Finite Group Databasebeta

← #626011

Subgroups — #626011

Order 512.

3470 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2566C211yesmaximal, characteristic, complemented
4C2×#2566C211yesmaximal, characteristic, complemented
5C2×#2660C211yesmaximal, characteristic, complemented
6C2×#2328C211yesmaximal, characteristic
7#15505C218yesmaximal, complemented
8#15325C218yesmaximal, complemented
9#15244C211yesmaximal, characteristic, complemented
10#15244C211yesmaximal, characteristic, complemented
11#15244C211yesmaximal, characteristic, complemented
12#15155C211yesmaximal, characteristic, complemented
13#14713C211yesmaximal, characteristic
14#14687C211yesmaximal, characteristic, complemented
15#14354C211yesmaximal, characteristic, complemented
16#14272C211yesmaximal, characteristic
17#13759C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C22⋊Q8C2214yes—
22C22×C4×D4C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#2660—28no—
25#2566—28no—
26#2566—28no—
27#2563—28no—
28#2542—28no—
29C2×#399C2214yes—
30C2×#399C2218yes—
31C2×#398C2214yes—
32C2×#396C2214yes—
33C2×#396C2214yes—
34C2×#395C2214yes—
35C2×#393C2214yes—
36#2328—28no—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2218yes—
48C2×#386C2214yes—
49C2×#386C2214yes—
50C2×#385C2214yes—
51C2×#385C2214yes—
52C2×#384C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C2×#382C2211yescharacteristic
59C2×#382C2211yescharacteristic
60C2×#382C2211yescharacteristic
61C2×#382C2211yescharacteristic
62C2×#382C2211yescharacteristic
63C2×#382C2211yescharacteristic
64C2×#382C2211yescharacteristic
65C2×#380C2214yes—
66C2×C4×C4⋊C4C2211yescharacteristic
67C2×C4×C4⋊C4C2211yescharacteristic
68C2×#376C2211yescharacteristic
69C2×#376C2211yescharacteristic
70C2×C43C2211yescharacteristic
71#1444C2218yes—
72#1439C2218yes—
73#1439C2218yes—
74#1439C2218yes—
75#1438C2218yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2218yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C23×Q8C2312yes—
82C23×D4C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2314yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—24no—
90C2×#127—24no—
91C2×#127—24no—
92C2×#127Q818yescomplemented
93C2×#125—24no—
94C2×#125—24no—
95C2×#125—24no—
96C2×C22⋊Q8—216no—
97C2×C22⋊Q8—416no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
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⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C42C2314yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#399—216no—
142#399—416no—
143#398—216no—
144#396—216no—
145#396—216no—
146#395—216no—
147#393—216no—
148#390—24no—
149#390—24no—
150#390—24no—
151#390—24no—
152#390—24no—
153#390—24no—
154#390—24no—
155#390—24no—
156#390—24no—
157#390—24no—
158#390—416no—
159#386—216no—
160#386—216no—
161#385—216no—
162#385—216no—
163#384—24no—
164#384—24no—
165#384—24no—
166#384—24no—
167#384—24no—
168#384—24no—
169#382—24no—
170#382—24no—
171#382—24no—
172#382—24no—
173#382—24no—
174#382—24no—
175#382—24no—
176#380—216no—
177C4×C4⋊C4D418yescomplemented
178C4×C4⋊C4D418yescomplemented
179#376—24no—
180#376D418yes—
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—
197C25C2412yes—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—48no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—28no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—28no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—48no—
217C22×D4—48no—
218C22×D4C2×Q814yes—
219C22×D4C2×Q814yes—
220C22×D4C2×Q8116yes—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—48no—
228C23×C4—48no—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C4⋊Q8—416no—
245#127—232no—
246#125—416no—
247C22⋊Q8—464no—
248C4×D4—416no—
249C4×D4—416no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—44no—
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—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—216no—
304C2×C22⋊C4—216no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—48no—
312C2×C22⋊C4—48no—
313C2×C22⋊C4—48no—
314C2×C22⋊C4—48no—
315C2×C22⋊C4—48no—
316C2×C22⋊C4—48no—
317C2×C22⋊C4—48no—
318C2×C22⋊C4—48no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—22no—
329C2×C42—28no—
330C2×C42—28no—
331C2×C42—44no—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×Q814yes—
342C2×C42#4114yes—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—44no—
353#95C2×D418yes—
354#95C2×D418yes—
355#95C2×D418yes—
356#95C2×D418yes—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—28no—
366C24—28no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—28no—
372C24—28no—
373C24C22×Q814yes—
374C24C22×Q814yes—
375C24C2511yescharacteristic
376C2×Q8—416no—
377C2×Q8—48no—
378C2×Q8—88no—
379C2×Q8—88no—
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—232no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—432no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—816no—
397C2×D4—88no—
398C2×D4—88no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—28no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—88no—
451C22×C4—88no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×Q814yes—
470C22×C4C22×Q812yes—
471C22×C4C2×#4114yes—
472C22×C4C2×#4112yes—
473C22×C4H(5,𝔽2)18yes—
474C22×C42−1+412yes—
475C22×C42−1+412yes—
476C4⋊C4—216no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—416no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C22⋊C4—416no—
494C22⋊C4—416no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C22⋊C4—88no—
505C42—28no—
506C42—28no—
507C42—28no—
508C42—28no—
509C42—28no—
510C42—416no—
511C42C22⋊Q8116yescomplemented
512C42#127116yes—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—216no—
520C23—216no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—48no—
534C23—48no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—48no—
544C23—48no—
545C23—48no—
546C23—48no—
547C23C23×D411yescharacteristic
548C23C23×D411yescharacteristic
549C23C23×D411yescharacteristic
550C23C23×Q811yescharacteristic
551C23C22×#4111yescharacteristic
552C23C2×H(5,𝔽2)14yes—
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+411yescharacteristic
555C23#58514yes—
556Q8—816no—
557D4—416no—
558D4—416no—
559D4—832no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—816no—
619C2×C4—816no—
620C2×C4C2×C22⋊Q818yes—
621C2×C4C2×C22⋊Q814yes—
622C2×C4C2×#12714yes—
623C2×C4C2×#12714yes—
624C2×C4C2×#12714yes—
625C2×C4#53518yes—
626C2×C4#53614yes—
627C2×C4#54714yes—
628C2×C4#54714yes—
629C2×C4D4×Q814yes—
630C2×C4D4×Q814yes—
631C22—44no—
632C22—44no—
633C22—48no—
634C22—48no—
635C22—48no—
636C22—48no—
637C22—48no—
638C22—48no—
639C22—82no—
640C22—82no—
641C22—84no—
642C22—84no—
643C22—82no—
644C22—82no—
645C22—84no—
646C22—84no—
647C22—84no—
648C22—84no—
649C22C22×C22⋊Q811yescharacteristic
650C22C22×#12711yescharacteristic
651C22C2×#53512yes—
652C22C2×#53611yescharacteristic
653C22#344912yes—
654C22C2×#54711yescharacteristic
655C22C2×#54711yescharacteristic
656C22C2×D4×Q811yescharacteristic
657C22C2×D4×Q811yescharacteristic
658C22#349414yes—
659C22#349812yes—
660C22#349812yes—
661C22#350012yes—
662C22#350012yes—
663C22#350114yes—
664C22#354714yes—
665C22#355914yes—
666C4—24no—
667C4—28no—
668C4—24no—
669C4—28no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—82no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#254218yes—
681C2—84no—
682C2—84no—
683C2C2×#254211yescharacteristic
684C2#3527411yescharacteristic
685C2#3532911yescharacteristic
686C2#3552912yes—
687C2#3730412yes—
688C2#3786412yes—
689C2#3786912yes—
690C2#3961112yes—
691C2#3961812yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#127C22
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×#2563C2semidirect
2. C2×#2542C2semidirect
3. C2×#2566C2semidirect
4. C2×#2566C2semidirect
5. C2×#2660C2semidirect
7. #15505C2semidirect
8. #15325C2semidirect
9. #15244C2semidirect
10. #15244C2semidirect
11. #15244C2semidirect
12. #15155C2semidirect
14. #14687C2semidirect
15. #14354C2semidirect
17. #13759C2semidirect
92. C2×#127Q8semidirect
177. C4×C4⋊C4D4semidirect
178. C4×C4⋊C4D4semidirect
511. C42C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25422
C2×#25662
C2×#26602
C2×#23282
#155052
#153252
#152442
#151552
#147132
#146872
#143542
#142722
#137592