Finite Group Databasebeta

← #722865

Subgroups — #722865

Order 512.

2780 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic
2C22×#400C211yesmaximal, characteristic, complemented
3C2×#2345C211yesmaximal, characteristic, complemented
4C2×#2322C211yesmaximal, characteristic
5#17074C212yesmaximal, complemented
6#16899C212yesmaximal, complemented
7#15819C214yesmaximal, complemented
8#15787C214yesmaximal, complemented
9#15156C212yesmaximal, complemented
10#15103C212yesmaximal, complemented
11#14935C212yesmaximal, complemented
12#14921C212yesmaximal
13#14339C211yesmaximal, characteristic, complemented
14#14044C212yesmaximal, complemented
15#13996C212yesmaximal
16#13965C211yesmaximal, characteristic, complemented
17#13960C211yesmaximal, characteristic
18C22×#123C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2487—28no—
23C2×#400—28no—
24C2×#400C2212yes—
25C2×#400C2211yescharacteristic
26C2×#400C2212yes—
27C2×#400C2214yes—
28C2×#400C2211yescharacteristic
29C2×#399C2211yescharacteristic
30C2×#399C2211yescharacteristic
31#2345—28no—
32#2322—28no—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#387C2212yes—
38C2×#387C2214yes—
39C2×#386C2211yescharacteristic
40C2×#386C2212yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#385C2211yescharacteristic
44C2×#385C2212yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#381C2211yescharacteristic
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56C22×#95C2212yes—
57C22×#95C2212yes—
58C22×#95C2211yescharacteristic
59#1445C2214yes—
60#1444C2214yes—
61#1443C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1439C2214yes—
74#1439C2212yes—
75#1439C2214yes—
76#1439C2214yes—
77#1437C2214yes—
78#1436C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×D4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C2×#123—24no—
95C2×#123—24no—
96C2×#123Q818yes—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—24no—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4D418yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2314yes—
134C22×C42C2312yes—
135#400—216no—
136#400—28no—
137#400—28no—
138#400—24no—
139#400—24no—
140#399—24no—
141#399—24no—
142#389—28no—
143#389—28no—
144#389—28no—
145#388—28no—
146#387—28no—
147#387—216no—
148#386—28no—
149#386—24no—
150#386—44no—
151#385—28no—
152#385—24no—
153#385—28no—
154#385—28no—
155#385—44no—
156#381—24no—
157#381—44no—
158#380—24no—
159#380—24no—
160#380—28no—
161#380—28no—
162C4×C4⋊C4—28no—
163C4×C22⋊C4—28no—
164C4×C22⋊C4D418yes—
165C2×#95—28no—
166C2×#95—28no—
167C2×#95—24no—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C25C2411yescharacteristic
198C25C2411yescharacteristic
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4C2×Q814yes—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4—44no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2×D418yes—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×Q814yes—
234C23×C4#4114yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250#123—232no—
251C4×D4—416no—
252C2×C4⋊C4—232no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
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—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—416no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
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⋊C4C2×D418yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×Q818yes—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D414yes—
321C2×C22⋊C4C2×Q818yes—
322C2×C22⋊C4C2×Q814yes—
323C2×C22⋊C4#4114yes—
324C2×C42—22no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—28no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42C2×D418yes—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—22no—
344#95—22no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95C2×D418yes—
361C24—24no—
362C24—24no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—21no—
369C24—24no—
370C24—24no—
371C24C22×D412yes—
372C24C22×D412yes—
373C24C22×Q812yes—
374C24C22×Q812yes—
375C24C2×#4112yes—
376C24C2×#4112yes—
377C24H(5,𝔽2)12yes—
378C242−1+412yes—
379C24C2511yescharacteristic
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C2×D4—44no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—216no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—48no—
471C22×C4—44no—
472C22×C4—84no—
473C22×C4—84no—
474C22×C4—88no—
475C22×C4—88no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—88no—
479C22×C4—88no—
480C22×C4#121116yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C22×Q814yes—
490C22×C4C22×Q812yes—
491C22×C4C22×Q814yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4114yes—
495C4⋊C4—216no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—216no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—44no—
504C4⋊C4—84no—
505C4⋊C4—88no—
506C4⋊C4—84no—
507C4⋊C4—84no—
508C4⋊C4—88no—
509C4⋊C4—88no—
510C22⋊C4—28no—
511C22⋊C4—216no—
512C22⋊C4—44no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—44no—
516C22⋊C4—84no—
517C22⋊C4—84no—
518C22⋊C4—84no—
519C22⋊C4—88no—
520C22⋊C4C22⋊Q8116yes—
521C42—48no—
522C42—44no—
523C42—88no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—28no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—28no—
538C23—41no—
539C23—41no—
540C23—42no—
541C23—41no—
542C23—41no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23C2×#12114yes—
552C23C2×C22⋊Q814yes—
553C23#53714yes—
554C23D4×Q814yes—
555C23#55014yes—
556C23#55414yes—
557C23#55614yes—
558C23C23×D411yescharacteristic
559C23C23×D411yescharacteristic
560C23C23×Q811yescharacteristic
561C23C22×#4111yescharacteristic
562C23C22×#4111yescharacteristic
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×2−1+411yescharacteristic
565C23#58514yes—
566C23#58514yes—
567D4—416no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—416no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—416no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—88no—
647C2×C4—816no—
648C2×C4—88no—
649C2×C4—816no—
650C2×C4C2×#12114yes—
651C2×C4C2×#12114yes—
652C2×C4C2×C22⋊Q818yes—
653C22—24no—
654C22—24no—
655C22—28no—
656C22—28no—
657C22—28no—
658C22—28no—
659C22—28no—
660C22—28no—
661C22—42no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—81no—
666C22—81no—
667C22—82no—
668C22—82no—
669C22—82no—
670C22#253818yes—
671C22C22×#12111yescharacteristic
672C22C22×C22⋊Q811yescharacteristic
673C22C2×#53711yescharacteristic
674C22#344912yes—
675C22C2×D4×Q811yescharacteristic
676C22C2×#55011yescharacteristic
677C22C2×#55411yescharacteristic
678C22C2×#55611yescharacteristic
679C22#348512yes—
680C22#348512yes—
681C22#349112yes—
682C22#350012yes—
683C22#350414yes—
684C22#351012yes—
685C22#352114yes—
686C22#355314yes—
687C22#356014yes—
688C4—24no—
689C4—24no—
690C4—28no—
691C4—44no—
692C4—44no—
693C4—44no—
694C4—42no—
695C4—44no—
696C4—81no—
697C4—82no—
698C4—81no—
699C4—84no—
700C4—82no—
701C4—82no—
702C4—84no—
703C2—28no—
704C2—82no—
705C2C2×#253811yescharacteristic
706C2#3508911yescharacteristic
707C2#3561711yescharacteristic
708C2#3603112yes—
709C2#3912912yes—
710C2#3958912yes—
711C2#4137212yes—
712C2#4351612yes—
713C2#4446312yes—
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
2. C22×#400C2semidirect
3. C2×#2345C2semidirect
5. #17074C2semidirect
6. #16899C2semidirect
7. #15819C2semidirect
8. #15787C2semidirect
9. #15156C2semidirect
10. #15103C2semidirect
11. #14935C2semidirect
13. #14339C2semidirect
14. #14044C2semidirect
16. #13965C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C22×#4002
C2×#23452
C2×#23222
#170742
#168992
#158192
#157872
#151562
#151032
#149352
#149212
#143392
#140442
#139962
#139652
#139602