Finite Group Databasebeta

← #164448

Subgroups — #164448

Order 512.

967 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36376C211yesmaximal, characteristic
2#21998C211yesmaximal, characteristic, complemented
3#20567C211yesmaximal, characteristic, complemented
4#7710C211yesmaximal, characteristic, complemented
5#7696C211yesmaximal, characteristic, complemented
6#7687C211yesmaximal, characteristic, complemented
7#7687C211yesmaximal, characteristic, complemented
8#7670C211yesmaximal, characteristic, complemented
9#7656C211yesmaximal, characteristic, complemented
10#7631C211yesmaximal, characteristic, complemented
11#7624C211yesmaximal, characteristic, complemented
12#7609C211yesmaximal, characteristic, complemented
13#7590C211yesmaximal, characteristic, complemented
14#7575C211yesmaximal, characteristic, complemented
15#7563C211yesmaximal, characteristic, complemented
16C2×#552C411yescharacteristic
17C2×#548C411yescharacteristic
18C2×#436—21no—
19#2861C2211yescharacteristic
20#2857C2211yescharacteristic
21#2850C2211yescharacteristic
22#2565C411yescharacteristic
23#2556—21no—
24#2556—21no—
25#2545C411yescharacteristic
26#2521—21no—
27#2521—21no—
28#2501C411yescharacteristic
29#2499C411yescharacteristic
30#2434C411yescharacteristic
31#2432C411yescharacteristic
32#2425—21no—
33#2425—21no—
34#2359—21no—
35#2359—21no—
36C4×C4⋊Q8C2211yescharacteristic
37#2297C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1923—21no—
41#1916—21no—
42#1870—21no—
43#1851—21no—
44#1839C2211yescharacteristic
45#1831C2211yescharacteristic
46#1817C2211yescharacteristic
47#1797C2211yescharacteristic
48C8×C4⋊C4—21no—
49#1767C2211yescharacteristic
50#1753—21no—
51#1753—21no—
52C4×C22⋊C8C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55#1744C2211yescharacteristic
56#1728C2211yescharacteristic
57#1728C2211yescharacteristic
58#1728C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1272C2211yescharacteristic
68#1272C2211yescharacteristic
69#1272C2211yescharacteristic
70#1272C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×#41C2×C411yescharacteristic
76C23×Q8C2×C411yescharacteristic
77C22×M16—21no—
78C22×M16—21no—
79C22×M16—21no—
80#552—44no—
81#548—44no—
82C2×C4⋊Q8C2×C411yescharacteristic
83C2×C4⋊Q8C2×C411yescharacteristic
84C2×#126—21no—
85C2×#125—21no—
86C2×#125C2×C411yescharacteristic
87C2×#124—21no—
88C2×#124C2×C411yescharacteristic
89C2×#124C2×C411yescharacteristic
90C2×#123—21no—
91C2×#123—21no—
92C2×#123—21no—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8C2×C411yescharacteristic
95C2×C22⋊Q8C2×C411yescharacteristic
96C2×C22⋊Q8C2×C411yescharacteristic
97C2×#121—21no—
98C2×#121C2×C411yescharacteristic
99C2×C4×Q8C2×C411yescharacteristic
100C2×C4×Q8C2×C411yescharacteristic
101C2×C4×D4—21no—
102C2×C4×D4C2×C411yescharacteristic
103C2×C4×D4D411yescharacteristic
104C2×C4×D4D411yescharacteristic
105C2×#117—21no—
106C2×#117C2×C411yescharacteristic
107C22×C4⋊C4C2×C411yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#436—28no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×#97—21no—
120C2×C4×C8—21no—
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124#399—21no—
125#399—21no—
126#398—21no—
127#398—21no—
128#397—21no—
129#397—21no—
130#396—21no—
131#396—21no—
132#395—21no—
133#395—21no—
134#394—21no—
135#394—21no—
136#391—21no—
137#391—21no—
138#391C2×C411yescharacteristic
139#391C2×C411yescharacteristic
140#391C2×C411yescharacteristic
141#391C2×C411yescharacteristic
142#391C2311yescharacteristic
143#391C2311yescharacteristic
144#391C2311yescharacteristic
145#390C2×C411yescharacteristic
146#390C2×C411yescharacteristic
147#390C2311yescharacteristic
148#389—21no—
149#389—21no—
150#389—21no—
151#389—21no—
152#388—21no—
153#388—21no—
154#388—21no—
155#388—21no—
156#388—21no—
157#388—21no—
158#388—21no—
159#388—21no—
160#387—21no—
161#387—21no—
162#387—21no—
163#387—21no—
164#386—21no—
165#386—21no—
166#386—21no—
167#386—21no—
168#385—21no—
169#385—21no—
170#385—21no—
171#385—21no—
172#384C2×C411yescharacteristic
173#384C2×C411yescharacteristic
174#382C2×C411yescharacteristic
175#380—21no—
176#380—21no—
177C4×C4⋊C4—21no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C22⋊C4—21no—
183C4×C22⋊C4—21no—
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4C2×C411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C4×C22⋊C4D411yescharacteristic
190C2×#95—21no—
191C2×#95—21no—
192C43C2311yescharacteristic
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336—21no—
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208#336C2311yescharacteristic
209C2×#41—22no—
210C2×#41—41no—
211C2×#41—42no—
212C2×#41—81no—
213C22×Q8—21no—
214C22×Q8—22no—
215C22×Q8—41no—
216C22×Q8—42no—
217C22×Q8—81no—
218C22×Q8C22×C411yescharacteristic
219C22×Q8C22×C411yescharacteristic
220C22×Q8C22×C411yescharacteristic
221C22×D4—21no—
222C22×D4—21no—
223C22×D4C22⋊C411yescharacteristic
224C22×D4C22×C411yescharacteristic
225C23×C4—21no—
226C23×C4—21no—
227C23×C4—21no—
228C23×C4C22×C411yescharacteristic
229C23×C4C22×C411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C2×M16—24no—
235C2×M16—24no—
236C2×M16—42no—
237C2×M16—81no—
238C2×M16—81no—
239C2×M16—81no—
240C2×M16—81no—
241C2×M16—81no—
242C2×M16—81no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8—21no—
249C22×C8—21no—
250C22×C8—21no—
251C4⋊Q8—22no—
252C4⋊Q8—41no—
253C4⋊Q8—81no—
254#126—44no—
255#125—42no—
256#125—44no—
257#124—42no—
258#124—44no—
259#124—81no—
260#123—44no—
261#123—44no—
262#123—44no—
263C22⋊Q8—44no—
264C22⋊Q8—42no—
265C22⋊Q8—42no—
266C22⋊Q8—42no—
267#121—44no—
268#121—42no—
269C4×Q8—42no—
270C4×Q8—42no—
271C4×D4—24no—
272C4×D4—24no—
273C4×D4—42no—
274C4×D4—44no—
275#117—44no—
276#117—42no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—42no—
294C2×C4⋊C4C22⋊C411yescharacteristic
295C2×C4⋊C4C22⋊C411yescharacteristic
296C2×C4⋊C4C22⋊C411yescharacteristic
297C2×C4⋊C4C22×C411yescharacteristic
298C2×C4⋊C4C22×C411yescharacteristic
299C2×C4⋊C4C22×C411yescharacteristic
300C2×C4⋊C4C22×C411yescharacteristic
301C2×C22⋊C4—21no—
302C2×C22⋊C4—21no—
303C2×C22⋊C4—21no—
304C2×C22⋊C4—21no—
305C2×C22⋊C4—21no—
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—21no—
311C2×C22⋊C4—21no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—21no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—21no—
316C2×C22⋊C4C22⋊C411yescharacteristic
317C2×C22⋊C4C22⋊C411yescharacteristic
318C2×C22⋊C4C22⋊C411yescharacteristic
319C2×C22⋊C4C22⋊C411yescharacteristic
320C2×C22⋊C4C22⋊C411yescharacteristic
321C2×C22⋊C4C22⋊C411yescharacteristic
322C2×C42—21no—
323C2×C42—21no—
324C2×C42—21no—
325C2×C42—81no—
326C2×C42—81no—
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330C2×C42C22×C411yescharacteristic
331C2×C42C2×D411yescharacteristic
332C2×C42C2×D411yescharacteristic
333C2×C42C2×D411yescharacteristic
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2411yescharacteristic
340C4⋊C8—24no—
341C4⋊C8—81no—
342C4⋊C8—81no—
343C22⋊C8—24no—
344C22⋊C8—24no—
345C22⋊C8—42no—
346C22⋊C8—81no—
347C22⋊C8—81no—
348#97—81no—
349C4×C8—24no—
350C4×C8—81no—
351C4×C8—81no—
352C4×C8—81no—
353#95—21no—
354#95—21no—
355#95—21no—
356#95—21no—
357#95—21no—
358#95—21no—
359#95—21no—
360#95—21no—
361#95—21no—
362#95—21no—
363#95—21no—
364#95—21no—
365#95—21no—
366#95—21no—
367#95—21no—
368#95—21no—
369#95C22⋊C411yescharacteristic
370#95C22⋊C411yescharacteristic
371#95C22⋊C411yescharacteristic
372#95C22⋊C411yescharacteristic
373#95C22⋊C411yescharacteristic
374#95C22⋊C411yescharacteristic
375C24—21no—
376C24C2×C22⋊C411yescharacteristic
377C24C2×C22⋊C411yescharacteristic
378#41—84no—
379#41—84no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—42no—
386C2×Q8—41no—
387C2×Q8—41no—
388C2×Q8—41no—
389C2×Q8—41no—
390C2×Q8—41no—
391C2×Q8—41no—
392C2×Q8—42no—
393C2×Q8—81no—
394C2×Q8—84no—
395C2×Q8—84no—
396C2×Q8#13112yes—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—22no—
401C2×D4—41no—
402C2×D4—42no—
403C2×D4—41no—
404C2×D4—41no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4—42no—
409C2×D4—81no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—81no—
449C22×C4—81no—
450C22×C4C2×C22⋊C411yescharacteristic
451C22×C4C2×C22⋊C411yescharacteristic
452C22×C4C2×C22⋊C411yescharacteristic
453C22×C4C2×C22⋊C411yescharacteristic
454C22×C4C2×C22⋊C411yescharacteristic
455C22×C4C2×C22⋊C411yescharacteristic
456C22×C4C2×C22⋊C411yescharacteristic
457C22×C4C2×C22⋊C411yescharacteristic
458C22×C4C2×C22⋊C411yescharacteristic
459C22×C4C2×C22⋊C411yescharacteristic
460C22×C4C23×C411yescharacteristic
461C22×C4C22×D411yescharacteristic
462C22×C4C22×D411yescharacteristic
463M16—84no—
464M16—84no—
465M16—84no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—41no—
473C2×C8—41no—
474C2×C8—41no—
475C2×C8—41no—
476C2×C8—41no—
477C2×C8—41no—
478C2×C8—41no—
479C2×C8—41no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—41no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—41no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—81no—
511C4⋊C4—81no—
512C4⋊C4#10014yes—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—22no—
516C22⋊C4—22no—
517C22⋊C4—41no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C22⋊C4—41no—
528C22⋊C4—42no—
529C22⋊C4—42no—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—42no—
533C22⋊C4—81no—
534C22⋊C4—81no—
535C22⋊C4#10414yes—
536C42—21no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—22no—
543C42—22no—
544C42—41no—
545C42—41no—
546C42—41no—
547C42—41no—
548C42—41no—
549C42—41no—
550C42—41no—
551C42—41no—
552C42—42no—
553C42—42no—
554C42—42no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—41no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23C2×#10412yes—
576C23C22×C22⋊C411yescharacteristic
577Q8—42no—
578Q8—42no—
579Q8—84no—
580Q8—81no—
581Q8—81no—
582Q8—81no—
583Q8—81no—
584D4—44no—
585D4—81no—
586D4—81no—
587D4—84no—
588D4—84no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—81no—
651C2×C4—81no—
652C2×C4—84no—
653C2×C4—84no—
654C2×C4—84no—
655C2×C4C2×#10012yes—
656C2×C4C2×#10012yes—
657C2×C4C2×#10012yes—
658C2×C4C2×#10412yes—
659C2×C4C2×#10412yes—
660C2×C4#42112yes—
661C2×C4C2×#13111yescharacteristic
662C2×C4#56811yescharacteristic
663C2×C4#57711yescharacteristic
664C2×C4#57711yescharacteristic
665C2×C4#57811yescharacteristic
666C2×C4#57811yescharacteristic
667C8—42no—
668C8—42no—
669C8—42no—
670C8—42no—
671C8—81no—
672C8—81no—
673C8—81no—
674C8—81no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—41no—
679C22—41no—
680C22—41no—
681C22—41no—
682C22—41no—
683C22—41no—
684C22—41no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—81no—
690C22—81no—
691C22—81no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22#288011yescharacteristic
697C22#288311yescharacteristic
698C22C22×#10011yescharacteristic
699C22#289811yescharacteristic
700C22#289811yescharacteristic
701C22C22×#10411yescharacteristic
702C22C2×#42111yescharacteristic
703C4—21no—
704C4—21no—
705C4—22no—
706C4—22no—
707C4—22no—
708C4—21no—
709C4—21no—
710C4—22no—
711C4—21no—
712C4—21no—
713C4—22no—
714C4—22no—
715C4—42no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—81no—
721C4—81no—
722C4—81no—
723C4—81no—
724C4—81no—
725C4—81no—
726C2—42no—
727C2—81no—
728C2—161no—
729C2#1742111yescharacteristic
730C2#1751911yescharacteristic
731C2#1752511yescharacteristic
732C2#1754811yescharacteristic
733C2#1759811yescharacteristic
734C2#1764711yescharacteristic
735C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#548C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21998C2semidirect
3. #20567C2semidirect
4. #7710C2semidirect
5. #7696C2semidirect
6. #7687C2semidirect
7. #7687C2semidirect
8. #7670C2semidirect
9. #7656C2semidirect
10. #7631C2semidirect
11. #7624C2semidirect
12. #7609C2semidirect
13. #7590C2semidirect
14. #7575C2semidirect
15. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#363762
#219982
#205672
#77102
#76962
#76872
#76702
#76562
#76312
#76242
#76092
#75902
#75752
#75632