8C2312yes—
109C2×C4×D4—21no—
110C2×C4×D4—21no—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D411yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×#117—22no—
115C22×C4⋊C4—22no—
116C22×C4⋊C4—22no—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C42C2312yes—
120#491—24no—
121#491—44no—
122#432—44no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2312yes—
128C2×#103C2312yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2312yes—
132C2×#103C2312yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×#97C2311yescharacteristic
136C2×#97C2312yes—
137C2×#97C2311yescharacteristic
138C2×#97C2312yes—
139C2×C4×C8C2312yes—
140#400—24no—
141#400—22no—
142#400—22no—
143#398—22no—
144#398—22no—
145#398—22no—
146#398—22no—
147#396—24no—
148#393—24no—
149#391—24no—
150#391C2312yes—
151#391C2314yes—
152#390—22no—
153#389—22no—
154#389—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—21no—
159#389—21no—
160#389—22no—
161#389—22no—
162#389—21no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389C2311yescharacteristic
167#389C2311yescharacteristic
168#389C2312yes—
169#389C2312yes—
170#388—24no—
171#387—21no—
172#387—21no—
173#387—22no—
174#387—22no—
175#387—24no—
176#385—22no—
177#385—22no—
178#385—22no—
179#385—22no—
180#385—22no—
181#385—22no—
182#384—21no—
183#384—21no—
184#384—21no—
185#384—21no—
186#384C2311yescharacteristic
187#384C2311yescharacteristic
188#384C2311yescharacteristic
189#384C2312yes—
190#384C2312yes—
191#382—22no—
192#382—22no—
193#382C2311yescharacteristic
194#382C2312yes—
195#380—22no—
196#380—22no—
197#380—22no—
198#380—22no—
199#380—22no—
200#380C2312yes—
201#380C2314yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—21no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4D411yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2311yescharacteristic
215C4×C22⋊C4D412yes—
216C4×C22⋊C4D412yes—
217C4×C22⋊C4C2312yes—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2314yes—
224#340C2314yes—
225#340C2312yes—
226#340C2314yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#340C2312yes—
231#336C2312yes—
232#336C2314yes—
233C22×Q8—22no—
234C22×Q8C2412yes—
235C22×D4C2×D411yescharacteristic
236C23×C4—22no—
237C23×C4—22no—
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2412yes—
241C22×C8C2412yes—
242C22×C8C2414yes—
243C22×C8C2412yes—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8—44no—
247C4⋊Q8—44no—
248C4⋊Q8—82no—
249#125—24no—
250#125—24no—
251#125—24no—
252#125—22no—
253#125—24no—
254#125—42no—
255#125—42no—
256#125—48no—
257#125—48no—
258#125#4114yes—
259C22⋊Q8—44no—
260C22⋊Q8—44no—
261C4×Q8—24no—
262C4×Q8—24no—
263C4×D4—24no—
264C4×D4—24no—
265C4×D4—42no—
266C4×D4—44no—
267C4×D4—44no—
268#117—84no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—48no—
303C2×C4⋊C4—48no—
304C2×C4⋊C4C2×D411yescharacteristic
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C4⋊C4C2411yescharacteristic
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2411yescharacteristic
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4C2×D412yes—
320C2×C22⋊C4C2×D412yes—
321C2×C22⋊C4C2×D412yes—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2412yes—
335C2×C42C2411yescharacteristic
336C2×C42C2411yescharacteristic
337C2×C42C2411yescharacteristic
338#107—44no—
339#106—44no—
340C4⋊C8—24no—
341C4⋊C8—44no—
342#103—24no—
343#103—24no—
344#103—42no—
345#103—42no—
346#103—44no—
347#103—44no—
348C22⋊C8—44no—
349C22⋊C8—44no—
350#97—22no—
351#97—22no—
352#97—24no—
353#97—42no—
354C4×C8—24no—
355#95—24no—
356#95—24no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—22no—
362#95—22no—
363#95—24no—
364#95—21no—
365#95—21no—
366#95—24no—
367#95—22no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95C2×D412yes—
374#95C2×D412yes—
375#95C2×D412yes—
376C24C22×D412yes—
377C2×Q8—22no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—22no—
381C2×Q8—22no—
382C2×Q8—22no—
383C2×Q8—44no—
384C2×Q8—82no—
385C2×Q8—82no—
386C2×D4—22no—
387C2×D4—22no—
388C2×D4#13714yes—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—82no—
413C22×C4—82no—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D412yes—
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423C22×C4C2511yescharacteristic
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—24no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—24no—
433C4⋊C4—22no—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—44no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—44no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—82no—
472C4⋊C4—81no—
473C4⋊C4—82no—
474C4⋊C4—84no—
475C4⋊C4—82no—
476C4⋊C4—82no—
477C4⋊C4C2×#4112yes—
478C4⋊C4C2×#4112yes—
479C22⋊C4—24no—
480C22⋊C4—24no—
481C22⋊C4—42no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—84no—
485C22⋊C4—84no—
486C42—24no—
487C42—24no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—22no—
492C42—22no—
493C42—41no—
494C42—41no—
495C42—44no—
496C42—42no—
497C42—42no—
498C42—84no—
499C42C2×#4112yes—
500C42H(5,𝔽2)12yes—
501C422−1+414yes—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23C2×#13714yes—
509C23C23×D411yescharacteristic
510Q8—42no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—164no—
515D4—28no—
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—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—48no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
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—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—84no—
568C2×C4—88no—
569C2×C4—88no—
570C2×C4#53612yes—
571C2×C4#53612yes—
572C2×C4#54612yes—
573C2×C4#54612yes—
574C2×C4C2×#13712yes—
575C2×C4#57512yes—
576C2×C4C22×#4111yescharacteristic
577C2×C4C2×H(5,𝔽2)11yescharacteristic
578C2×C4C2×2−1+412yes—
579C2×C4#58512yes—
580C8—42no—
581C8—42no—
582C8—44no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22—42no—
590C22C2×#53611yescharacteristic
591C22C2×#54611yescharacteristic
592C22#350011yescharacteristic
593C22C22×#13711yescharacteristic
594C22C2×#57511yescharacteristic
595C22#358511yescharacteristic
596C22#358611yescharacteristic
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—42no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—82no—
608C4—81no—
609C4—81no—
610C4—84no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4#323614yes—
615C4#351812yes—
616C4#354712yes—
617C2—44no—
618C2#3895811yescharacteristic
619C2C2×#323611yescharacteristic
620C2#6088611yescharacteristic
621C2#6116211yescharacteristic
622C2#6116511yescharacteristic
623C2#6257711yescharacteristic
624C2#6259011yescharacteristic
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. #33297C2semidirect
4. #31764C2semidirect
5. #31762C2semidirect
6. #29852C2semidirect
7. #29213C2semidirect
8. #28725C2semidirect
9. #27881C2semidirect
10. #26027C2semidirect
11. #26023C2semidirect
12. #25912C2semidirect
15. #21838C2semidirect
16. C4×#491C2semidirect

Maximal subgroups

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

subgroupindex
#361942
D4×#1252
#332972
#317642
#317622
#298522
#292132
#287252
#278812
#260272
#260232
#259122
#249712
#249692
#218382
C4×#4912
#212702