Finite Group Databasebeta

← #6795709

Subgroups — #6795709

Order 512.

1369 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#34941C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic, complemented
4#31715C214yesmaximal, complemented
5#31107C214yesmaximal, complemented
6#29938C212yesmaximal, complemented
7#29290C211yesmaximal, characteristic, complemented
8#29206C211yesmaximal, characteristic, complemented
9#28836C212yesmaximal, complemented
10#27738C212yesmaximal, complemented
11#27649C212yesmaximal, complemented
12#26297C212yesmaximal
13#25015C212yesmaximal
14#24711C212yesmaximal, complemented
15#22799C211yesmaximal, characteristic
16#21988C212yesmaximal, complemented
17C4×#477C211yesmaximal, characteristic, complemented
18C2×#485C2214yes—
19C2×#477C2211yescharacteristic
20C2×#475C2212yes—
21C2×#432C2212yes—
22#2868C2211yescharacteristic
23#2857C2212yes—
24#2574—22no—
25#2571C2212yes—
26#2567C2211yescharacteristic
27#2563—21no—
28#2563C2211yescharacteristic
29#2552C2211yescharacteristic
30#2538—21no—
31#2538C2211yescharacteristic
32#2508—21no—
33#2497—22no—
34#2497C2212yes—
35#2492—22no—
36#2454—21no—
37#2439C2214yes—
38#2423C2212yes—
39#2396—22no—
40#2369—21no—
41#2355—21no—
42#2354C2212yes—
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#125—21no—
46C4×C22⋊Q8C2212yes—
47#2293—21no—
48#2285C2212yes—
49#2285C2212yes—
50C42×Q8C2211yescharacteristic
51C42×D4C2211yescharacteristic
52#2099C2214yes—
53#2087C2214yes—
54#2085C2214yes—
55#2084C2214yes—
56#2077C2214yes—
57#2077C2214yes—
58#2075C2214yes—
59#2074C2214yes—
60#2058C2214yes—
61#2035C2214yes—
62#1990C2212yes—
63#1988C2212yes—
64#1987C2212yes—
65#1987C2211yescharacteristic
66#1987C2212yes—
67#1954C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2212yes—
71#1948C2212yes—
72#1947C2212yes—
73#1947C2211yescharacteristic
74#1947C2212yes—
75#1944C2214yes—
76#1935C2214yes—
77#1924C2214yes—
78#1921C2212yes—
79#1863C2212yes—
80#1854C2214yes—
81#1854C2214yes—
82#1847C2214yes—
83#1844C2212yes—
84#1842C2211yescharacteristic
85#1836C2214yes—
86#1825C2214yes—
87#1803C2211yescharacteristic
88#1803C2212yes—
89#1803C2212yes—
90#1797C2212yes—
91C4×#107C2212yes—
92#1770C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C2×C4⋊Q8—21no—
96C2×C4⋊Q8—21no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#125—21no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2312yes—
103C2×C4×Q8—21no—
104C2×C4×Q8—21no—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×Q8C2311yescharacteristic
107C2×C4×Q8C2311yescharacteristic
108C2×C4×D4D411yescharacteristic
109C2×C4×D4D411yescharacteristic
110C2×C4×D4C2311yescharacteristic
111C2×#117—22no—
112C2×#117C2314yes—
113C22×C42C2312yes—
114#485—48no—
115#477—24no—
116#475—44no—
117#432—28no—
118C2×#107C2312yes—
119C2×#107C2314yes—
120C2×#107C2312yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123C2×#106C2314yes—
124C2×C4⋊C8C2311yescharacteristic
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2312yes—
128C2×C4⋊C8C2311yescharacteristic
129C2×#103C2314yes—
130C2×#103C2312yes—
131C2×#103C2312yes—
132C2×C22⋊C8C2314yes—
133C2×C22⋊C8C2314yes—
134C2×#97C2314yes—
135C2×C4×C8C2312yes—
136#400—24no—
137#400—22no—
138#399—22no—
139#398—22no—
140#398—22no—
141#396—22no—
142#396—22no—
143#396—24no—
144#395—22no—
145#393—24no—
146#393—22no—
147#391—21no—
148#391—21no—
149#391—22no—
150#391—24no—
151#391C2312yes—
152#390—22no—
153#390—21no—
154#390—21no—
155#389—21no—
156#389—21no—
157#389—21no—
158#389—22no—
159#389—21no—
160#389—21no—
161#389—21no—
162#389—22no—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#389C2312yes—
167#388—22no—
168#388—22no—
169#388—24no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—24no—
174#385—22no—
175#384—21no—
176#384—21no—
177#384C2311yescharacteristic
178#384C2311yescharacteristic
179#384C2312yes—
180#384C2312yes—
181#384C2312yes—
182#384C2312yes—
183#382—21no—
184#382—22no—
185#382—22no—
186#382C2314yes—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—22no—
191#380C2312yes—
192#380C2312yes—
193#380C2314yes—
194#380C2312yes—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—22no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—21no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4D411yescharacteristic
206C4×C4⋊C4D411yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2312yes—
209C4×C4⋊C4C2311yescharacteristic
210C4×C4⋊C4C2311yescharacteristic
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4D412yes—
215C4×C22⋊C4D412yes—
216C4×C22⋊C4C2312yes—
217C43—21no—
218C43C2311yescharacteristic
219#340C2314yes—
220#340C2314yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2314yes—
225#340C2314yes—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2312yes—
230#336C2314yes—
231#336C2312yes—
232C22×Q8—21no—
233C22×Q8—21no—
234C22×Q8—22no—
235C22×Q8C2411yescharacteristic
236C22×D4C2×D411yescharacteristic
237C23×C4C2×D412yes—
238C23×C4C2×D412yes—
239C23×C4C2412yes—
240C22×C8C2412yes—
241C22×C8C2414yes—
242C22×C8C2412yes—
243C4⋊Q8—22no—
244C4⋊Q8—22no—
245C4⋊Q8—22no—
246C4⋊Q8—22no—
247C4⋊Q8—42no—
248C4⋊Q8—42no—
249C4⋊Q8—44no—
250C4⋊Q8—82no—
251#125—44no—
252C22⋊Q8—24no—
253C22⋊Q8—42no—
254C22⋊Q8—84no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—22no—
260C4×Q8—41no—
261C4×Q8—44no—
262C4×Q8—82no—
263C4×D4—24no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—42no—
267#117—44no—
268#117—44no—
269#117—84no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D411yescharacteristic
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2411yescharacteristic
299C2×C4⋊C4C2411yescharacteristic
300C2×C4⋊C4C2411yescharacteristic
301C2×C4⋊C4C2412yes—
302C2×C4⋊C4C2412yes—
303C2×C4⋊C4C2412yes—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C22⋊C4C2×D412yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—21no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—21no—
321C2×C42—21no—
322C2×C42—24no—
323C2×C42—21no—
324C2×C42C2×D411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42C2×D412yes—
327C2×C42C2×D412yes—
328C2×C42C2×D411yescharacteristic
329C2×C42C2×D411yescharacteristic
330C2×C42C2411yescharacteristic
331C2×C42C2412yes—
332C2×C42C2411yescharacteristic
333C2×C42C2411yescharacteristic
334#107—24no—
335#107—42no—
336#107—44no—
337#106—44no—
338#106—44no—
339#106—44no—
340C4⋊C8—24no—
341C4⋊C8—24no—
342C4⋊C8—22no—
343C4⋊C8—42no—
344C4⋊C8#4114yes—
345#103—28no—
346#103—24no—
347#103—24no—
348C22⋊C8—28no—
349C22⋊C8—44no—
350#97—28no—
351C4×C8—24no—
352#95—22no—
353#95—22no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—24no—
358#95—24no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—24no—
363#95—22no—
364#95—22no—
365#95—22no—
366#95—22no—
367#95—22no—
368#95—24no—
369#95C2×D412yes—
370#95C2×D412yes—
371#95C2×D412yes—
372C24C22×D412yes—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—21no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—42no—
379C2×Q8—42no—
380C2×Q8—42no—
381C2×Q8—42no—
382C2×Q8—44no—
383C2×Q8—81no—
384C2×Q8—81no—
385C2×Q8—82no—
386C2×Q8—82no—
387C2×Q8H(5,𝔽2)12yes—
388C2×D4—22no—
389C2×D4—22no—
390C2×D4—22no—
391C22×C4—21no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D411yescharacteristic
416C22×C4C22×D411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C2511yescharacteristic
422C2×C8—22no—
423C2×C8—22no—
424C2×C8—24no—
425C2×C8—22no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—24no—
429C2×C8—22no—
430C2×C8C2×#4114yes—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—24no—
437C4⋊C4—22no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—22no—
450C4⋊C4—42no—
451C4⋊C4—44no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—44no—
456C4⋊C4—44no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—44no—
462C4⋊C4—44no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—82no—
466C4⋊C4—82no—
467C4⋊C4—82no—
468C4⋊C4—81no—
469C4⋊C4—82no—
470C4⋊C4—84no—
471C4⋊C4—82no—
472C4⋊C4—82no—
473C22⋊C4—24no—
474C22⋊C4—24no—
475C22⋊C4—42no—
476C22⋊C4—44no—
477C22⋊C4—82no—
478C22⋊C4—84no—
479C22⋊C4—82no—
480C22⋊C4—82no—
481C22⋊C4—84no—
482C42—24no—
483C42—24no—
484C42—22no—
485C42—22no—
486C42—22no—
487C42—21no—
488C42—22no—
489C42—21no—
490C42—41no—
491C42—42no—
492C42—42no—
493C42—42no—
494C42—44no—
495C42—42no—
496C42—42no—
497C42—44no—
498C42—81no—
499C42—84no—
500C42#13714yes—
501C42C2×#4112yes—
502C422−1+412yes—
503C422−1+412yes—
504C422−1+412yes—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23C23×D411yescharacteristic
513Q8—22no—
514Q8—22no—
515Q8—41no—
516Q8—41no—
517Q8—84no—
518Q8—84no—
519Q8—164no—
520D4—44no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—21no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—48no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—84no—
572C2×C4#53612yes—
573C2×C4#53612yes—
574C2×C4#54712yes—
575C2×C4#54712yes—
576C2×C4C2×#13712yes—
577C2×C4C2×#13712yes—
578C2×C4C2×#13712yes—
579C2×C4#57512yes—
580C2×C4#57712yes—
581C2×C4C22×#4111yescharacteristic
582C2×C4C2×H(5,𝔽2)11yescharacteristic
583C2×C4C2×2−1+411yescharacteristic
584C2×C4C2×2−1+411yescharacteristic
585C2×C4C2×2−1+411yescharacteristic
586C2×C4#58511yescharacteristic
587C8—24no—
588C8—44no—
589C8—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22C2×#53611yescharacteristic
598C22C2×#54711yescharacteristic
599C22#348811yescharacteristic
600C22C22×#13711yescharacteristic
601C22C2×#57511yescharacteristic
602C22C2×#57711yescharacteristic
603C22#358511yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—42no—
612C4—84no—
613C4—81no—
614C4—82no—
615C4—82no—
616C4—81no—
617C4—82no—
618C4—81no—
619C4—82no—
620C4#323714yes—
621C4#335614yes—
622C4#351511yescharacteristic
623C4#352211yescharacteristic
624C4#354411yescharacteristic
625C4#355511yescharacteristic
626C2—82no—
627C2#3892111yescharacteristic
628C2C2×#323711yescharacteristic
629C2#6089211yescharacteristic
630C2C2×#335611yescharacteristic
631C2#6125611yescharacteristic
632C2#6196511yescharacteristic
633C2#6198711yescharacteristic
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. #33196C2semidirect
4. #31715C2semidirect
5. #31107C2semidirect
6. #29938C2semidirect
7. #29290C2semidirect
8. #29206C2semidirect
9. #28836C2semidirect
10. #27738C2semidirect
11. #27649C2semidirect
14. #24711C2semidirect
16. #21988C2semidirect
17. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#349412
#331962
#317152
#311072
#299382
#292902
#292062
#288362
#277382
#276492
#262972
#250152
#247112
#227992
#219882
C4×#4772