Finite Group Databasebeta

← #164619

Subgroups — #164619

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36513C211yesmaximal, characteristic
2#22592C211yesmaximal, characteristic, complemented
3#20813C211yesmaximal, characteristic, complemented
4#7733C211yesmaximal, characteristic, complemented
5#7730C211yesmaximal, characteristic
6#7710C211yesmaximal, characteristic
7#7709C211yesmaximal, characteristic, complemented
8#7686C211yesmaximal, characteristic, complemented
9#7672C211yesmaximal, characteristic, complemented
10#7624C211yesmaximal, characteristic, complemented
11#7618C211yesmaximal, characteristic, complemented
12#7617C211yesmaximal, characteristic, complemented
13#7601C211yesmaximal, characteristic, complemented
14#7600C211yesmaximal, characteristic, complemented
15#7571C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17#2863C2211yescharacteristic
18#2860C2211yescharacteristic
19#2850C2211yescharacteristic
20#2601—21no—
21#2587—21no—
22#2570C411yescharacteristic
23#2545C411yescharacteristic
24#2529—21no—
25#2525—21no—
26#2518—21no—
27#2470—21no—
28#2429C411yescharacteristic
29#2426C411yescharacteristic
30#2422C411yescharacteristic
31#2421C411yescharacteristic
32#2374—21no—
33#2373—21no—
34#2367C411yescharacteristic
35#2351C411yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×#124C2211yescharacteristic
38C4×C22⋊Q8C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1917—21no—
41#1916—21no—
42#1851—21no—
43#1843—21no—
44#1839C2211yescharacteristic
45#1831C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1817C2211yescharacteristic
49#1798—21no—
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1768—21no—
53#1767C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C4×C4⋊C8C2211yescharacteristic
56#1753—21no—
57#1745C2211yescharacteristic
58#1728C2211yescharacteristic
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
75C22×M16—21no—
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×#126—21no—
78C2×#124C2×C411yescharacteristic
79C2×#124C2×C411yescharacteristic
80C2×#123—21no—
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×#121C2×C411yescharacteristic
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×Q8C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×D4C2×C411yescharacteristic
89C2×#117D411yescharacteristic
90C2×#117D411yescharacteristic
91C22×C42C2311yescharacteristic
92#433—44no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#400—21no—
109#400—21no—
110#398—21no—
111#398—21no—
112#397—21no—
113#397—21no—
114#396—21no—
115#396—21no—
116#396—21no—
117#396—21no—
118#394—21no—
119#394—21no—
120#391C2×C411yescharacteristic
121#391C2×C411yescharacteristic
122#391C2311yescharacteristic
123#391C2311yescharacteristic
124#390—21no—
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2311yescharacteristic
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2311yescharacteristic
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#388—21no—
150#388—21no—
151#388—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#386—21no—
160#386—21no—
161#385—21no—
162#385—21no—
163#385—21no—
164#384C2×C411yescharacteristic
165#383—21no—
166#382—21no—
167#382C2×C411yescharacteristic
168#382C2×C411yescharacteristic
169#380C2×C411yescharacteristic
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187#376D411yescharacteristic
188#376D411yescharacteristic
189C43C2311yescharacteristic
190C43C2311yescharacteristic
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×Q8C22×C411yescharacteristic
208C22×Q8C22×C411yescharacteristic
209C22×D4—21no—
210C22×D4C22×C411yescharacteristic
211C23×C4—21no—
212C23×C4C22×C411yescharacteristic
213C23×C4C2×D411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C2×M16—42no—
216C2×M16—42no—
217C2×M16—81no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C4⋊Q8—22no—
227C4⋊Q8—41no—
228#126—42no—
229#126—42no—
230#124—22no—
231#124—41no—
232#124—42no—
233#123—44no—
234C22⋊Q8—22no—
235C22⋊Q8—41no—
236C22⋊Q8—81no—
237#121—81no—
238C4×Q8—42no—
239C4×Q8—81no—
240C4×Q8—81no—
241C4×Q8—81no—
242C4×D4—81no—
243#117—22no—
244#117—22no—
245#117—42no—
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⋊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×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4C22⋊C411yescharacteristic
286C2×C22⋊C4C22⋊C411yescharacteristic
287C2×C22⋊C4C22⋊C411yescharacteristic
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42—21no—
294C2×C42—42no—
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
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×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2411yescharacteristic
311C4⋊C8—42no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314C4⋊C8—42no—
315C4⋊C8—42no—
316C4⋊C8—42no—
317C4⋊C8—42no—
318C4⋊C8—81no—
319C22⋊C8—42no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322#97—42no—
323#97—42no—
324C4×C8—42no—
325C4×C8—81no—
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#95—21no—
344#95—21no—
345#95C22⋊C411yescharacteristic
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353C24—21no—
354C24C2×C22⋊C411yescharacteristic
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8#13112yes—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—42no—
368C2×D4—81no—
369C2×D4—81no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—41no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—81no—
391C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C23×C411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406M16—84no—
407C2×C8—22no—
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—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—41no—
437C4⋊C4—42no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—42no—
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—42no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—81no—
475C22⋊C4—81no—
476C22⋊C4—81no—
477C22⋊C4—81no—
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—81no—
500C42—81no—
501C42—81no—
502C42—81no—
503C42AGL(1,ℤ/8)12yes—
504C42#13712yes—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23C22×C22⋊C411yescharacteristic
520Q8—42no—
521Q8—42no—
522Q8—81no—
523Q8—81no—
524D4—81no—
525D4—81no—
526D4—162no—
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—22no—
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—21no—
562C2×C4—21no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
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—41no—
586C2×C4—41no—
587C2×C4—81no—
588C2×C4—81no—
589C2×C4—162no—
590C2×C4#41012yes—
591C2×C4#41712yes—
592C2×C4#41712yes—
593C2×C4#42112yes—
594C2×C4C2×#13111yescharacteristic
595C2×C4#56811yescharacteristic
596C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
597C2×C4C2×#13711yescharacteristic
598C2×C4#57711yescharacteristic
599C2×C4#57811yescharacteristic
600C8—42no—
601C8—81no—
602C8—81no—
603C8—81no—
604C8—81no—
605C8—81no—
606C8—81no—
607C8—81no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—81no—
616C22—81no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—81no—
621C22—81no—
622C22#288011yescharacteristic
623C22C2×#41011yescharacteristic
624C22#288811yescharacteristic
625C22C2×#41711yescharacteristic
626C22#289811yescharacteristic
627C22C2×#42111yescharacteristic
628C22#290111yescharacteristic
629C4—21no—
630C4—21no—
631C4—22no—
632C4—22no—
633C4—22no—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—22no—
639C4—41no—
640C4—42no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—41no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—81no—
653C2—81no—
654C2—161no—
655C2#1741511yescharacteristic
656C2#1751511yescharacteristic
657C2#1752411yescharacteristic
658C2#1756311yescharacteristic
659C2#1760911yescharacteristic
660C2#1768111yescharacteristic
661C2#1776411yescharacteristic
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. #22592C2semidirect
3. #20813C2semidirect
4. #7733C2semidirect
7. #7709C2semidirect
8. #7686C2semidirect
9. #7672C2semidirect
10. #7624C2semidirect
11. #7618C2semidirect
12. #7617C2semidirect
13. #7601C2semidirect
14. #7600C2semidirect
15. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#365132
#225922
#208132
#77332
#77302
#77102
#77092
#76862
#76722
#76242
#76182
#76172
#76012
#76002
#75712