Finite Group Databasebeta

← #6798775

Subgroups — #6798775

Order 512.

1428 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
2#36132C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#31765C214yesmaximal, complemented
5#31763C214yesmaximal, complemented
6#29852C212yesmaximal, complemented
7#29213C211yesmaximal, characteristic, complemented
8#28725C211yesmaximal, characteristic, complemented
9#27880C212yesmaximal, complemented
10#26026C212yesmaximal, complemented
11#26022C212yesmaximal, complemented
12#25910C212yesmaximal, complemented
13#24972C212yesmaximal
14#24970C212yesmaximal
15#21839C212yesmaximal, complemented
16C4×#491C211yesmaximal, characteristic, complemented
17#21270C211yesmaximal, characteristic
18C2×#533C2212yes—
19C2×#491C2211yescharacteristic
20C2×#491C2212yes—
21C2×#432C2212yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2851C2212yes—
26#2571—21no—
27#2563—21no—
28#2551C2211yescharacteristic
29#2551C2211yescharacteristic
30#2538C2212yes—
31#2497—22no—
32#2492C2214yes—
33#2465—22no—
34#2465—22no—
35#2452—21no—
36#2452—21no—
37#2439—22no—
38#2439C2212yes—
39#2439C2212yes—
40#2369—21no—
41#2365—21no—
42#2365—21no—
43#2355C2212yes—
44#2354—21no—
45#2354C2211yescharacteristic
46#2354C2211yescharacteristic
47C4×C4⋊Q8C2211yescharacteristic
48C4×#125C2211yescharacteristic
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51#2077C2214yes—
52#2075C2214yes—
53#2074C2214yes—
54#2068C2214yes—
55#2066C2214yes—
56#2058C2212yes—
57#2057C2212yes—
58#2036C2214yes—
59#2035C2214yes—
60#2024C2214yes—
61#2024C2214yes—
62#1993C2214yes—
63#1992C2212yes—
64#1992C2212yes—
65#1992C2212yes—
66#1992C2212yes—
67#1992C2211yescharacteristic
68#1988C2214yes—
69#1987C2214yes—
70#1960C2211yescharacteristic
71#1954C2212yes—
72#1948C2212yes—
73#1947C2212yes—
74#1939C2212yes—
75#1935C2212yes—
76#1935C2212yes—
77#1924C2214yes—
78#1921C2214yes—
79#1919C2214yes—
80#1873C2214yes—
81#1873C2214yes—
82#1847C2212yes—
83#1836C2211yescharacteristic
84#1834C2212yes—
85#1834C2212yes—
86#1825C2214yes—
87#1825C2214yes—
88#1803C2212yes—
89#1803C2212yes—
90#1767C2211yescharacteristic
91C4×#103C2212yes—
92C4×#103C2212yes—
93#1749C2212yes—
94C4×#97C2211yescharacteristic
95#533—28no—
96#533—44no—
97C2×C4⋊Q8—21no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8—21no—
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×C4⋊Q8C2312yes—
104C2×C4⋊Q8C2312yes—
105C2×#125C2312yes—
106C2×#125C2311yescharacteristic
107C2×C22⋊Q8—22no—
108C2×C22⋊Q8—22no—
109C2×C22⋊Q8C2314yes—
110C2×C4×Q8—22no—
111C2×C4×Q8—22no—
112C2×C4×Q8C2312yes—
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C2×#117C2312yes—
118C2×#117C2312yes—
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#396—24no—
142#396—22no—
143#396—22no—
144#395—22no—
145#395—22no—
146#393—24no—
147#393—22no—
148#393—22no—
149#391—22no—
150#391—22no—
151#391—24no—
152#391C2312yes—
153#391C2314yes—
154#390—22no—
155#390—21no—
156#390—21no—
157#389—22no—
158#389—21no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389—22no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—21no—
168#389C2311yescharacteristic
169#389C2311yescharacteristic
170#389C2312yes—
171#389C2312yes—
172#388—24no—
173#388—22no—
174#388—22no—
175#387—21no—
176#387—21no—
177#387—22no—
178#387—22no—
179#387—24no—
180#387—22no—
181#387—22no—
182#385—22no—
183#385—22no—
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#380C2312yes—
199#380C2314yes—
200#380C2312yes—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—21no—
203C4×C4⋊C4—21no—
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×Q8—22no—
235C22×Q8—22no—
236C22×Q8C2412yes—
237C22×D4C2×D411yescharacteristic
238C23×C4C2×D412yes—
239C23×C4C2×D412yes—
240C23×C4C2412yes—
241C22×C8C2412yes—
242C22×C8C2414yes—
243C22×C8C2412yes—
244C4⋊Q8—24no—
245C4⋊Q8—24no—
246C4⋊Q8—22no—
247C4⋊Q8—22no—
248C4⋊Q8—42no—
249C4⋊Q8—42no—
250C4⋊Q8—44no—
251C4⋊Q8—44no—
252C4⋊Q8—44no—
253C4⋊Q8—44no—
254C4⋊Q8—82no—
255#125—24no—
256#125—22no—
257#125—24no—
258#125#4114yes—
259C22⋊Q8—48no—
260C22⋊Q8—48no—
261C22⋊Q8—44no—
262C22⋊Q8—44no—
263C4×Q8—24no—
264C4×Q8—24no—
265C4×Q8—48no—
266C4×Q8—48no—
267C4×D4—24no—
268C4×D4—24no—
269C4×D4—42no—
270#117—24no—
271#117—24no—
272#117—42no—
273#117—42no—
274#117—84no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—21no—
300C2×C4⋊C4C2×D411yescharacteristic
301C2×C4⋊C4C2×D411yescharacteristic
302C2×C4⋊C4C2×D411yescharacteristic
303C2×C4⋊C4C2412yes—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2414yes—
307C2×C4⋊C4C2411yescharacteristic
308C2×C4⋊C4C2411yescharacteristic
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2411yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4C2×D412yes—
318C2×C22⋊C4C2×D412yes—
319C2×C22⋊C4C2×D412yes—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42—24no—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2×D412yes—
331C2×C42C2×D412yes—
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2411yescharacteristic
335C2×C42C2412yes—
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—22no—
358#95—22no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—21no—
367#95—21no—
368#95—24no—
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—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×Q8—82no—
391C2×Q8—82no—
392C2×D4—22no—
393C2×D4—22no—
394C2×D4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4C22×D411yescharacteristic
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D412yes—
417C22×C4C22×D412yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D412yes—
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—21no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—24no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—44no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—42no—
463C4⋊C4—44no—
464C4⋊C4—42no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—84no—
470C4⋊C4—82no—
471C4⋊C4—82no—
472C4⋊C4#13714yes—
473C4⋊C4C2×#4112yes—
474C4⋊C4C2×#4112yes—
475C22⋊C4—24no—
476C22⋊C4—24no—
477C22⋊C4—42no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—82no—
481C22⋊C4—82no—
482C22⋊C4—84no—
483C22⋊C4—84no—
484C42—24no—
485C42—24no—
486C42—22no—
487C42—22no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—41no—
492C42—41no—
493C42—44no—
494C42—44no—
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—
508C23—22no—
509C23C23×D411yescharacteristic
510Q8—42no—
511Q8—42no—
512Q8—42no—
513Q8—42no—
514Q8—88no—
515Q8—88no—
516Q8—164no—
517D4—44no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—48no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—41no—
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—42no—
560C2×C4—44no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—84no—
569C2×C4#53612yes—
570C2×C4#53612yes—
571C2×C4#54612yes—
572C2×C4#54612yes—
573C2×C4C2×#13714yes—
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—42no—
584C22—42no—
585C22—42no—
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—24no—
600C4—22no—
601C4—42no—
602C4—41no—
603C4—41no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—82no—
608C4—81no—
609C4—84no—
610C4—81no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4#323714yes—
615C4#351812yes—
616C4#354712yes—
617C2—82no—
618C2#3895811yescharacteristic
619C2C2×#323711yescharacteristic
620C2#6088611yescharacteristic
621C2#6116211yescharacteristic
622C2#6116511yescharacteristic
623C2#6258011yescharacteristic
624C2#6258911yescharacteristic
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. #31765C2semidirect
5. #31763C2semidirect
6. #29852C2semidirect
7. #29213C2semidirect
8. #28725C2semidirect
9. #27880C2semidirect
10. #26026C2semidirect
11. #26022C2semidirect
12. #25910C2semidirect
15. #21839C2semidirect
16. C4×#491C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#361322
#332972
#317652
#317632
#298522
#292132
#287252
#278802
#260262
#260222
#259102
#249722
#249702
#218392
C4×#4912
#212702