Finite Group Databasebeta

← #164746

Subgroups — #164746

Order 512.

758 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36306C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3#20785C211yesmaximal, characteristic, complemented
4#7716C211yesmaximal, characteristic, complemented
5#7714C211yesmaximal, characteristic
6#7693C211yesmaximal, characteristic, complemented
7#7669C211yesmaximal, characteristic, complemented
8#7658C211yesmaximal, characteristic
9#7656C211yesmaximal, characteristic, complemented
10#7644C211yesmaximal, characteristic, complemented
11#7643C211yesmaximal, characteristic, complemented
12#7618C211yesmaximal, characteristic, complemented
13#7581C211yesmaximal, characteristic, complemented
14#7575C211yesmaximal, characteristic, complemented
15#7567C211yesmaximal, characteristic, complemented
16#2863C2211yescharacteristic
17#2861C2211yescharacteristic
18#2854C2211yescharacteristic
19#2589—21no—
20#2532—21no—
21#2532—21no—
22#2527—21no—
23#2514C411yescharacteristic
24#2510—21no—
25#2498—21no—
26#2453C411yescharacteristic
27#2448C411yescharacteristic
28#2447C411yescharacteristic
29#2442C411yescharacteristic
30#2436C411yescharacteristic
31#2361C411yescharacteristic
32#2359—21no—
33#2353—21no—
34Q8×C22⋊C4C411yescharacteristic
35C4×#124C2211yescharacteristic
36#2295C2211yescharacteristic
37C2×#391C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1939—21no—
40#1923—21no—
41#1923—21no—
42#1839C2211yescharacteristic
43#1839C2211yescharacteristic
44#1832—21no—
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1818—21no—
48#1797C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51C8×C4⋊C4—21no—
52#1768—21no—
53#1750—21no—
54C4×C22⋊C8C2211yescharacteristic
55#1745C2211yescharacteristic
56#1744C2211yescharacteristic
57#1744C2211yescharacteristic
58#1744C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×Q8C2×C411yescharacteristic
76C2×#125C2×C411yescharacteristic
77C2×#124C2×C411yescharacteristic
78C2×#124C2×C411yescharacteristic
79C2×#123—21no—
80C2×#121C2×C411yescharacteristic
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×#117D411yescharacteristic
86C2×#117D411yescharacteristic
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C42C2311yescharacteristic
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×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103#401—21no—
104#400—21no—
105#400—21no—
106#400—21no—
107#399—21no—
108#398—21no—
109#398—21no—
110#397—21no—
111#397—21no—
112#397—21no—
113#393—21no—
114#393—21no—
115#391—21no—
116#391—21no—
117#391—21no—
118#391—21no—
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#391C2×C411yescharacteristic
124#391C2×C411yescharacteristic
125#391C2311yescharacteristic
126#391C2311yescharacteristic
127#390C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#390C2311yescharacteristic
130#389—21no—
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2311yescharacteristic
136#388—21no—
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#386—21no—
155#386—21no—
156#386—21no—
157#386—21no—
158#385—21no—
159#385—21no—
160#385—21no—
161#383—21no—
162#383—21no—
163#382C2×C411yescharacteristic
164#381—21no—
165#380—21no—
166#380—21no—
167#380C2×C411yescharacteristic
168C4×C4⋊C4—21no—
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C22⋊C4—21no—
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—
185C2×#95—21no—
186C43C2311yescharacteristic
187C43C2311yescharacteristic
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#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C22×Q8—21no—
207C22×Q8—22no—
208C22×Q8—41no—
209C22×Q8—42no—
210C22×Q8—81no—
211C22×Q8C22×C411yescharacteristic
212C22×Q8C22×C411yescharacteristic
213C22×D4—21no—
214C22×D4C22×C411yescharacteristic
215C23×C4—21no—
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—
229#125—81no—
230#124—22no—
231#124—41no—
232#124—81no—
233#123—42no—
234#123—42no—
235#121—42no—
236C4×Q8—22no—
237C4×Q8—41no—
238C4×Q8—81no—
239C4×D4—44no—
240C4×D4—81no—
241#117—22no—
242#117—22no—
243#117—42no—
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—81no—
255C2×C4⋊C4C22⋊C411yescharacteristic
256C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
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⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4C22⋊C411yescharacteristic
278C2×C22⋊C4C22⋊C411yescharacteristic
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C42—21no—
282C2×C42—21no—
283C2×C42—21no—
284C2×C42—42no—
285C2×C42C22×C411yescharacteristic
286C2×C42C22×C411yescharacteristic
287C2×C42C22×C411yescharacteristic
288C2×C42C22×C411yescharacteristic
289C2×C42C22×C411yescharacteristic
290C2×C42C2×D411yescharacteristic
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2411yescharacteristic
301C4⋊C8—42no—
302C4⋊C8—42no—
303C4⋊C8—42no—
304C4⋊C8—81no—
305C4⋊C8—81no—
306C22⋊C8—42no—
307C22⋊C8—42no—
308C22⋊C8—42no—
309C22⋊C8—81no—
310#97—42no—
311C4×C8—42no—
312C4×C8—42no—
313C4×C8—42no—
314C4×C8—42no—
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
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348C24—21no—
349C24C2×C22⋊C411yescharacteristic
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—42no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—42no—
359C2×Q8—81no—
360C2×Q8—84no—
361C2×Q8—84no—
362C2×Q8#13112yes—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—42no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—81no—
392C22×C4—81no—
393C22×C4—81no—
394C22×C4—81no—
395C22×C4—81no—
396C22×C4C2×C22⋊C411yescharacteristic
397C22×C4C2×C22⋊C411yescharacteristic
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×C4#13512yes—
408C22×C4#13512yes—
409C22×C4C23×C411yescharacteristic
410C22×C4C22×D411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C2×C8—22no—
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—
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—21no—
440C4⋊C4—21no—
441C4⋊C4—41no—
442C4⋊C4—42no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—81no—
454C4⋊C4—81no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—41no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C22⋊C4—42no—
464C22⋊C4—42no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—81no—
472C22⋊C4—81no—
473C22⋊C4—81no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—22no—
489C42—22no—
490C42—22no—
491C42—41no—
492C42—41no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—42no—
499C42—42no—
500C42—81no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23#41612yes—
515C23C22×C22⋊C411yescharacteristic
516Q8—42no—
517Q8—42no—
518Q8—84no—
519Q8—81no—
520Q8—81no—
521D4—81no—
522D4—81no—
523D4—84no—
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—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
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—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—81no—
586C2×C4—81no—
587C2×C4—84no—
588C2×C4—162no—
589C2×C4—162no—
590C2×C4#41012yes—
591C2×C4#41612yes—
592C2×C4#42112yes—
593C2×C4C2×#13111yescharacteristic
594C2×C4#56811yescharacteristic
595C2×C4C2×#13511yescharacteristic
596C2×C4C2×#13511yescharacteristic
597C2×C4#57711yescharacteristic
598C2×C4#57811yescharacteristic
599C8—42no—
600C8—81no—
601C8—81no—
602C8—81no—
603C8—81no—
604C8—81no—
605C8—81no—
606C8—81no—
607C22—22no—
608C22—22no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22#288011yescharacteristic
622C22C2×#41011yescharacteristic
623C22#288811yescharacteristic
624C22C2×#41611yescharacteristic
625C22#289811yescharacteristic
626C22C2×#42111yescharacteristic
627C22#290111yescharacteristic
628C4—21no—
629C4—21no—
630C4—22no—
631C4—22no—
632C4—21no—
633C4—21no—
634C4—22no—
635C4—21no—
636C4—21no—
637C4—42no—
638C4—41no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—81no—
652C2—42no—
653C2—161no—
654C2#1741211yescharacteristic
655C2#1751011yescharacteristic
656C2#1752511yescharacteristic
657C2#1756711yescharacteristic
658C2#1760511yescharacteristic
659C2#1767911yescharacteristic
660C2#1772611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×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
2. #22632C2semidirect
3. #20785C2semidirect
4. #7716C2semidirect
6. #7693C2semidirect
7. #7669C2semidirect
9. #7656C2semidirect
10. #7644C2semidirect
11. #7643C2semidirect
12. #7618C2semidirect
13. #7581C2semidirect
14. #7575C2semidirect
15. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#363062
#226322
#207852
#77162
#77142
#76932
#76692
#76582
#76562
#76442
#76432
#76182
#75812
#75752
#75672