Finite Group Databasebeta

← #726317

Subgroups — #726317

Order 512.

2656 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2628C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic, complemented
4C2×#2323C211yesmaximal, characteristic
5#17023C212yesmaximal
6#16993C212yesmaximal, complemented
7#15813C214yesmaximal, complemented
8#15408C214yesmaximal
9#15179C212yesmaximal
10#15176C212yesmaximal
11#15137C212yesmaximal, complemented
12#14880C212yesmaximal, complemented
13#14142C212yesmaximal, complemented
14#13955C211yesmaximal, characteristic, complemented
15#13942C211yesmaximal, characteristic, complemented
16#13876C212yesmaximal
17#13852C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C22×#117C2211yescharacteristic
23#2628—28no—
24#2567—28no—
25C2×#400C2212yes—
26C2×#399C2212yes—
27C2×#398C2214yes—
28C2×#395C2212yes—
29C2×#395C2212yes—
30C2×#395C2212yes—
31C2×#393C2212yes—
32#2365—28no—
33#2323—28no—
34C2×#391C2211yescharacteristic
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#388C2212yes—
44C2×#388C2212yes—
45C2×#388C2214yes—
46C2×#387C2212yes—
47C2×#387C2214yes—
48C2×#384C2212yes—
49C2×#382C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2212yes—
57#1444C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2211yescharacteristic
65#1441C2211yescharacteristic
66#1441C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2212yes—
71#1440C2214yes—
72#1439C2212yes—
73#1439C2211yescharacteristic
74#1439C2211yescharacteristic
75#1439C2214yes—
76#1439C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2214yes—
84#1434C2214yes—
85#1434C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8D418yes—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—28no—
99C2×C4×Q8—24no—
100C2×C4×Q8—48no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—24no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
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×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#400—28no—
137#399—28no—
138#398—48no—
139#395—28no—
140#395—28no—
141#395—28no—
142#393—28no—
143#391—28no—
144#391—28no—
145#391—24no—
146#391—28no—
147#389—216no—
148#389—24no—
149#389—28no—
150#389—28no—
151#389—28no—
152#388—28no—
153#388—28no—
154#388—48no—
155#387—28no—
156#387—216no—
157#384—28no—
158#382—24no—
159#380—24no—
160#380—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4D418yes—
165C4×C22⋊C4—28no—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8C2×D418yes—
208C22×Q8#4114yes—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4#4114yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C4⋊Q8—232no—
233C22⋊Q8—232no—
234C4×Q8—432no—
235C4×Q8—416no—
236#117—416no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
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—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D414yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—28no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C2×D418yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42#4114yes—
328#95—24no—
329#95—28no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95C2×D418yes—
358C24—21no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24C2×#4112yes—
369C24C2511yescharacteristic
370C2×Q8—216no—
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—416no—
375C2×Q8—48no—
376C2×Q8—416no—
377C2×Q8—48no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
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—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4C22×D414yes—
467C22×C4C22×D418yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4114yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C42−1+414yes—
484C22×C42−1+414yes—
485C22×C42−1+412yes—
486C22×C42−1+412yes—
487C4⋊C4—216no—
488C4⋊C4—28no—
489C4⋊C4—216no—
490C4⋊C4—216no—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—416no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—84no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C4⋊C4—88no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—48no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C42—48no—
513C42—416no—
514C42—44no—
515C42—48no—
516C42—48no—
517C42—48no—
518C42#121116yes—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—41no—
534C23—41no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23C23×D411yescharacteristic
543C23C23×D411yescharacteristic
544C23C22×#4111yescharacteristic
545C23C22×#4111yescharacteristic
546C23C22×#4111yescharacteristic
547C23C2×H(5,𝔽2)12yes—
548C23C2×2−1+411yescharacteristic
549C23C2×2−1+411yescharacteristic
550C23C2×2−1+412yes—
551C23C2×2−1+412yes—
552C23#58512yes—
553Q8—432no—
554Q8—416no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
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—28no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—416no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4C2×#12114yes—
646C2×C4C2×#12114yes—
647C2×C4C2×#12114yes—
648C2×C4#52514yes—
649C2×C4#53614yes—
650C2×C4#54814yes—
651C2×C4#54814yes—
652C2×C4#55214yes—
653C2×C4#55214yes—
654C22—44no—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—42no—
659C22—81no—
660C22—81no—
661C22—82no—
662C22—81no—
663C22—82no—
664C22—81no—
665C22C22×#12111yescharacteristic
666C22C2×#52511yescharacteristic
667C22C2×#53611yescharacteristic
668C22#346012yes—
669C22C2×#54811yescharacteristic
670C22C2×#54811yescharacteristic
671C22C2×#55211yescharacteristic
672C22C2×#55211yescharacteristic
673C22#348411yescharacteristic
674C22#348612yes—
675C22#348711yescharacteristic
676C22#349212yes—
677C22#349512yes—
678C22#349612yes—
679C22#349911yescharacteristic
680C22#349911yescharacteristic
681C22#350912yes—
682C22#351812yes—
683C22#352212yes—
684C22#355012yes—
685C22#355212yes—
686C22#355512yes—
687C22#356312yes—
688C4—24no—
689C4—24no—
690C4—44no—
691C4—44no—
692C4—42no—
693C4—44no—
694C4—48no—
695C4—44no—
696C4—44no—
697C4—44no—
698C4—81no—
699C4—81no—
700C4—82no—
701C4—84no—
702C4—82no—
703C4#253618yes—
704C2—82no—
705C2C2×#253611yescharacteristic
706C2#3516811yescharacteristic
707C2#3566711yescharacteristic
708C2#3569311yescharacteristic
709C2#3603011yescharacteristic
710C2#3798112yes—
711C2#3827312yes—
712C2#4242711yescharacteristic
713C2#4287911yescharacteristic
714C2#4489612yes—
715C2#4510912yes—
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×#2567C2semidirect
3. C2×#2365C2semidirect
6. #16993C2semidirect
7. #15813C2semidirect
11. #15137C2semidirect
12. #14880C2semidirect
13. #14142C2semidirect
14. #13955C2semidirect
15. #13942C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#26282
C2×#23652
C2×#23232
#170232
#169932
#158132
#154082
#151792
#151762
#151372
#148802
#141422
#139552
#139422
#138762
#138522