Finite Group Databasebeta

← #729698

Subgroups — #729698

Order 512.

2738 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2581C211yesmaximal, characteristic, complemented
3C2×#2357C211yesmaximal, characteristic
4C2×#2326C211yesmaximal, characteristic
5#16394C212yesmaximal, complemented
6#16262C212yesmaximal, complemented
7#15839C214yesmaximal, complemented
8#15774C214yesmaximal, complemented
9#15226C212yesmaximal
10#15138C212yesmaximal
11#14841C212yesmaximal, complemented
12#14763C212yesmaximal, complemented
13#14277C211yesmaximal, characteristic, complemented
14#14142C212yesmaximal, complemented
15#14005C212yesmaximal, complemented
16#13926C211yesmaximal, characteristic, complemented
17#13850C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2581—28no—
24#2538—28no—
25C2×#400C2212yes—
26C2×#397C2214yes—
27C2×#396C2214yes—
28C2×#395C2212yes—
29#2357—28no—
30#2326—28no—
31C2×#391C2211yescharacteristic
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2214yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#381C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×#376C2211yescharacteristic
56#1446C2211yescharacteristic
57#1446C2211yescharacteristic
58#1445C2211yescharacteristic
59#1445C2211yescharacteristic
60#1444C2212yes—
61#1444C2214yes—
62#1444C2212yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1442C2212yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1440C2214yes—
73#1440C2214yes—
74#1440C2214yes—
75#1440C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2214yes—
84#1434C2214yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×D4C2311yescharacteristic
90C24×C4C2312yes—
91C2×#124—24no—
92C2×#124—24no—
93C2×#124—24no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—48no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×D4—24no—
103C2×C4×D4—24no—
104C2×C4×D4D418yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136#400—28no—
137#397—48no—
138#396—216no—
139#395—28no—
140#391—28no—
141#391—28no—
142#391—24no—
143#390—24no—
144#390—28no—
145#390—28no—
146#389—28no—
147#389—28no—
148#389—28no—
149#389—48no—
150#388—28no—
151#388—28no—
152#388—216no—
153#387—216no—
154#385—28no—
155#385—28no—
156#384—24no—
157#384—24no—
158#382D418yes—
159#381—28no—
160#380—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C22⋊C4—28no—
164#376—24no—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C25C2412yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—42no—
207C22×Q8#4114yes—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4C2×D414yes—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4C2×D418yes—
220C23×C4C2412yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236#124—232no—
237C22⋊Q8—432no—
238C4×Q8—416no—
239C4×Q8—416no—
240C4×D4—232no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4C2×D418yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
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×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4C2×D418yes—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42#4114yes—
327C2×C42#4114yes—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—42no—
349#95—42no—
350#95—44no—
351#95—42no—
352#95—42no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95C2×D418yes—
357#95C2×D418yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24C22×D414yes—
369C24C2511yescharacteristic
370C2×Q8—28no—
371C2×Q8—28no—
372C2×Q8—28no—
373C2×Q8—28no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×D4—28no—
383C2×D4—28no—
384C2×D4—28no—
385C2×D4—44no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—216no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4C22×D418yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)14yes—
487C22×C42−1+414yes—
488C22×C42−1+412yes—
489C22×C42−1+412yes—
490C4⋊C4—28no—
491C4⋊C4—216no—
492C4⋊C4—48no—
493C4⋊C4—416no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—216no—
508C22⋊C4—416no—
509C22⋊C4—416no—
510C22⋊C4—88no—
511C22⋊C4—84no—
512C22⋊C4—84no—
513C22⋊C4—88no—
514C42—28no—
515C42—48no—
516C42—48no—
517C42—44no—
518C42—48no—
519C42—88no—
520C42#121116yes—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23C23×D411yescharacteristic
545C23C23×D411yescharacteristic
546C23C22×#4111yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+411yescharacteristic
551C23C2×2−1+411yescharacteristic
552C23C2×2−1+412yes—
553C23#58512yes—
554C23#58512yes—
555Q8—416no—
556Q8—416no—
557D4—416no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—816no—
646C2×C4C2×#12114yes—
647C2×C4C2×#12114yes—
648C2×C4C2×#12114yes—
649C2×C4#52514yes—
650C2×C4#53614yes—
651C2×C4#54714yes—
652C2×C4#54814yes—
653C2×C4#55214yes—
654C2×C4#55614yes—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22C22×#12111yescharacteristic
667C22C2×#52511yescharacteristic
668C22C2×#53611yescharacteristic
669C22#346112yes—
670C22C2×#54711yescharacteristic
671C22C2×#54811yescharacteristic
672C22C2×#55211yescharacteristic
673C22C2×#55611yescharacteristic
674C22#348411yescharacteristic
675C22#348511yescharacteristic
676C22#348812yes—
677C22#348912yes—
678C22#349412yes—
679C22#349911yescharacteristic
680C22#350011yescharacteristic
681C22#350612yes—
682C22#350712yes—
683C22#351612yes—
684C22#351712yes—
685C22#355012yes—
686C22#355212yes—
687C22#355712yes—
688C22#356512yes—
689C4—24no—
690C4—24no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—48no—
696C4—44no—
697C4—44no—
698C4—82no—
699C4—84no—
700C4—81no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4#241518yes—
705C2—84no—
706C2C2×#241511yescharacteristic
707C2#3530811yescharacteristic
708C2#3551411yescharacteristic
709C2#3570411yescharacteristic
710C2#3584611yescharacteristic
711C2#3823112yes—
712C2#3980012yes—
713C2#4242711yescharacteristic
714C2#4264711yescharacteristic
715C2#4496412yes—
716C2#4531312yes—
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. C2×#2581C2semidirect
5. #16394C2semidirect
6. #16262C2semidirect
7. #15839C2semidirect
8. #15774C2semidirect
11. #14841C2semidirect
12. #14763C2semidirect
13. #14277C2semidirect
14. #14142C2semidirect
15. #14005C2semidirect
16. #13926C2semidirect
17. #13850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#25812
C2×#23572
C2×#23262
#163942
#162622
#158392
#157742
#152262
#151382
#148412
#147632
#142772
#141422
#140052
#139262
#138502