Finite Group Databasebeta

← #164283

Subgroups — #164283

Order 512.

775 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36279C211yesmaximal, characteristic
2#22641C211yesmaximal, characteristic
3#20468C211yesmaximal, characteristic
4#7702C211yesmaximal, characteristic, complemented
5#7701C211yesmaximal, characteristic
6#7689C211yesmaximal, characteristic, complemented
7#7682C211yesmaximal, characteristic, complemented
8#7675C211yesmaximal, characteristic
9#7673C211yesmaximal, characteristic, complemented
10#7630C211yesmaximal, characteristic, complemented
11#7629C211yesmaximal, characteristic
12#7624C211yesmaximal, characteristic, complemented
13#7581C211yesmaximal, characteristic, complemented
14#7575C211yesmaximal, characteristic
15#7571C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17#2863C2211yescharacteristic
18#2862C2211yescharacteristic
19#2855C2211yescharacteristic
20#2612—21no—
21#2605—21no—
22#2586—21no—
23#2520—21no—
24#2511C411yescharacteristic
25#2505—21no—
26#2451C411yescharacteristic
27#2446C411yescharacteristic
28#2446C411yescharacteristic
29#2440C411yescharacteristic
30#2438—21no—
31#2434C411yescharacteristic
32#2368—21no—
33#2360C411yescharacteristic
34#2343—21no—
35Q8×C22⋊C4C411yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37#2295C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1940—21no—
41#1924—21no—
42#1923—21no—
43#1839C2211yescharacteristic
44#1839C2211yescharacteristic
45#1797C2211yescharacteristic
46#1771—21no—
47#1768—21no—
48#1767C2211yescharacteristic
49#1767C2211yescharacteristic
50C4×C4⋊C8C2211yescharacteristic
51C4×C4⋊C8C2211yescharacteristic
52#1750—21no—
53#1749C2211yescharacteristic
54#1749C2211yescharacteristic
55C4×C22⋊C8C2211yescharacteristic
56#1745C2211yescharacteristic
57#1744C2211yescharacteristic
58#1726—21no—
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1272C2211yescharacteristic
68#1272C2211yescharacteristic
69#1272C2211yescharacteristic
70#1272C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C23×Q8C2×C411yescharacteristic
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×Q8C2×C411yescharacteristic
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C42C2311yescharacteristic
88#433—28no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107#400—21no—
108#400—21no—
109#400—21no—
110#400—21no—
111#398—21no—
112#398—21no—
113#398—21no—
114#398—21no—
115#395—21no—
116#395—21no—
117#395—21no—
118#395—21no—
119#391—21no—
120#391—21no—
121#391—21no—
122#391—21no—
123#391—21no—
124#391—21no—
125#391—21no—
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#391C2×C411yescharacteristic
132#391C2×C411yescharacteristic
133#391C2×C411yescharacteristic
134#391C2311yescharacteristic
135#391C2311yescharacteristic
136#391C2311yescharacteristic
137#389—21no—
138#389—21no—
139#389—21no—
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#389C2311yescharacteristic
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#387—21no—
160#385—21no—
161#385—21no—
162#385—21no—
163#385—21no—
164#385—21no—
165#384C2×C411yescharacteristic
166#383—21no—
167#382—21no—
168#382—21no—
169#381—21no—
170#380—21no—
171#380C2×C411yescharacteristic
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4C2×C411yescharacteristic
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186#376—21no—
187#376D411yescharacteristic
188#376D411yescharacteristic
189C2×#95—21no—
190C2×#95—21no—
191C43C2311yescharacteristic
192C43C2311yescharacteristic
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—22no—
210C22×Q8—41no—
211C22×Q8—42no—
212C22×Q8—81no—
213C22×Q8C22×C411yescharacteristic
214C22×Q8C22×C411yescharacteristic
215C22×Q8C22×C411yescharacteristic
216C23×C4—21no—
217C23×C4—21no—
218C23×C4C22×C411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C4⋊Q8—22no—
230C4⋊Q8—41no—
231C4⋊Q8—81no—
232C4⋊Q8—81no—
233C22⋊Q8—42no—
234C22⋊Q8—44no—
235C4×Q8—22no—
236C4×Q8—41no—
237C4×Q8—81no—
238C4×Q8—81no—
239#117—24no—
240#117—22no—
241#117—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—81no—
260C2×C4⋊C4C22⋊C411yescharacteristic
261C2×C4⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C4⋊C4C22×C411yescharacteristic
266C2×C4⋊C4C22×C411yescharacteristic
267C2×C4⋊C4C22×C411yescharacteristic
268C2×C4⋊C4C22×C411yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4C22⋊C411yescharacteristic
272C2×C22⋊C4C22⋊C411yescharacteristic
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—42no—
280C2×C42C22×C411yescharacteristic
281C2×C42C22×C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C2×D411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2411yescharacteristic
296C4⋊C8—24no—
297C4⋊C8—24no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301C22⋊C8—24no—
302C22⋊C8—24no—
303C22⋊C8—42no—
304C22⋊C8—81no—
305#97—24no—
306#97—42no—
307#97—42no—
308#97—42no—
309#97—81no—
310C4×C8—24no—
311C4×C8—42no—
312C4×C8—42no—
313C4×C8—81no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—21no—
320#95—21no—
321#95—21no—
322#95—21no—
323#95—21no—
324#95—21no—
325#95—21no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351C24C2×C22⋊C411yescharacteristic
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—42no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×Q8—41no—
364C2×Q8—42no—
365C2×Q8—42no—
366C2×Q8—81no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×Q8—84no—
370C2×Q8—84no—
371C2×Q8#13112yes—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—81no—
396C22×C4—81no—
397C22×C4—81no—
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—42no—
454C4⋊C4—42no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C4⋊C4—81no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—42no—
468C22⋊C4—42no—
469C22⋊C4#10414yes—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—22no—
481C42—22no—
482C42—22no—
483C42—22no—
484C42—41no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—42no—
491C42—81no—
492C42—81no—
493C42—81no—
494C42#13712yes—
495C42#13712yes—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23C2×#10412yes—
503C23C22×C22⋊C411yescharacteristic
504Q8—42no—
505Q8—42no—
506Q8—84no—
507Q8—81no—
508Q8—81no—
509Q8—81no—
510Q8—81no—
511Q8—162no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—84no—
577C2×C4—162no—
578C2×C4#41012yes—
579C2×C4#41912yes—
580C2×C4#41912yes—
581C2×C4C2×#10412yes—
582C2×C4C2×#10412yes—
583C2×C4C2×#13111yescharacteristic
584C2×C4#56811yescharacteristic
585C2×C4C2×#13711yescharacteristic
586C2×C4C2×#13711yescharacteristic
587C2×C4#57711yescharacteristic
588C2×C4#57811yescharacteristic
589C8—42no—
590C8—42no—
591C8—42no—
592C8—42no—
593C8—42no—
594C8—81no—
595C8—81no—
596C8—81no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22#288011yescharacteristic
605C22C2×#41011yescharacteristic
606C22#288811yescharacteristic
607C22C2×#41911yescharacteristic
608C22#289811yescharacteristic
609C22C22×#10411yescharacteristic
610C22#290111yescharacteristic
611C4—21no—
612C4—21no—
613C4—22no—
614C4—22no—
615C4—22no—
616C4—22no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—22no—
622C4—42no—
623C4—41no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—41no—
628C4—41no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C4—81no—
634C4—81no—
635C4—81no—
636C2—42no—
637C2#1741211yescharacteristic
638C2#1751411yescharacteristic
639C2#1751911yescharacteristic
640C2#1754511yescharacteristic
641C2#1761211yescharacteristic
642C2#1765211yescharacteristic
643C2#1776011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #7702C2semidirect
6. #7689C2semidirect
7. #7682C2semidirect
9. #7673C2semidirect
10. #7630C2semidirect
12. #7624C2semidirect
13. #7581C2semidirect
15. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#362792
#226412
#204682
#77022
#77012
#76892
#76822
#76752
#76732
#76302
#76292
#76242
#75812
#75752
#75712