Finite Group Databasebeta

← #162345

Subgroups — #162345

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 643 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35026C211yesmaximal, characteristic
2#22788C211yesmaximal, characteristic
3#22008C211yesmaximal, characteristic
4#7731C211yesmaximal, characteristic, complemented
5#7715C211yesmaximal, characteristic, complemented
6#7711C211yesmaximal, characteristic, complemented
7#7695C211yesmaximal, characteristic, complemented
8#7693C211yesmaximal, characteristic, complemented
9#7677C211yesmaximal, characteristic, complemented
10#7673C211yesmaximal, characteristic, complemented
11#7657C211yesmaximal, characteristic, complemented
12#7600C211yesmaximal, characteristic, complemented
13#7593C211yesmaximal, characteristic, complemented
14#7583C211yesmaximal, characteristic, complemented
15#7576C211yesmaximal, characteristic, complemented
16#2864C2211yescharacteristic
17#2863C2211yescharacteristic
18#2855C2211yescharacteristic
19#2851C2211yescharacteristic
20#2655—21no—
21#2652—21no—
22#2585—21no—
23#2573—21no—
24#2509C411yescharacteristic
25#2503C411yescharacteristic
26#2494C411yescharacteristic
27#2493—21no—
28#2486C411yescharacteristic
29#2453C411yescharacteristic
30#2442C411yescharacteristic
31#2421C411yescharacteristic
32#2419—21no—
33#2417C411yescharacteristic
34#2322—21no—
35#2313—21no—
36C4×C22⋊Q8C2211yescharacteristic
37C2×#389C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1954—21no—
40#1939—21no—
41#1924—21no—
42#1923—21no—
43#1923—21no—
44#1916—21no—
45#1870—21no—
46#1839C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1797C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1753—21no—
55C4×C22⋊C8C2211yescharacteristic
56#1745C2211yescharacteristic
57#1744C2211yescharacteristic
58#1728C2211yescharacteristic
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—
76C2×#125C2×C411yescharacteristic
77C2×#123—21no—
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×C4×Q8C2×C411yescharacteristic
81C2×C4×D4C2×C411yescharacteristic
82C2×C4×D4D411yescharacteristic
83C2×C4×D4D411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4C2×C411yescharacteristic
86C22×C4⋊C4C2×C411yescharacteristic
87C22×C42C2311yescharacteristic
88C22×C42C2311yescharacteristic
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×#97—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104#401—21no—
105#400—21no—
106#400—21no—
107#400—21no—
108#400—21no—
109#400—21no—
110#400—21no—
111#400—21no—
112#398—21no—
113#398—21no—
114#398—21no—
115#398—21no—
116#397—21no—
117#397—21no—
118#397—21no—
119#396—21no—
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#391C2311yescharacteristic
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
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#388—21no—
139#388—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#387—21no—
154#387—21no—
155#387—21no—
156#386—21no—
157#385—21no—
158#385—21no—
159#385—21no—
160#385—21no—
161#385—21no—
162#383—21no—
163#382C2×C411yescharacteristic
164#382C2×C411yescharacteristic
165#382C2×C411yescharacteristic
166#381—21no—
167#381—21no—
168#381—21no—
169#381—21no—
170#380—21no—
171#380—21no—
172#380C2×C411yescharacteristic
173#380C2×C411yescharacteristic
174#380C2×C411yescharacteristic
175#380C2×C411yescharacteristic
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C22⋊C4—21no—
182C4×C22⋊C4C2×C411yescharacteristic
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C2×#95—21no—
189C43C2311yescharacteristic
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#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C22×Q8C22×C411yescharacteristic
207C22×D4C22⋊C411yescharacteristic
208C23×C4—21no—
209C23×C4—21no—
210C23×C4C22×C411yescharacteristic
211C23×C4C22×C411yescharacteristic
212C23×C4C2×D411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C2×M16—42no—
217C2×M16—42no—
218C2×M16—81no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227#125—81no—
228#123—44no—
229C22⋊Q8—24no—
230C22⋊Q8—81no—
231C4×Q8—81no—
232C4×D4—24no—
233C4×D4—42no—
234C4×D4—41no—
235C4×D4—41no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
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—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—81no—
257C2×C4⋊C4C22⋊C411yescharacteristic
258C2×C4⋊C4C22⋊C411yescharacteristic
259C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4C22⋊C411yescharacteristic
274C2×C22⋊C4C22⋊C411yescharacteristic
275C2×C22⋊C4C22⋊C411yescharacteristic
276C2×C22⋊C4C22⋊C411yescharacteristic
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—81no—
282C2×C42—81no—
283C2×C42C22×C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
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—42no—
297C4⋊C8—81no—
298C4⋊C8—81no—
299C4⋊C8—81no—
300C22⋊C8—42no—
301C22⋊C8—42no—
302C22⋊C8—81no—
303C22⋊C8—81no—
304C22⋊C8—81no—
305C22⋊C8—81no—
306C22⋊C8—81no—
307#97—42no—
308C4×C8—42no—
309C4×C8—81no—
310C4×C8—81no—
311#95—21no—
312#95—21no—
313#95—21no—
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#95C22⋊C411yescharacteristic
341#95C22⋊C411yescharacteristic
342#95C22⋊C411yescharacteristic
343#95C22⋊C411yescharacteristic
344#95C22⋊C411yescharacteristic
345#95C22⋊C411yescharacteristic
346C24C2×C22⋊C411yescharacteristic
347C24C2×C22⋊C411yescharacteristic
348C2×Q8—22no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—41no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—22no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—81no—
388C22×C4—81no—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
398C22×C4C2×C22⋊C411yescharacteristic
399C22×C4#13114yes—
400C22×C4C23×C411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D411yescharacteristic
403M16—84no—
404C2×C8—22no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
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—41no—
426C2×C8—41no—
427C2×C8—41no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—41no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—42no—
450C4⋊C4—81no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—81no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—42no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C42—22no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—22no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—41no—
482C42—41no—
483C42—41no—
484C42—41no—
485C42—81no—
486C42—81no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23—21no—
500C23#40812yes—
501C23C22×C22⋊C411yescharacteristic
502Q8—82no—
503D4—44no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
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—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—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
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—81no—
566C2×C4—81no—
567C2×C4—81no—
568C2×C4—81no—
569C2×C4—81no—
570C2×C4—81no—
571C2×C4—82no—
572C2×C4—84no—
573C2×C4—162no—
574C2×C4#40812yes—
575C2×C4#42112yes—
576C2×C4C2×#13112yes—
577C2×C4#57511yescharacteristic
578C2×C4#57511yescharacteristic
579C2×C4#57711yescharacteristic
580C2×C4#57811yescharacteristic
581C8—42no—
582C8—81no—
583C8—81no—
584C8—81no—
585C8—81no—
586C8—81no—
587C8—81no—
588C8—81no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22C2×#40811yescharacteristic
604C22#288311yescharacteristic
605C22#288911yescharacteristic
606C22#289611yescharacteristic
607C22#289811yescharacteristic
608C22C2×#42111yescharacteristic
609C22#290111yescharacteristic
610C4—22no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—22no—
616C4—22no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—81no—
631C4—81no—
632C4—81no—
633C2—42no—
634C2—81no—
635C2#1738711yescharacteristic
636C2#1746911yescharacteristic
637C2#1748011yescharacteristic
638C2#1760011yescharacteristic
639C2#1762311yescharacteristic
640C2#1768211yescharacteristic
641C2#1776711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7731C2semidirect
5. #7715C2semidirect
6. #7711C2semidirect
7. #7695C2semidirect
8. #7693C2semidirect
9. #7677C2semidirect
10. #7673C2semidirect
11. #7657C2semidirect
12. #7600C2semidirect
13. #7593C2semidirect
14. #7583C2semidirect
15. #7576C2semidirect

Maximal subgroups

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

subgroupindex
#350262
#227882
#220082
#77312
#77152
#77112
#76952
#76932
#76772
#76732
#76572
#76002
#75932
#75832
#75762