Finite Group Databasebeta

← #164228

Subgroups — #164228

Order 512.

791 conjugacy classes of subgroups.

The subgroup lattice has 641 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36280C211yesmaximal, characteristic
2#22591C211yesmaximal, characteristic
3#20368C211yesmaximal, characteristic
4#7709C211yesmaximal, characteristic, complemented
5#7703C211yesmaximal, characteristic, complemented
6#7693C211yesmaximal, characteristic, complemented
7#7681C211yesmaximal, characteristic
8#7680C211yesmaximal, characteristic, complemented
9#7655C211yesmaximal, characteristic
10#7638C211yesmaximal, characteristic, complemented
11#7630C211yesmaximal, characteristic, complemented
12#7607C211yesmaximal, characteristic
13#7588C211yesmaximal, characteristic, complemented
14#7581C211yesmaximal, characteristic, complemented
15C2×#1271C211yesmaximal, characteristic
16C2×#405—21no—
17#2863C2211yescharacteristic
18#2859C2211yescharacteristic
19#2855C2211yescharacteristic
20#2612—21no—
21#2606—21no—
22#2585—21no—
23#2511C411yescharacteristic
24#2507C411yescharacteristic
25#2507C411yescharacteristic
26#2502C411yescharacteristic
27#2473—21no—
28#2462—21no—
29#2451C411yescharacteristic
30#2440C411yescharacteristic
31#2438—21no—
32#2360—21no—
33#2360C411yescharacteristic
34#2345C411yescharacteristic
35#2343—21no—
36C4×C4⋊Q8C2211yescharacteristic
37#2295C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1917—21no—
41#1917—21no—
42#1850—21no—
43#1839C2211yescharacteristic
44#1831C2211yescharacteristic
45#1817C2211yescharacteristic
46#1767C2211yescharacteristic
47#1752—21no—
48#1746—21no—
49#1746—21no—
50#1745C2211yescharacteristic
51#1745C2211yescharacteristic
52#1745C2211yescharacteristic
53#1744C2211yescharacteristic
54#1744C2211yescharacteristic
55#1729—21no—
56#1728C2211yescharacteristic
57C2×#336C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C22×M16—21no—
76C23×C8—21no—
77C23×C8—21no—
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#125—21no—
80C2×C22⋊Q8—21no—
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×C4×Q8—21no—
83C2×C4×Q8C2×C411yescharacteristic
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C4⋊C4C2×C411yescharacteristic
89C22×C42C2311yescharacteristic
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C22⋊C8—21no—
93#405—28no—
94C2×#97—21no—
95C2×#97—21no—
96C2×#97—21no—
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100#400—21no—
101#400—21no—
102#398—21no—
103#398—21no—
104#398—21no—
105#398—21no—
106#398—21no—
107#398—21no—
108#398—21no—
109#398—21no—
110#395—21no—
111#395—21no—
112#391—21no—
113#391—21no—
114#391C2×C411yescharacteristic
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2311yescharacteristic
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#389—21no—
127#389C2×C411yescharacteristic
128#389C2×C411yescharacteristic
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2311yescharacteristic
136#389C2311yescharacteristic
137#389C2311yescharacteristic
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
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#385—21no—
154#385—21no—
155#385—21no—
156#385—21no—
157#385—21no—
158#384C2×C411yescharacteristic
159#384C2×C411yescharacteristic
160#384C2×C411yescharacteristic
161#383—21no—
162#382—21no—
163#381—21no—
164#380—21no—
165#380C2×C411yescharacteristic
166C4×C4⋊C4—21no—
167C4×C4⋊C4—21no—
168C4×C4⋊C4C2×C411yescharacteristic
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4D411yescharacteristic
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4C2×C411yescharacteristic
178C4×C22⋊C4C2×C411yescharacteristic
179C4×C22⋊C4D411yescharacteristic
180C4×C22⋊C4D411yescharacteristic
181#376—21no—
182#376D411yescharacteristic
183#376D411yescharacteristic
184C2×#95—21no—
185C43C2311yescharacteristic
186C43C2311yescharacteristic
187#336—21no—
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
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×Q8C22×C411yescharacteristic
209C23×C4—21no—
210C23×C4—21no—
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C2×M16—24no—
215C2×M16—42no—
216C2×M16—42no—
217C22×C8—24no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—42no—
229C22×C8—42no—
230C22×C8—81no—
231C4⋊Q8—22no—
232C4⋊Q8—41no—
233#125—81no—
234#125—81no—
235C22⋊Q8—44no—
236C22⋊Q8—42no—
237C4×Q8—22no—
238C4×Q8—41no—
239C4×Q8—81no—
240C4×Q8—81no—
241#117—22no—
242#117—22no—
243#117—24no—
244C2×C4⋊C4—22no—
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—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—81no—
270C2×C4⋊C4—81no—
271C2×C4⋊C4C22⋊C411yescharacteristic
272C2×C4⋊C4C22⋊C411yescharacteristic
273C2×C4⋊C4C22⋊C411yescharacteristic
274C2×C4⋊C4C22⋊C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C4⋊C4C22×C411yescharacteristic
281C2×C4⋊C4C22×C411yescharacteristic
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4C22⋊C411yescharacteristic
285C2×C22⋊C4C22⋊C411yescharacteristic
286C2×C22⋊C4C22⋊C411yescharacteristic
287C2×C22⋊C4C22⋊C411yescharacteristic
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—42no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2411yescharacteristic
309C4⋊C8—42no—
310C4⋊C8—81no—
311C22⋊C8—42no—
312#97—24no—
313#97—24no—
314#97—42no—
315#97—81no—
316C4×C8—24no—
317C4×C8—24no—
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—42no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—42no—
358C2×Q8—81no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4#13712yes—
395C22×C4#13712yes—
396C22×C4C23×C411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399M16—48no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—22no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—48no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
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—84no—
426C4⋊C4—21no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—42no—
436C4⋊C4—42no—
437C4⋊C4—42no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—42no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—81no—
454C4⋊C4—81no—
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—
464C4⋊C4#13112yes—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4#10414yes—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—22no—
484C42—22no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—41no—
493C42—42no—
494C42—81no—
495C42—81no—
496C42—81no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23—21no—
501C23#41012yes—
502C23#41912yes—
503C23C2×#10412yes—
504C23C22×C22⋊C411yescharacteristic
505Q8—84no—
506Q8—81no—
507Q8—81no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
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—24no—
521C2×C4—24no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
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—84no—
575C2×C4—162no—
576C2×C4#41912yes—
577C2×C4C2×#10412yes—
578C2×C4C2×#10412yes—
579C2×C4C2×#13111yescharacteristic
580C2×C4#56811yescharacteristic
581C2×C4C2×#13711yescharacteristic
582C2×C4C2×#13711yescharacteristic
583C2×C4#57611yescharacteristic
584C2×C4#57711yescharacteristic
585C8—42no—
586C8—42no—
587C8—42no—
588C8—42no—
589C8—42no—
590C8—81no—
591C8—81no—
592C8—81no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22#150614yes—
600C22#288011yescharacteristic
601C22C2×#41011yescharacteristic
602C22#288911yescharacteristic
603C22C2×#41911yescharacteristic
604C22#289711yescharacteristic
605C22C22×#10411yescharacteristic
606C22#290111yescharacteristic
607C4—21no—
608C4—21no—
609C4—22no—
610C4—21no—
611C4—21no—
612C4—22no—
613C4—21no—
614C4—21no—
615C4—22no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—42no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—81no—
632C2—24no—
633C2#1741711yescharacteristic
634C2#1751411yescharacteristic
635C2#1751711yescharacteristic
636C2C2×#150611yescharacteristic
637C2#1760411yescharacteristic
638C2#1765311yescharacteristic
639C2#1776111yescharacteristic
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. #7709C2semidirect
5. #7703C2semidirect
6. #7693C2semidirect
8. #7680C2semidirect
10. #7638C2semidirect
11. #7630C2semidirect
13. #7588C2semidirect
14. #7581C2semidirect

Maximal subgroups

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

subgroupindex
#362802
#225912
#203682
#77092
#77032
#76932
#76812
#76802
#76552
#76382
#76302
#76072
#75882
#75812
C2×#12712