Finite Group Databasebeta

← #5486446

Subgroups — #5486446

Order 512.

1730 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17289C211yesmaximal, characteristic, complemented
2#17285C212yesmaximal, complemented
3#17251C212yesmaximal, complemented
4#16668C212yesmaximal
5#16666C212yesmaximal
6#16622C212yesmaximal, complemented
7#16617C212yesmaximal, complemented
8#16098C212yesmaximal, complemented
9#16093C211yesmaximal, characteristic, complemented
10#16065C212yesmaximal, complemented
11#15989C212yesmaximal
12#15987C212yesmaximal
13#15466C211yesmaximal, characteristic
14#15265C211yesmaximal, characteristic
15#14923C212yesmaximal
16#14919C212yesmaximal
17#14668C211yesmaximal, characteristic, complemented
18#14663C211yesmaximal, characteristic, complemented
19#14159C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#398C2211yescharacteristic
24C2×#395C2211yescharacteristic
25C2×#393C2211yescharacteristic
26C2×#391C2211yescharacteristic
27C2×#387C2212yes—
28C2×#387C2212yes—
29C2×#385C2212yes—
30C2×#383C2212yes—
31C2×#382C2212yes—
32C2×#381C2212yes—
33C2×#381C2212yes—
34C2×C4×C22⋊C4C2211yescharacteristic
35C22×#95C2211yescharacteristic
36#1446C2212yes—
37#1446C2212yes—
38#1446C2211yescharacteristic
39#1446C2211yescharacteristic
40#1446C2211yescharacteristic
41#1446C2211yescharacteristic
42#1445C2212yes—
43#1445C2212yes—
44#1445C2212yes—
45#1445C2212yes—
46#1445C2212yes—
47#1445C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1445C2211yescharacteristic
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1443C2212yes—
57#1443C2212yes—
58#1443C2212yes—
59#1443C2212yes—
60#1443C2212yes—
61#1443C2212yes—
62#1443C2212yes—
63#1443C2212yes—
64#1443C2212yes—
65#1443C2212yes—
66#1443C2212yes—
67#1442C2212yes—
68#1442C2212yes—
69#1442C2212yes—
70#1442C2212yes—
71#1442C2212yes—
72#1442C2212yes—
73#1442C2212yes—
74#1441C2212yes—
75#1441C2212yes—
76#1441C2212yes—
77#1441C2212yes—
78#1441C2212yes—
79#1441C2212yes—
80#1441C2212yes—
81#1441C2212yes—
82#1441C2212yes—
83#1441C2212yes—
84#1441C2212yes—
85#1441C2212yes—
86#1439C2212yes—
87#1439C2212yes—
88#1439C2212yes—
89#1439C2212yes—
90#1439C2212yes—
91#1439C2212yes—
92#1437C2211yescharacteristic
93#1437C2211yescharacteristic
94#1436C2212yes—
95#1436C2212yes—
96#1436C2212yes—
97#1436C2212yes—
98#1436C2211yescharacteristic
99#1436C2212yes—
100#1436C2211yescharacteristic
101#1434C2212yes—
102C22⋊#95C2212yes—
103C22⋊#95C2212yes—
104C22⋊#95C2212yes—
105C22⋊#95C2212yes—
106C22⋊#95C2212yes—
107C23×Q8C2311yescharacteristic
108C23×Q8C2311yescharacteristic
109C24×C4C2311yescharacteristic
110C24×C4C2312yes—
111C2×C4⋊Q8—24no—
112C2×C4⋊Q8—44no—
113C2×C22⋊Q8—24no—
114C2×C22⋊Q8—42no—
115C2×C22⋊Q8—42no—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2312yes—
139#399—42no—
140#398—42no—
141#395—42no—
142#393—42no—
143#391—24no—
144#387—44no—
145#387—44no—
146#385—44no—
147#383—44no—
148#382—44no—
149#381—44no—
150#381—44no—
151C4×C22⋊C4—42no—
152C2×#95—42no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2311yescharacteristic
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2311yescharacteristic
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C2×#95C2312yes—
208C2×#95C2312yes—
209C2×#95C2311yescharacteristic
210C2×#95C2311yescharacteristic
211C2×#95C2312yes—
212C2×#95C2311yescharacteristic
213C25C2411yescharacteristic
214C22×Q8—22no—
215C22×Q8—41no—
216C22×Q8—41no—
217C22×Q8—42no—
218C22×Q8—42no—
219C22×Q8—42no—
220C22×Q8—42no—
221C22×Q8—41no—
222C22×Q8—41no—
223C22×Q8#4114yes—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2412yes—
254C4⋊Q8—416no—
255C22⋊Q8—416no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—41no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—41no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4—41no—
294C2×C4⋊C4—42no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—41no—
308C2×C22⋊C4—41no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—42no—
332#95—41no—
333#95—41no—
334#95—42no—
335#95—42no—
336#95—42no—
337#95—42no—
338#95—42no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—41no—
357#95—41no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95—42no—
362#95—42no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95—42no—
367#95—42no—
368#95—42no—
369#95—42no—
370#95—42no—
371#95—42no—
372#95—42no—
373#95—42no—
374#95—42no—
375#95—42no—
376#95—42no—
377#95—41no—
378#95—42no—
379#95—41no—
380#95—42no—
381#95—41no—
382#95—41no—
383C24—21no—
384C24—22no—
385C24—21no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24H(5,𝔽2)12yes—
391C242−1+414yes—
392C24C2511yescharacteristic
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—82no—
398C2×Q8—82no—
399C2×Q8—84no—
400C2×Q8—84no—
401C2×Q8—84no—
402C2×Q8—84no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
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—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—22no—
510C22×C4—22no—
511C22×C4—22no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—82no—
516C22×C4—82no—
517C22×C4—84no—
518C22×C4—84no—
519C22×C4—84no—
520C22×C4—84no—
521C22×C4—84no—
522C22×C4—84no—
523C22×C4C2×#4112yes—
524C22×C4C2×#4112yes—
525C22×C4C2×#4112yes—
526C22×C4H(5,𝔽2)14yes—
527C22×C4H(5,𝔽2)12yes—
528C22×C42−1+414yes—
529C4⋊C4—48no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—48no—
533C4⋊C4—48no—
534C4⋊C4—44no—
535C4⋊C4—48no—
536C4⋊C4—82no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C22⋊C4—48no—
543C22⋊C4—48no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—84no—
549C42—28no—
550C42—48no—
551C42—82no—
552C23—22no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—41no—
566C23—42no—
567C23—42no—
568C23—41no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23#56314yes—
573C23C22×#4111yescharacteristic
574C23C2×H(5,𝔽2)11yescharacteristic
575C23C2×H(5,𝔽2)11yescharacteristic
576C23C2×H(5,𝔽2)12yes—
577C23C2×2−1+412yes—
578C23C2×2−1+412yes—
579C23#58512yes—
580C23#58512yes—
581C23#58512yes—
582Q8—88no—
583Q8—88no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—88no—
699C2×C4—816no—
700C22—24no—
701C22—28no—
702C22—44no—
703C22—44no—
704C22—42no—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—44no—
709C22#349011yescharacteristic
710C22#349011yescharacteristic
711C22#349111yescharacteristic
712C22#352312yes—
713C22#352412yes—
714C22C2×#56311yescharacteristic
715C22#352911yescharacteristic
716C22#353312yes—
717C22#353512yes—
718C22#353512yes—
719C22#353812yes—
720C22#354212yes—
721C22#354212yes—
722C22#354411yescharacteristic
723C22#355111yescharacteristic
724C22#355412yes—
725C22#355712yes—
726C22#356112yes—
727C22#356412yes—
728C22#356412yes—
729C22#356412yes—
730C4—44no—
731C4—42no—
732C4—42no—
733C4—42no—
734C4—42no—
735C4—44no—
736C4—82no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—81no—
745C4—81no—
746C4—82no—
747C4—82no—
748C2—44no—
749C2#4620511yescharacteristic
750C2#4627511yescharacteristic
751C2#4699711yescharacteristic
752C2#5166712yes—
753C2#5167112yes—
754C2#5236812yes—
755C2#5238912yes—
756C2#5566612yes—
757C2#5617212yes—
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. #17289C2semidirect
2. #17285C2semidirect
3. #17251C2semidirect
6. #16622C2semidirect
7. #16617C2semidirect
8. #16098C2semidirect
9. #16093C2semidirect
10. #16065C2semidirect
17. #14668C2semidirect
18. #14663C2semidirect

Maximal subgroups

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

subgroupindex
#172892
#172852
#172512
#166682
#166662
#166222
#166172
#160982
#160932
#160652
#159892
#159872
#154662
#152652
#149232
#149192
#146682
#146632
#141592