Finite Group Databasebeta

← #626008

Subgroups — #626008

Order 512.

3210 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic
3C2×#2565C211yesmaximal, characteristic, complemented
4C2×#2566C211yesmaximal, characteristic
5C2×#2660C211yesmaximal, characteristic, complemented
6C2×#2326C211yesmaximal, characteristic
7#15765C218yesmaximal, complemented
8#15430C218yesmaximal, complemented
9#15244C211yesmaximal, characteristic
10#15240C211yesmaximal, characteristic, complemented
11#15199C211yesmaximal, characteristic, complemented
12#15155C211yesmaximal, characteristic
13#14708C211yesmaximal, characteristic
14#14692C211yesmaximal, characteristic, complemented
15#14354C211yesmaximal, characteristic, complemented
16#14273C211yesmaximal, characteristic
17#13759C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#124C2211yescharacteristic
21C22×C22⋊Q8C2214yes—
22C22×C4×Q8C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#2660—28no—
25#2567—28no—
26#2566—28no—
27#2565—28no—
28#2542—28no—
29C2×#399C2214yes—
30C2×#399C2218yes—
31C2×#398C2214yes—
32C2×#398C2214yes—
33C2×#396C2214yes—
34C2×#395C2214yes—
35C2×#395C2214yes—
36#2326—28no—
37C2×#391C2211yescharacteristic
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#389C2214yes—
48C2×#388C2214yes—
49C2×#387C2214yes—
50C2×#385C2214yes—
51C2×#385C2214yes—
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C2×#384C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C2×#384C2218yes—
62C2×#382C2211yescharacteristic
63C2×#382C2211yescharacteristic
64C2×#382C2211yescharacteristic
65C2×#382C2211yescharacteristic
66C2×C4×C4⋊C4C2211yescharacteristic
67C2×C4×C4⋊C4C2211yescharacteristic
68C2×#376C2211yescharacteristic
69C2×#376C2211yescharacteristic
70C2×C43C2211yescharacteristic
71#1445C2218yes—
72#1444C2218yes—
73#1444C2218yes—
74#1439C2218yes—
75#1439C2218yes—
76#1437C2218yes—
77#1436C2214yes—
78#1436C2214yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×Q8C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2312yes—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×#125—24no—
89C2×#125—24no—
90C2×#125—24no—
91C2×#124—24no—
92C2×#124—24no—
93C2×#124Q818yescomplemented
94C2×C22⋊Q8—216no—
95C2×C22⋊Q8—416no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#399—216no—
140#399—416no—
141#398—216no—
142#398—216no—
143#396—216no—
144#395—216no—
145#395—216no—
146#391—24no—
147#391—24no—
148#391—24no—
149#391—24no—
150#391—24no—
151#390—24no—
152#390—24no—
153#390—24no—
154#390—24no—
155#390—24no—
156#389—216no—
157#388—216no—
158#387—216no—
159#385—216no—
160#385—216no—
161#384—24no—
162#384—24no—
163#384—24no—
164#384—24no—
165#384—24no—
166#384—24no—
167#384—24no—
168#384—24no—
169#384—24no—
170#384—416no—
171#382—24no—
172#382—24no—
173#382—24no—
174#382—24no—
175C4×C4⋊C4D418yes—
176C4×C4⋊C4D418yes—
177#376—24no—
178#376D418yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C43—24no—
196C25C2412yes—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—48no—
207C22×Q8C2×Q814yes—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C2×Q814yes—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—48no—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C4⋊Q8—416no—
236#125—416no—
237#124—232no—
238C22⋊Q8—464no—
239C4×Q8—416no—
240C4×D4—416no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—216no—
277C2×C4⋊C4—216no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4—48no—
286C2×C4⋊C4—48no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—48no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—48no—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D418yes—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—216no—
302C2×C22⋊C4—216no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—48no—
306C2×C22⋊C4—48no—
307C2×C22⋊C4—48no—
308C2×C22⋊C4C2×Q8116yes—
309C2×C42—28no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—28no—
315C2×C42—28no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×Q814yes—
335C2×C42#4114yes—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95C2×D418yes—
349#95C2×D418yes—
350#95C2×D418yes—
351#95C2×D418yes—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—28no—
357C24—28no—
358C24—22no—
359C24—22no—
360C24C22×Q814yes—
361C24C2511yescharacteristic
362C2×Q8—28no—
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—416no—
366C2×Q8—44no—
367C2×Q8—48no—
368C2×Q8—44no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—88no—
372C2×Q8—88no—
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—44no—
377C2×D4—44no—
378C2×D4—44no—
379C2×D4—44no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—28no—
427C22×C4—28no—
428C22×C4—28no—
429C22×C4—28no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—416no—
433C22×C4—48no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×Q814yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4112yes—
456C22×C4H(5,𝔽2)18yes—
457C22×C42−1+418yes—
458C22×C42−1+412yes—
459C22×C42−1+412yes—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—416no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—44no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—416no—
474C4⋊C4—416no—
475C4⋊C4—88no—
476C4⋊C4—88no—
477C4⋊C4—88no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C22⋊C4—232no—
482C22⋊C4—416no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—416no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C42—232no—
489C42—28no—
490C42—28no—
491C42—28no—
492C42—28no—
493C42—28no—
494C42—48no—
495C42C22⋊Q8116yescomplemented
496C42#127116yes—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—216no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—48no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—48no—
513C23—48no—
514C23C23×D411yescharacteristic
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C23×Q811yescharacteristic
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)14yes—
520C23C2×2−1+411yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23C2×2−1+414yes—
523Q8—416no—
524Q8—816no—
525D4—416no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—28no—
531C2×C4—28no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—816no—
591C2×C4C2×C22⋊Q818yes—
592C2×C4C2×C22⋊Q814yes—
593C2×C4C2×#12714yes—
594C2×C4C2×#12714yes—
595C2×C4C2×#12714yes—
596C2×C4#53518yes—
597C2×C4#53618yes—
598C2×C4#53614yes—
599C2×C4#54714yes—
600C2×C4#54714yes—
601C2×C4D4×Q814yes—
602C2×C4D4×Q814yes—
603C22—44no—
604C22—48no—
605C22—48no—
606C22—48no—
607C22—84no—
608C22—82no—
609C22—84no—
610C22—84no—
611C22—82no—
612C22C22×C22⋊Q811yescharacteristic
613C22C22×#12711yescharacteristic
614C22C2×#53512yes—
615C22C2×#53611yescharacteristic
616C22C2×#53612yes—
617C22C2×#54711yescharacteristic
618C22C2×#54711yescharacteristic
619C22C2×D4×Q811yescharacteristic
620C22C2×D4×Q811yescharacteristic
621C22#349214yes—
622C22#349812yes—
623C22#349812yes—
624C22#349912yes—
625C22#349912yes—
626C22#350314yes—
627C22#354414yes—
628C22#355214yes—
629C4—24no—
630C4—28no—
631C4—28no—
632C4—28no—
633C4—24no—
634C4—44no—
635C4—44no—
636C4—44no—
637C4—44no—
638C4—82no—
639C4—82no—
640C4—82no—
641C4—82no—
642C4—82no—
643C4—82no—
644C4#254218yes—
645C2—84no—
646C2C2×#254211yescharacteristic
647C2#3527411yescharacteristic
648C2#3531711yescharacteristic
649C2#3553212yes—
650C2#3730312yes—
651C2#3786212yes—
652C2#3786312yes—
653C2#3961612yes—
654C2#3961712yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2567C2semidirect
3. C2×#2565C2semidirect
5. C2×#2660C2semidirect
7. #15765C2semidirect
8. #15430C2semidirect
10. #15240C2semidirect
11. #15199C2semidirect
14. #14692C2semidirect
15. #14354C2semidirect
17. #13759C2semidirect
93. C2×#124Q8semidirect
495. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25422
C2×#25652
C2×#25662
C2×#26602
C2×#23262
#157652
#154302
#152442
#152402
#151992
#151552
#147082
#146922
#143542
#142732
#137592