Finite Group Databasebeta

← #4456929

Subgroups — #4456929

Order 512.

2438 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2551C211yesmaximal, characteristic, complemented
2C2×#2562C211yesmaximal, characteristic
3#17009C212yesmaximal, complemented
4#17006C212yesmaximal, complemented
5#16833C214yesmaximal, complemented
6#16602C212yesmaximal, complemented
7#16506C212yesmaximal, complemented
8#16374C214yesmaximal
9#15183C212yesmaximal, complemented
10#15132C212yesmaximal, complemented
11#15125C212yesmaximal, complemented
12#15125C212yesmaximal, complemented
13#14283C211yesmaximal, characteristic, complemented
14#14160C211yesmaximal, characteristic, complemented
15#13906C211yesmaximal, characteristic
16#13847C211yesmaximal, characteristic
17#13797C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C22×#125C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2562—28no—
23#2551—28no—
24C2×#400C2212yes—
25C2×#397C2214yes—
26C2×#395C2214yes—
27C2×#394C2212yes—
28C2×#393C2214yes—
29C2×#393C2214yes—
30C2×#390C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#384C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×#376C2211yescharacteristic
48#1446C2214yes—
49#1446C2214yes—
50#1445C2212yes—
51#1445C2214yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2211yescharacteristic
67#1439C2211yescharacteristic
68#1439C2211yescharacteristic
69#1439C2211yescharacteristic
70#1439C2214yes—
71#1439C2212yes—
72#1439C2212yes—
73#1438C2212yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2212yes—
84#1434C2214yes—
85C23×Q8C2314yes—
86C23×D4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—44no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8—48no—
94C2×#125—24no—
95C2×#125—24no—
96C2×#125—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#400—28no—
133#397—48no—
134#395—216no—
135#394—28no—
136#393—48no—
137#393—48no—
138#390—44no—
139#389—28no—
140#389—28no—
141#388—28no—
142#388—28no—
143#384—24no—
144#380—24no—
145#380—24no—
146#380—24no—
147#380—24no—
148#380—44no—
149#380—44no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4D418yes—
153C4×C22⋊C4—44no—
154C4×C22⋊C4—44no—
155#376—24no—
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C25C2411yescharacteristic
186C25C2411yescharacteristic
187C22×Q8—28no—
188C22×Q8—28no—
189C22×Q8—44no—
190C22×Q8—44no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8—44no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—41no—
200C22×D4—41no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C4⋊Q8—432no—
227C4⋊Q8—432no—
228#125—416no—
229C4×D4—416no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
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—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—44no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C2×D418yes—
315C2×C42C2×D414yes—
316C2×C42#4114yes—
317C2×C42#4114yes—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—42no—
336#95—42no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95C2×D418yes—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—21no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24H(5,𝔽2)12yes—
368C242−1+412yes—
369C24C2511yescharacteristic
370C2×Q8—416no—
371C2×Q8—416no—
372C2×Q8—88no—
373C2×Q8—88no—
374C2×Q8—88no—
375C2×Q8—88no—
376C2×Q8—88no—
377C2×D4—28no—
378C2×D4—44no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
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—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4H(5,𝔽2)14yes—
482C22×C4H(5,𝔽2)14yes—
483C22×C42−1+414yes—
484C22×C42−1+414yes—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—28no—
488C4⋊C4—28no—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—416no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—84no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—44no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C42—48no—
513C42—48no—
514C42—48no—
515C42—84no—
516C42#124116yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23C23×D411yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4111yescharacteristic
566C23C2×H(5,𝔽2)12yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)11yescharacteristic
569C23C2×2−1+412yes—
570C23C2×2−1+412yes—
571C23C2×2−1+411yescharacteristic
572C23#58511yescharacteristic
573C23#58511yescharacteristic
574Q8—832no—
575D4—416no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4C2×#12414yes—
663C2×C4C2×#12414yes—
664C2×C4C2×#12414yes—
665C2×C4#53514yes—
666C2×C4#53614yes—
667C2×C4#54218yes—
668C2×C4#54218yes—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—42no—
678C22—42no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22—81no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—81no—
690C22—81no—
691C22C22×#12411yescharacteristic
692C22C2×#53511yescharacteristic
693C22C2×#53611yescharacteristic
694C22C2×#54212yes—
695C22C2×#54212yes—
696C22#348411yescharacteristic
697C22#348611yescharacteristic
698C22#348811yescharacteristic
699C22#348911yescharacteristic
700C22#349211yescharacteristic
701C22#349311yescharacteristic
702C22#349411yescharacteristic
703C22#349511yescharacteristic
704C22#351112yes—
705C22#351112yes—
706C22#353712yes—
707C22#353812yes—
708C22#354112yes—
709C22#354212yes—
710C22#354412yes—
711C22#354712yes—
712C22#355212yes—
713C22#355512yes—
714C4—24no—
715C4—24no—
716C4—44no—
717C4—44no—
718C4—44no—
719C4—44no—
720C4—82no—
721C4—81no—
722C4—81no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—84no—
727C4—84no—
728C4#264018yes—
729C2—82no—
730C2—82no—
731C2C2×#264011yescharacteristic
732C2#3554211yescharacteristic
733C2#3559211yescharacteristic
734C2#3734012yes—
735C2#3734912yes—
736C2#4245812yes—
737C2#4248011yescharacteristic
738C2#4248511yescharacteristic
739C2#4266012yes—
740C2#4280811yescharacteristic
741C2#4281311yescharacteristic
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×#2551C2semidirect
3. #17009C2semidirect
4. #17006C2semidirect
5. #16833C2semidirect
6. #16602C2semidirect
7. #16506C2semidirect
9. #15183C2semidirect
10. #15132C2semidirect
11. #15125C2semidirect
12. #15125C2semidirect
13. #14283C2semidirect
14. #14160C2semidirect
17. #13797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25512
C2×#25622
#170092
#170062
#168332
#166022
#165062
#163742
#151832
#151322
#151252
#142832
#141602
#139062
#138472
#137972