Finite Group Databasebeta

← #164756

Subgroups — #164756

Order 512.

785 conjugacy classes of subgroups.

The subgroup lattice has 695 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36460C211yesmaximal, characteristic
2#22589C211yesmaximal, characteristic, complemented
3#20785C211yesmaximal, characteristic, complemented
4#7726C211yesmaximal, characteristic, complemented
5#7724C211yesmaximal, characteristic
6#7692C211yesmaximal, characteristic, complemented
7#7672C211yesmaximal, characteristic, complemented
8#7666C211yesmaximal, characteristic, complemented
9#7664C211yesmaximal, characteristic
10#7649C211yesmaximal, characteristic, complemented
11#7648C211yesmaximal, characteristic, complemented
12#7618C211yesmaximal, characteristic, complemented
13#7589C211yesmaximal, characteristic, complemented
14#7584C211yesmaximal, characteristic, complemented
15#7567C211yesmaximal, characteristic, complemented
16#2860C2211yescharacteristic
17#2860C2211yescharacteristic
18#2856C2211yescharacteristic
19#2571C411yescharacteristic
20#2559—21no—
21#2559—21no—
22#2548C411yescharacteristic
23#2526—21no—
24#2526—21no—
25#2512—21no—
26#2500—21no—
27#2426C411yescharacteristic
28#2426C411yescharacteristic
29#2420C411yescharacteristic
30#2420C411yescharacteristic
31#2366C411yescharacteristic
32#2363—21no—
33#2357—21no—
34Q8×C4⋊C4C411yescharacteristic
35C4×#124C2211yescharacteristic
36C4×#124C2211yescharacteristic
37C4×C22⋊Q8C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1916—21no—
40#1916—21no—
41#1850—21no—
42#1839C2211yescharacteristic
43#1839C2211yescharacteristic
44#1839C2211yescharacteristic
45#1832—21no—
46#1831C2211yescharacteristic
47#1818—21no—
48#1817C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1768—21no—
52C4×C4⋊C8C2211yescharacteristic
53C8×C22⋊C4—21no—
54#1750—21no—
55#1745C2211yescharacteristic
56#1744C2211yescharacteristic
57#1744C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1270C2211yescharacteristic
68#1270C2211yescharacteristic
69#1270C2211yescharacteristic
70#1270C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×C8—21no—
76C2×#127C2×C411yescharacteristic
77C2×#126—21no—
78C2×#126—21no—
79C2×#124C2×C411yescharacteristic
80C2×#124C2×C411yescharacteristic
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×#121—21no—
83C2×#121—21no—
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×D4—21no—
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×#117D411yescharacteristic
93C2×#117D411yescharacteristic
94C22×C42C2311yescharacteristic
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106#400—21no—
107#400—21no—
108#397—21no—
109#397—21no—
110#396—21no—
111#396—21no—
112#394—21no—
113#394—21no—
114#394—21no—
115#394—21no—
116#394—21no—
117#394—21no—
118#391C2311yescharacteristic
119#390—21no—
120#390—21no—
121#390—21no—
122#390—21no—
123#390—21no—
124#390—21no—
125#390—21no—
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#390C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#390C2311yescharacteristic
131#390C2311yescharacteristic
132#389—21no—
133#389—21no—
134#389—21no—
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2×C411yescharacteristic
138#389C2×C411yescharacteristic
139#389C2311yescharacteristic
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#386—21no—
154#386—21no—
155#386—21no—
156#386—21no—
157#385—21no—
158#384—21no—
159#384C2×C411yescharacteristic
160#382—21no—
161#382C2×C411yescharacteristic
162#382C2×C411yescharacteristic
163#381—21no—
164#381—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⋊C4D411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
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#376D411yescharacteristic
183#376D411yescharacteristic
184C43C2311yescharacteristic
185C43C2311yescharacteristic
186#336—21no—
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#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C22×Q8C22×C411yescharacteristic
207C22×D4—21no—
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4C22×C411yescharacteristic
212C22×D4C22×C411yescharacteristic
213C23×C4—21no—
214C23×C4—21no—
215C23×C4C22×C411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
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—81no—
229#127—42no—
230#126—81no—
231#126—81no—
232#126—81no—
233#126—81no—
234#124—22no—
235#124—22no—
236#124—41no—
237#124—41no—
238C22⋊Q8—22no—
239C22⋊Q8—41no—
240#121—42no—
241#121—42no—
242#121—81no—
243#121—81no—
244#121—81no—
245#121—81no—
246C4×Q8—42no—
247C4×Q8—81no—
248C4×Q8—81no—
249C4×D4—44no—
250C4×D4—81no—
251C4×D4—81no—
252#117—22no—
253#117—22no—
254#117—42no—
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⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C22⋊C4C22⋊C411yescharacteristic
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—42no—
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C22×C411yescharacteristic
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2411yescharacteristic
322C4⋊C8—42no—
323C4⋊C8—42no—
324C4⋊C8—42no—
325C4⋊C8—81no—
326C22⋊C8—42no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329#97—42no—
330C4×C8—42no—
331C4×C8—42no—
332C4×C8—81no—
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#95—21no—
346#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354C24—21no—
355C24—21no—
356C24C2×C22⋊C411yescharacteristic
357C2×Q8—21no—
358C2×Q8—41no—
359C2×Q8—41no—
360C2×Q8#13112yes—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—81no—
376C2×D4—81no—
377C2×D4—81no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4—81no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—42no—
398C22×C4—42no—
399C22×C4—81no—
400C22×C4—81no—
401C22×C4—81no—
402C22×C4—81no—
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4#13512yes—
415C22×C4#13512yes—
416C22×C4C23×C411yescharacteristic
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C2×C8—22no—
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—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—84no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—41no—
450C4⋊C4—42no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—81no—
460C4⋊C4—81no—
461C4⋊C4—81no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—22no—
469C22⋊C4—41no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—42no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—41no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—42no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—42no—
519C42—42no—
520C42—81no—
521C42—81no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23#41612yes—
543C23C22×C22⋊C411yescharacteristic
544Q8—42no—
545Q8—42no—
546D4—81no—
547D4—81no—
548D4—81no—
549D4—81no—
550D4—84no—
551D4—162no—
552D4—162no—
553D4—162no—
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—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—162no—
615C2×C4—162no—
616C2×C4#41012yes—
617C2×C4#41612yes—
618C2×C4#42112yes—
619C2×C4C2×#13111yescharacteristic
620C2×C4#56811yescharacteristic
621C2×C4C2×#13511yescharacteristic
622C2×C4C2×#13511yescharacteristic
623C2×C4#57611yescharacteristic
624C2×C4#57711yescharacteristic
625C8—42no—
626C8—81no—
627C8—81no—
628C8—81no—
629C8—81no—
630C8—81no—
631C8—81no—
632C8—81no—
633C22—22no—
634C22—22no—
635C22—22no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—81no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22#288011yescharacteristic
655C22C2×#41011yescharacteristic
656C22#288811yescharacteristic
657C22C2×#41611yescharacteristic
658C22#289711yescharacteristic
659C22C2×#42111yescharacteristic
660C22#290111yescharacteristic
661C4—21no—
662C4—21no—
663C4—22no—
664C4—21no—
665C4—21no—
666C4—22no—
667C4—21no—
668C4—21no—
669C4—22no—
670C4—41no—
671C4—42no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—81no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—81no—
684C2—42no—
685C2—161no—
686C2—161no—
687C2#1741511yescharacteristic
688C2#1751011yescharacteristic
689C2#1752611yescharacteristic
690C2#1756711yescharacteristic
691C2#1760411yescharacteristic
692C2#1767911yescharacteristic
693C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2856C22
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. #22589C2semidirect
3. #20785C2semidirect
4. #7726C2semidirect
6. #7692C2semidirect
7. #7672C2semidirect
8. #7666C2semidirect
10. #7649C2semidirect
11. #7648C2semidirect
12. #7618C2semidirect
13. #7589C2semidirect
14. #7584C2semidirect
15. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364602
#225892
#207852
#77262
#77242
#76922
#76722
#76662
#76642
#76492
#76482
#76182
#75892
#75842
#75672