Finite Group Databasebeta

← #729125

Subgroups — #729125

Order 512.

2592 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2582C211yesmaximal, characteristic, complemented
2C2×#2396C211yesmaximal, characteristic
3C2×#2356C211yesmaximal, characteristic, complemented
4C2×#2323C211yesmaximal, characteristic
5#16678C212yesmaximal, complemented
6#16665C212yesmaximal
7#15447C214yesmaximal
8#15347C214yesmaximal, complemented
9#15192C212yesmaximal, complemented
10#15182C212yesmaximal
11#14910C212yesmaximal
12#14901C212yesmaximal, complemented
13#14340C211yesmaximal, characteristic
14#14151C212yesmaximal, complemented
15#13994C212yesmaximal
16#13925C211yesmaximal, characteristic, complemented
17#13910C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2582—28no—
23#2396—28no—
24C2×#400C2211yescharacteristic
25C2×#399C2211yescharacteristic
26C2×#398C2214yes—
27C2×#398C2212yes—
28C2×#396C2212yes—
29C2×#395C2211yescharacteristic
30C2×#393C2211yescharacteristic
31#2356—28no—
32#2323—28no—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#389C2214yes—
43C2×#388C2212yes—
44C2×#388C2212yes—
45C2×#388C2214yes—
46C2×#387C2212yes—
47C2×#387C2214yes—
48C2×#384C2211yescharacteristic
49C2×#384C2212yes—
50C2×#383C2212yes—
51C2×#382C2211yescharacteristic
52C2×#380C2212yes—
53C2×#380C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2212yes—
59C2×#376C2212yes—
60#1446C2212yes—
61#1445C2212yes—
62#1445C2214yes—
63#1444C2212yes—
64#1444C2212yes—
65#1443C2214yes—
66#1443C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1440C2214yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2214yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86#1434C2214yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8D418yes—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—24no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#400—24no—
137#399—24no—
138#398—216no—
139#398—28no—
140#396—28no—
141#395—24no—
142#393—24no—
143#391—24no—
144#391—28no—
145#391—28no—
146#391—28no—
147#389—28no—
148#389—24no—
149#389—28no—
150#389—28no—
151#389—48no—
152#389—48no—
153#388—28no—
154#388—28no—
155#388—48no—
156#387—216no—
157#387—28no—
158#384—28no—
159#384—24no—
160#383—28no—
161#382—24no—
162#380—28no—
163#380—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—28no—
167C4×C4⋊C4D418yes—
168C4×C22⋊C4—28no—
169#376—28no—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C25C2411yescharacteristic
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—42no—
210C22×Q8C2×D414yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4#4114yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C22⋊Q8—232no—
235C4×Q8—232no—
236C4×Q8—416no—
237#117—416no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D418yes—
293C2×C4⋊C4C2×D418yes—
294C2×C4⋊C4#4114yes—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—28no—
327C2×C42—28no—
328C2×C42—24no—
329C2×C42—44no—
330C2×C42C2×D418yes—
331C2×C42C2×D418yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42#4114yes—
335#95—24no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95C2×D414yes—
362#95C2×D414yes—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—21no—
371C24—21no—
372C24—22no—
373C24C2×#4112yes—
374C24C2511yescharacteristic
375C2×Q8—28no—
376C2×Q8—28no—
377C2×Q8—28no—
378C2×Q8—28no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D414yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C42−1+414yes—
489C22×C42−1+414yes—
490C22×C42−1+412yes—
491C22×C42−1+412yes—
492C4⋊C4—216no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—48no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—48no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—48no—
508C4⋊C4—84no—
509C4⋊C4—84no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C22⋊C4—216no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—44no—
516C22⋊C4—44no—
517C22⋊C4—48no—
518C22⋊C4—84no—
519C42—216no—
520C42—216no—
521C42—44no—
522C42—48no—
523C42—416no—
524C42—84no—
525C42#121116yes—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23C23×D411yescharacteristic
550C23C23×D411yescharacteristic
551C23C22×#4111yescharacteristic
552C23C22×#4111yescharacteristic
553C23C22×#4111yescharacteristic
554C23C2×H(5,𝔽2)12yes—
555C23C2×2−1+411yescharacteristic
556C23C2×2−1+411yescharacteristic
557C23C2×2−1+412yes—
558C23C2×2−1+412yes—
559C23#58512yes—
560Q8—416no—
561Q8—416no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—44no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—416no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4C2×#12114yes—
652C2×C4C2×#12114yes—
653C2×C4C2×#12114yes—
654C2×C4#52514yes—
655C2×C4#53614yes—
656C2×C4#53718yes—
657C2×C4#54714yes—
658C2×C4#54814yes—
659C2×C4#55214yes—
660C2×C4#55614yes—
661C22—42no—
662C22—44no—
663C22—42no—
664C22—44no—
665C22—42no—
666C22—82no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—82no—
671C22—81no—
672C22C22×#12111yescharacteristic
673C22C2×#52511yescharacteristic
674C22C2×#53611yescharacteristic
675C22C2×#53712yes—
676C22C2×#54711yescharacteristic
677C22C2×#54811yescharacteristic
678C22C2×#55211yescharacteristic
679C22C2×#55611yescharacteristic
680C22#348611yescharacteristic
681C22#348711yescharacteristic
682C22#348912yes—
683C22#349512yes—
684C22#349811yescharacteristic
685C22#349911yescharacteristic
686C22#350612yes—
687C22#350912yes—
688C22#351112yes—
689C22#351312yes—
690C22#352312yes—
691C22#354412yes—
692C22#355012yes—
693C22#355512yes—
694C22#356312yes—
695C4—24no—
696C4—28no—
697C4—24no—
698C4—44no—
699C4—44no—
700C4—44no—
701C4—42no—
702C4—44no—
703C4—44no—
704C4—84no—
705C4—82no—
706C4—84no—
707C4—81no—
708C4—81no—
709C4—82no—
710C4#241518yes—
711C2—82no—
712C2C2×#241511yescharacteristic
713C2#3527311yescharacteristic
714C2#3566111yescharacteristic
715C2#3574911yescharacteristic
716C2#3584411yescharacteristic
717C2#3804312yes—
718C2#3972712yes—
719C2#4278911yescharacteristic
720C2#4287911yescharacteristic
721C2#4495612yes—
722C2#4506512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2582C2semidirect
3. C2×#2356C2semidirect
5. #16678C2semidirect
8. #15347C2semidirect
9. #15192C2semidirect
12. #14901C2semidirect
14. #14151C2semidirect
16. #13925C2semidirect
17. #13910C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25822
C2×#23962
C2×#23562
C2×#23232
#166782
#166652
#154472
#153472
#151922
#151822
#149102
#149012
#143402
#141512
#139942
#139252
#139102