Finite Group Databasebeta

← #164775

Subgroups — #164775

Order 512.

735 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36385C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3#20781C211yesmaximal, characteristic, complemented
4#7730C211yesmaximal, characteristic
5#7716C211yesmaximal, characteristic, complemented
6#7682C211yesmaximal, characteristic, complemented
7#7670C211yesmaximal, characteristic
8#7662C211yesmaximal, characteristic, complemented
9#7656C211yesmaximal, characteristic, complemented
10#7648C211yesmaximal, characteristic, complemented
11#7644C211yesmaximal, characteristic, complemented
12#7614C211yesmaximal, characteristic, complemented
13#7583C211yesmaximal, characteristic, complemented
14#7581C211yesmaximal, characteristic, complemented
15#7560C211yesmaximal, characteristic, complemented
16#2862C2211yescharacteristic
17#2861C2211yescharacteristic
18#2850C2211yescharacteristic
19#2609—21no—
20#2574—21no—
21#2550C411yescharacteristic
22#2545C411yescharacteristic
23#2532—21no—
24#2529—21no—
25#2518—21no—
26#2491C411yescharacteristic
27#2489C411yescharacteristic
28#2474—21no—
29#2429C411yescharacteristic
30#2427C411yescharacteristic
31#2372—21no—
32#2367C411yescharacteristic
33#2356C411yescharacteristic
34#2346—21no—
35C4×C22⋊Q8C2211yescharacteristic
36#2295C2211yescharacteristic
37#2293C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1939—21no—
40#1923—21no—
41#1923—21no—
42#1839C2211yescharacteristic
43#1839C2211yescharacteristic
44#1832—21no—
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1798—21no—
50#1797C2211yescharacteristic
51C8×C4⋊C4—21no—
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1750—21no—
55C4×C22⋊C8C2211yescharacteristic
56#1746—21no—
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
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×#125C2×C411yescharacteristic
78C2×#124—21no—
79C2×#124C2×C411yescharacteristic
80C2×#124C2×C411yescharacteristic
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×#121C2×C411yescharacteristic
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×#117—21no—
88C2×#117D411yescharacteristic
89C2×#117D411yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊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—
104C2×C4×C8—21no—
105#400—21no—
106#398—21no—
107#398—21no—
108#397—21no—
109#397—21no—
110#396—21no—
111#396—21no—
112#396—21no—
113#395—21no—
114#394—21no—
115#394—21no—
116#393—21no—
117#391—21no—
118#391—21no—
119#391—21no—
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2×C411yescharacteristic
123#391C2×C411yescharacteristic
124#391C2311yescharacteristic
125#391C2311yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#390C2311yescharacteristic
129#389—21no—
130#389—21no—
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2311yescharacteristic
134#388—21no—
135#388—21no—
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#388—21no—
148#388—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#385—21no—
158#385—21no—
159#383—21no—
160#382—21no—
161#382—21no—
162#382—21no—
163#382C2×C411yescharacteristic
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⋊C4—21no—
178C4×C22⋊C4C2×C411yescharacteristic
179C4×C22⋊C4C2×C411yescharacteristic
180C4×C22⋊C4D411yescharacteristic
181C4×C22⋊C4D411yescharacteristic
182#376—21no—
183#376—21no—
184#376D411yescharacteristic
185#376D411yescharacteristic
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×Q8C22×C411yescharacteristic
208C22×Q8C22×C411yescharacteristic
209C22×D4—21no—
210C22×D4C22×C411yescharacteristic
211C23×C4—21no—
212C23×C4C22×C411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C22×C8—21no—
216C22×C8—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C4⋊Q8—81no—
224C4⋊Q8—81no—
225#125—81no—
226#124—42no—
227#124—44no—
228#124—81no—
229C22⋊Q8—22no—
230C22⋊Q8—41no—
231C22⋊Q8—81no—
232#121—81no—
233C4×Q8—22no—
234C4×Q8—41no—
235C4×Q8—42no—
236C4×D4—22no—
237C4×D4—41no—
238#117—22no—
239#117—22no—
240#117—42no—
241#117—42no—
242#117—42no—
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⋊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⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4C22⋊C411yescharacteristic
280C2×C22⋊C4C22⋊C411yescharacteristic
281C2×C22⋊C4C22⋊C411yescharacteristic
282C2×C22⋊C4C22⋊C411yescharacteristic
283C2×C42—21no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—42no—
289C2×C42C22×C411yescharacteristic
290C2×C42C22×C411yescharacteristic
291C2×C42C22×C411yescharacteristic
292C2×C42C22×C411yescharacteristic
293C2×C42C22×C411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2411yescharacteristic
305C4⋊C8—42no—
306C4⋊C8—42no—
307C4⋊C8—42no—
308C4⋊C8—81no—
309C4⋊C8—81no—
310C22⋊C8—42no—
311C22⋊C8—42no—
312C22⋊C8—42no—
313C22⋊C8—81no—
314#97—42no—
315C4×C8—42no—
316C4×C8—42no—
317C4×C8—42no—
318C4×C8—42no—
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—21no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—41no—
357C2×Q8—41no—
358C2×Q8—42no—
359C2×Q8—81no—
360C2×Q8—81no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4—42no—
366C2×D4—81no—
367C2×D4—81no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
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—81no—
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×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C23×C411yescharacteristic
402C22×C4C22×D411yescharacteristic
403C22×C4C22×D411yescharacteristic
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—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—41no—
433C4⋊C4—42no—
434C4⋊C4—42no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—42no—
442C4⋊C4—42no—
443C4⋊C4—81no—
444C4⋊C4—81no—
445C4⋊C4—81no—
446C4⋊C4—81no—
447C4⋊C4—81no—
448C4⋊C4—81no—
449C4⋊C4—81no—
450C4⋊C4#13112yes—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—41no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—42no—
465C22⋊C4—42no—
466C22⋊C4—42no—
467C22⋊C4—81no—
468C22⋊C4—81no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C22⋊C4—81no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—22no—
483C42—22no—
484C42—22no—
485C42—41no—
486C42—41no—
487C42—41no—
488C42—41no—
489C42—41no—
490C42—41no—
491C42—41no—
492C42—42no—
493C42—42no—
494C42—42no—
495C42—42no—
496C42—81no—
497C42#13512yes—
498C42#13512yes—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23C22×C22⋊C411yescharacteristic
514Q8—81no—
515Q8—81no—
516Q8—81no—
517Q8—81no—
518Q8—162no—
519D4—81no—
520D4—81no—
521D4—162no—
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—21no—
539C2×C4—22no—
540C2×C4—21no—
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—42no—
559C2×C4—42no—
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—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—81no—
581C2×C4—81no—
582C2×C4—81no—
583C2×C4—81no—
584C2×C4—84no—
585C2×C4#41012yes—
586C2×C4#41612yes—
587C2×C4#41612yes—
588C2×C4#42112yes—
589C2×C4C2×#13111yescharacteristic
590C2×C4#56811yescharacteristic
591C2×C4C2×#13511yescharacteristic
592C2×C4C2×#13511yescharacteristic
593C2×C4#57711yescharacteristic
594C2×C4#57811yescharacteristic
595C8—42no—
596C8—81no—
597C8—81no—
598C8—81no—
599C8—81no—
600C8—81no—
601C8—81no—
602C8—81no—
603C22—41no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22—81no—
617C22#288011yescharacteristic
618C22C2×#41011yescharacteristic
619C22#288811yescharacteristic
620C22C2×#41611yescharacteristic
621C22#289811yescharacteristic
622C22C2×#42111yescharacteristic
623C22#290111yescharacteristic
624C4—21no—
625C4—21no—
626C4—22no—
627C4—22no—
628C4—22no—
629C4—21no—
630C4—21no—
631C4—22no—
632C4—21no—
633C4—21no—
634C4—41no—
635C4—41no—
636C4—41no—
637C4—41no—
638C4—42no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—81no—
646C4—81no—
647C4—81no—
648C2—81no—
649C2—161no—
650C2#1741811yescharacteristic
651C2#1750811yescharacteristic
652C2#1752511yescharacteristic
653C2#1757611yescharacteristic
654C2#1760511yescharacteristic
655C2#1767911yescharacteristic
656C2#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. #20781C2semidirect
5. #7716C2semidirect
6. #7682C2semidirect
8. #7662C2semidirect
9. #7656C2semidirect
10. #7648C2semidirect
11. #7644C2semidirect
12. #7614C2semidirect
13. #7583C2semidirect
14. #7581C2semidirect
15. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#363852
#226322
#207812
#77302
#77162
#76822
#76702
#76622
#76562
#76482
#76442
#76142
#75832
#75812
#75602