Finite Group Databasebeta

← #627990

Subgroups — #627990

Order 512.

3182 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic, complemented
2C2×#2626C211yesmaximal, characteristic
3C2×#2312C211yesmaximal, characteristic
4C22×#388C211yesmaximal, characteristic
5#15758C214yesmaximal, complemented
6#15735C214yesmaximal, complemented
7#15632C214yesmaximal, complemented
8#15621C214yesmaximal, complemented
9#14918C212yesmaximal
10#14727C212yesmaximal
11#14702C211yesmaximal, characteristic, complemented
12#14697C211yesmaximal, characteristic, complemented
13#14339C211yesmaximal, characteristic, complemented
14#14044C212yesmaximal, complemented
15#13972C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2627—28no—
23#2626—28no—
24C2×#400C2212yescomplemented
25C2×#399C2212yes—
26C2×#398C2214yescomplemented
27C2×#397C2214yes—
28C2×#396C2212yes—
29C2×#396C2212yes—
30C2×#395C2212yescomplemented
31C2×#393C2212yes—
32#2312—28no—
33C2×#391C2212yescomplemented
34C2×#389C2212yescomplemented
35C2×#388—28no—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×#379C2212yes—
49C2×#379C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C22×#95C2211yescharacteristic
52C22×#95C2211yescharacteristic
53#1443C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1441C2212yescomplemented
57#1441C2214yescomplemented
58#1441C2212yescomplemented
59#1441C2214yescomplemented
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1436C2214yescomplemented
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71C4⋊#95C2214yescomplemented
72C4⋊#95C2212yescomplemented
73C4⋊#95C2212yescomplemented
74#1434C2214yescomplemented
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C26C2311yescharacteristic
80C23×Q8C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C22×C4⋊C4—24no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4D418yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42—24no—
128C22×C42—44no—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2314yes—
132#400—28no—
133#399—28no—
134#398—216no—
135#397—48no—
136#396—28no—
137#396—28no—
138#395—28no—
139#393—28no—
140#391—44no—
141#389—44no—
142#388—216no—
143#388—216no—
144#388—216no—
145#388—216no—
146#385—28no—
147#385—28no—
148#385—28no—
149#381—28no—
150#381—28no—
151#380—24no—
152#380—24no—
153#380D418yes—
154#379—28no—
155#379—28no—
156C4×C22⋊C4—28no—
157C2×#95—24no—
158C2×#95—24no—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C25—22no—
182C25—22no—
183C25—22no—
184C25—22no—
185C25—22no—
186C25—22no—
187C25—42no—
188C25—42no—
189C25—42no—
190C25—42no—
191C25C2×D414yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—42no—
198C22×Q8—42no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4#4114yes—
229C23×C4#4114yes—
230C23×C4#4114yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C22⋊Q8—416no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
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—24no—
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—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—232no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—232no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—416no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4C2×D418yes—
317C2×C22⋊C4C2×D418yes—
318C2×C22⋊C4C2×D418yes—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—28no—
324C2×C42—42no—
325C2×C42—42no—
326C2×C42—416no—
327C2×C42—44no—
328C2×C42—44no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
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—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95C2×D418yes—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—28no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—28no—
367C24—28no—
368C24—44no—
369C24—44no—
370C24—44no—
371C24—44no—
372C24—48no—
373C24—48no—
374C24—48no—
375C24—48no—
376C24—44no—
377C24—44no—
378C24—44no—
379C24—44no—
380C24—84no—
381C24—84no—
382C24—84no—
383C24—84no—
384C24—84no—
385C24—84no—
386C24—84no—
387C24—84no—
388C24C22×D414yes—
389C24C22×D412yes—
390C24C22×D412yes—
391C24C2×#4112yes—
392C24C2×#4112yes—
393C24C2×#4112yes—
394C242−1+412yes—
395C242−1+412yes—
396C24C2511yescharacteristic
397C2×Q8—48no—
398C2×Q8—44no—
399C2×Q8—82no—
400C2×Q8—82no—
401C2×Q8—84no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—28no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—28no—
440C22×C4—22no—
441C22×C4—24no—
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—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—84no—
494C22×C4—84no—
495C22×C4—84no—
496C22×C4—84no—
497C22×C4#121116yes—
498C22×C4C22×D414yes—
499C22×C4C22×D418yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C2×#4114yes—
505C22×C4C2×#4114yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4H(5,𝔽2)14yes—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—82no—
515C4⋊C4—82no—
516C4⋊C4—84no—
517C4⋊C4—84no—
518C4⋊C4—84no—
519C4⋊C4—88no—
520C4⋊C4—88no—
521C22⋊C4—232no—
522C22⋊C4—216no—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—416no—
526C22⋊C4—48no—
527C22⋊C4—48no—
528C22⋊C4—48no—
529C22⋊C4—88no—
530C22⋊C4—88no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—84no—
534C22⋊C4—84no—
535C42—48no—
536C42—48no—
537C42—88no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—416no—
562C23—416no—
563C23—416no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—88no—
581C23—88no—
582C23—88no—
583C23—88no—
584C23—816no—
585C23—816no—
586C23—816no—
587C23—816no—
588C23C2×#12114yes—
589C23#52514yes—
590C23#53614yes—
591C23#54714yes—
592C23#54814yes—
593C23#55214yes—
594C23#55614yes—
595C23C23×D411yescharacteristic
596C23C23×D411yescharacteristic
597C23C22×#4111yescharacteristic
598C23C22×#4111yescharacteristic
599C23C22×#4111yescharacteristic
600C23C2×H(5,𝔽2)12yes—
601C23C2×2−1+411yescharacteristic
602C23C2×2−1+411yescharacteristic
603C23#58512yes—
604C23#58512yes—
605C23#58512yes—
606Q8—88no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—416no—
631C2×C4—416no—
632C2×C4—416no—
633C2×C4—42no—
634C2×C4—416no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
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—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—88no—
685C2×C4—88no—
686C2×C4C2×#12114yes—
687C2×C4C2×#12114yes—
688C22—24no—
689C22—24no—
690C22—24no—
691C22—24no—
692C22—28no—
693C22—24no—
694C22—24no—
695C22—28no—
696C22—28no—
697C22—28no—
698C22—44no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—44no—
705C22—832no—
706C22—84no—
707C22—84no—
708C22—84no—
709C22—84no—
710C22#241518yes—
711C22C22×#12111yescharacteristic
712C22C2×#52511yescharacteristic
713C22C2×#53611yescharacteristic
714C22#345512yes—
715C22#346012yes—
716C22C2×#54711yescharacteristic
717C22C2×#54811yescharacteristic
718C22C2×#55211yescharacteristic
719C22C2×#55611yescharacteristic
720C22#348411yescharacteristic
721C22#348511yescharacteristic
722C22#348511yescharacteristic
723C22#348511yescharacteristic
724C22#349312yes—
725C22#349612yes—
726C22#350012yes—
727C22#350512yes—
728C22#350512yes—
729C22#351012yes—
730C22#353512yes—
731C22#353512yes—
732C22#356412yes—
733C22#356412yes—
734C4—24no—
735C4—24no—
736C4—44no—
737C4—44no—
738C4—44no—
739C4—44no—
740C4—48no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—84no—
747C2—28no—
748C2—84no—
749C2C2×#241511yescharacteristic
750C2#3509111yescharacteristic
751C2#3509311yescharacteristic
752C2#3535811yescharacteristic
753C2#3561711yescharacteristic
754C2#3672011yescharacteristic
755C2#3697011yescharacteristic
756C2#3937112yes—
757C2#3942312yes—
758C2#3950812yes—
759C2#3954312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2627C2semidirect
5. #15758C2semidirect
6. #15735C2semidirect
7. #15632C2semidirect
8. #15621C2semidirect
11. #14702C2semidirect
12. #14697C2semidirect
13. #14339C2semidirect
14. #14044C2semidirect
15. #13972C2semidirect
24. C2×#400C22semidirect
26. C2×#398C22semidirect
30. C2×#395C22semidirect
33. C2×#391C22semidirect
34. C2×#389C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
67. #1436C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect
71. C4⋊#95C22semidirect
72. C4⋊#95C22semidirect
73. C4⋊#95C22semidirect
74. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C2×#26262
C2×#23122
C22×#3882
#157582
#157352
#156322
#156212
#149182
#147272
#147022
#146972
#143392
#140442
#139722