Finite Group Databasebeta

← #164196

Subgroups — #164196

Order 512.

837 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36366C211yesmaximal, characteristic
2#22636C211yesmaximal, characteristic, complemented
3#20468C211yesmaximal, characteristic, complemented
4#7709C211yesmaximal, characteristic, complemented
5#7696C211yesmaximal, characteristic, complemented
6#7693C211yesmaximal, characteristic, complemented
7#7676C211yesmaximal, characteristic, complemented
8#7674C211yesmaximal, characteristic, complemented
9#7658C211yesmaximal, characteristic, complemented
10#7637C211yesmaximal, characteristic, complemented
11#7628C211yesmaximal, characteristic, complemented
12#7612C211yesmaximal, characteristic, complemented
13#7587C211yesmaximal, characteristic, complemented
14#7574C211yesmaximal, characteristic, complemented
15#7563C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17#2863C2211yescharacteristic
18#2861C2211yescharacteristic
19#2854C2211yescharacteristic
20#2598—21no—
21#2526—21no—
22#2516—21no—
23#2511C411yescharacteristic
24#2471—21no—
25#2471—21no—
26#2457—21no—
27#2454C411yescharacteristic
28#2447C411yescharacteristic
29#2445C411yescharacteristic
30#2441C411yescharacteristic
31#2431C411yescharacteristic
32#2363C411yescharacteristic
33#2359—21no—
34#2353—21no—
35#2339C411yescharacteristic
36C4×#127C2211yescharacteristic
37#2297C2211yescharacteristic
38#2293C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1939—21no—
41#1924—21no—
42#1924—21no—
43#1831C2211yescharacteristic
44#1831C2211yescharacteristic
45#1797C2211yescharacteristic
46#1770—21no—
47#1768—21no—
48#1767C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50#1750—21no—
51#1749C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1745C2211yescharacteristic
55#1745C2211yescharacteristic
56#1745C2211yescharacteristic
57#1744C2211yescharacteristic
58#1726—21no—
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
75C22×#41C2×C411yescharacteristic
76C2×#127C2×C411yescharacteristic
77C2×#125C2×C411yescharacteristic
78C2×#124C2×C411yescharacteristic
79C2×#123—21no—
80C2×#123—21no—
81C2×#121C2×C411yescharacteristic
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×#117—21no—
88C2×#117C2×C411yescharacteristic
89C2×#117D411yescharacteristic
90C2×#117D411yescharacteristic
91C22×C42C2311yescharacteristic
92#433—28no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111#400—21no—
112#400—21no—
113#399—21no—
114#399—21no—
115#397—21no—
116#397—21no—
117#396—21no—
118#396—21no—
119#394—21no—
120#394—21no—
121#392—21no—
122#392—21no—
123#391C2×C411yescharacteristic
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#391C2311yescharacteristic
128#390—21no—
129#390—21no—
130#390—21no—
131#390C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#390C2311yescharacteristic
136#390C2311yescharacteristic
137#389—21no—
138#389C2×C411yescharacteristic
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2311yescharacteristic
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#386—21no—
159#386—21no—
160#386—21no—
161#386—21no—
162#386—21no—
163#386—21no—
164#386—21no—
165#386—21no—
166#386—21no—
167#385—21no—
168#385—21no—
169#384C2×C411yescharacteristic
170#383—21no—
171#381—21no—
172#381—21no—
173#380—21no—
174#380C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4C2×C411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188#376D411yescharacteristic
189#376D411yescharacteristic
190C2×#95—21no—
191C43C2311yescharacteristic
192C43C2311yescharacteristic
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
207C2×#41—22no—
208C2×#41—41no—
209C2×#41—42no—
210C2×#41—81no—
211C22×Q8C22×C411yescharacteristic
212C22×D4—21no—
213C22×D4—21no—
214C22×D4—21no—
215C22×D4—21no—
216C22×D4C22×C411yescharacteristic
217C22×D4C22×C411yescharacteristic
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4C22×C411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C23×C4C2×D411yescharacteristic
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233#127—22no—
234#127—41no—
235#125—81no—
236#124—81no—
237#123—44no—
238#123—81no—
239#123—81no—
240#121—42no—
241C4×D4—22no—
242C4×D4—41no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—81no—
246C4×D4—81no—
247C4×D4—81no—
248C4×D4—81no—
249#117—24no—
250#117—41no—
251#117—41no—
252#117—81no—
253#117—81no—
254#117—81no—
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⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
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×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⋊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⋊C4C22⋊C411yescharacteristic
294C2×C22⋊C4C22⋊C411yescharacteristic
295C2×C22⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—42no—
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2411yescharacteristic
316C4⋊C8—24no—
317C4⋊C8—24no—
318C4⋊C8—42no—
319C4⋊C8—81no—
320C4⋊C8—81no—
321C22⋊C8—24no—
322C22⋊C8—24no—
323C22⋊C8—42no—
324C22⋊C8—81no—
325#97—24no—
326#97—42no—
327#97—42no—
328#97—42no—
329#97—42no—
330C4×C8—24no—
331C4×C8—42no—
332C4×C8—42no—
333C4×C8—42no—
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#95—21no—
347#95—21no—
348#95—21no—
349#95—21no—
350#95—21no—
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356#95C22⋊C411yescharacteristic
357#95C22⋊C411yescharacteristic
358#95C22⋊C411yescharacteristic
359C24—21no—
360C24—21no—
361C24C2×C22⋊C411yescharacteristic
362#41—84no—
363#41—84no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×D4—21no—
369C2×D4—21no—
370C2×D4—21no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—42no—
381C2×D4—42no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4—81no—
386C2×D4—81no—
387C2×D4#13112yes—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C2×C22⋊C411yescharacteristic
428C22×C4C2×C22⋊C411yescharacteristic
429C22×C4C23×C411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C22×C4C22×D411yescharacteristic
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—42no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4#10414yes—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—42no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—81no—
503C22⋊C4—81no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—22no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—41no—
524C42—41no—
525C42—41no—
526C42—81no—
527C42—81no—
528C23—22no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23C22×C22⋊C411yescharacteristic
550Q8—81no—
551Q8—81no—
552D4—42no—
553D4—42no—
554D4—84no—
555D4—81no—
556D4—81no—
557D4—84no—
558D4—84no—
559D4—162no—
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—21no—
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—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
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—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—84no—
620C2×C4—84no—
621C2×C4—162no—
622C2×C4—162no—
623C2×C4#41012yes—
624C2×C4C2×#10412yes—
625C2×C4C2×#10412yes—
626C2×C4C2×#10412yes—
627C2×C4C2×#13111yescharacteristic
628C2×C4#56811yescharacteristic
629C2×C4#57511yescharacteristic
630C2×C4#57511yescharacteristic
631C2×C4#57611yescharacteristic
632C2×C4#57711yescharacteristic
633C8—42no—
634C8—42no—
635C8—42no—
636C8—42no—
637C8—42no—
638C8—81no—
639C8—81no—
640C8—81no—
641C22—41no—
642C22—42no—
643C22—41no—
644C22—42no—
645C22—41no—
646C22—42no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22—81no—
662C22#288011yescharacteristic
663C22C2×#41011yescharacteristic
664C22#288811yescharacteristic
665C22#289611yescharacteristic
666C22#289711yescharacteristic
667C22C22×#10411yescharacteristic
668C22#290111yescharacteristic
669C4—22no—
670C4—21no—
671C4—21no—
672C4—22no—
673C4—22no—
674C4—21no—
675C4—21no—
676C4—22no—
677C4—21no—
678C4—21no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—81no—
691C4—81no—
692C2—81no—
693C2—82no—
694C2—161no—
695C2#1741211yescharacteristic
696C2#1751311yescharacteristic
697C2#1751711yescharacteristic
698C2#1754311yescharacteristic
699C2#1761011yescharacteristic
700C2#1765511yescharacteristic
701C2#1776611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2431C4
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. #22636C2semidirect
3. #20468C2semidirect
4. #7709C2semidirect
5. #7696C2semidirect
6. #7693C2semidirect
7. #7676C2semidirect
8. #7674C2semidirect
9. #7658C2semidirect
10. #7637C2semidirect
11. #7628C2semidirect
12. #7612C2semidirect
13. #7587C2semidirect
14. #7574C2semidirect
15. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#363662
#226362
#204682
#77092
#76962
#76932
#76762
#76742
#76582
#76372
#76282
#76122
#75872
#75742
#75632