Finite Group Databasebeta

← #637133

Subgroups — #637133

Order 512.

2498 conjugacy classes of subgroups.

The subgroup lattice has 687 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2C22×#382C211yesmaximal, characteristic
3#16689C214yesmaximal, complemented
4#16687C214yesmaximal, complemented
5#15242C212yesmaximal, complemented
6#15232C212yesmaximal, complemented
7#15228C212yesmaximal, complemented
8#15225C212yesmaximal, complemented
9#14936C212yesmaximal
10#14935C212yesmaximal
11#14685C211yesmaximal, characteristic
12#14275C212yesmaximal
13#14265C212yesmaximal
14#14130C211yesmaximal, characteristic
15#13907C211yesmaximal, characteristic
16#13880C211yesmaximal, characteristic
17C4×#381C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2442—28no—
23C2×#400C2212yes—
24C2×#398C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#384C2212yes—
34C2×#382—28no—
35C2×#382C2214yes—
36C2×#382C2212yes—
37C2×#382C2212yes—
38C2×#382C2211yescharacteristic
39C2×#381C2211yescharacteristic
40C2×#381C2211yescharacteristic
41C2×#381C2212yes—
42C2×#380C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C22×#95C2212yes—
49C22×#95C2212yes—
50#1445C2214yes—
51#1445C2214yes—
52#1445C2214yes—
53#1445C2214yes—
54#1444C2214yes—
55#1444C2212yes—
56#1444C2214yes—
57#1444C2214yes—
58#1443C2214yes—
59#1443C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2212yes—
76#1434C2214yes—
77#1434C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2211yescharacteristic
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C2×#125—24no—
91C2×#125—44no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42—24no—
119C22×C42—24no—
120C22×C42Q818yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126#400—28no—
127#398—28no—
128#389—24no—
129#389—24no—
130#389—44no—
131#389—44no—
132#385—28no—
133#385—28no—
134#385—44no—
135#385—44no—
136#384—44no—
137#382—24no—
138#382—28no—
139#382—28no—
140#382—216no—
141#381—24no—
142#381—24no—
143#381—44no—
144#380—28no—
145#380—24no—
146#380—24no—
147#380—28no—
148C4×C22⋊C4—24no—
149C4×C22⋊C4—24no—
150C2×#95—28no—
151C2×#95—28no—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25C2411yescharacteristic
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—44no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2×D414yes—
208C23×C4C2×D418yes—
209C23×C4C2×Q814yes—
210C23×C4C2×Q814yes—
211C23×C4C2×Q814yes—
212C23×C4#4114yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229#125—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
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—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—416no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D418yes—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4#4114yes—
287C2×C22⋊C4#4114yes—
288C2×C42—232no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42C2×Q818yes—
301C2×C42C2×Q818yes—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—28no—
306#95—22no—
307#95—22no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—42no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24C22×D412yes—
340C24C22×Q812yes—
341C24C2×#4112yes—
342C24C2×#4112yes—
343C24C2×#4112yes—
344C24H(5,𝔽2)12yes—
345C242−1+412yes—
346C24C2511yescharacteristic
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—216no—
352C22×C4—216no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—216no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—88no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4C22⋊Q8116yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×Q814yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q812yes—
459C22×C4C22×Q812yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C42−1+414yes—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—88no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C22⋊C4—28no—
483C22⋊C4—28no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C42—216no—
489C42—216no—
490C42—44no—
491C42—44no—
492C42—48no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23C2×C22⋊Q814yes—
510C23#53314yes—
511C23#54414yes—
512C23#54714yes—
513C23#55014yes—
514C23#55214yes—
515C23#55314yes—
516C23C23×D411yescharacteristic
517C23C23×Q811yescharacteristic
518C23C22×#4111yescharacteristic
519C23C22×#4111yescharacteristic
520C23C22×#4111yescharacteristic
521C23C2×H(5,𝔽2)12yes—
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×2−1+412yes—
524C23C2×2−1+411yescharacteristic
525C23#58512yes—
526C23#58512yes—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—232no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—416no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—416no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—816no—
617C2×C4—816no—
618C2×C4C2×C22⋊Q814yes—
619C2×C4C2×C22⋊Q814yes—
620C2×C4#53414yes—
621C2×C4#53614yes—
622C2×C4#53918yes—
623C2×C4#54418yes—
624C22—24no—
625C22—24no—
626C22—28no—
627C22—28no—
628C22—24no—
629C22—28no—
630C22—24no—
631C22—24no—
632C22—28no—
633C22—24no—
634C22#244118yes—
635C22C22×C22⋊Q811yescharacteristic
636C22C2×#53311yescharacteristic
637C22C2×#53411yescharacteristic
638C22C2×#53611yescharacteristic
639C22C2×#53912yes—
640C22C2×#54412yes—
641C22C2×#54411yescharacteristic
642C22#345812yes—
643C22C2×#54711yescharacteristic
644C22C2×#55011yescharacteristic
645C22C2×#55211yescharacteristic
646C22C2×#55311yescharacteristic
647C22#348511yescharacteristic
648C22#348711yescharacteristic
649C22#349312yes—
650C22#351412yes—
651C22#351412yes—
652C22#351512yes—
653C22#351512yes—
654C22#355812yes—
655C22#355912yes—
656C22#356112yes—
657C22#356312yes—
658C4—24no—
659C4—44no—
660C4—44no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—44no—
665C4—81no—
666C4—81no—
667C4—81no—
668C4—82no—
669C4—81no—
670C4—82no—
671C4—84no—
672C4—84no—
673C4#262718yes—
674C2—28no—
675C2C2×#262711yescharacteristic
676C2C2×#244111yescharacteristic
677C2#3513212yes—
678C2#3515111yescharacteristic
679C2#3756112yes—
680C2#3891811yescharacteristic
681C2#3891911yescharacteristic
682C2#3892912yes—
683C2#3913411yescharacteristic
684C2#3913511yescharacteristic
685C2#3914512yes—
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
3. #16689C2semidirect
4. #16687C2semidirect
5. #15242C2semidirect
6. #15232C2semidirect
7. #15228C2semidirect
8. #15225C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C22×#3822
#166892
#166872
#152422
#152322
#152282
#152252
#149362
#149352
#146852
#142752
#142652
#141302
#139072
#138802
C4×#3812