Finite Group Databasebeta

← #4422109

Subgroups — #4422109

Order 512.

2778 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic, complemented
2C2×#2535C211yesmaximal, characteristic
3C2×#2633C211yesmaximal, characteristic, complemented
4C2×#2357C211yesmaximal, characteristic, complemented
5#17022C212yesmaximal, complemented
6#16819C212yesmaximal, complemented
7#16394C212yesmaximal, complemented
8#16259C212yesmaximal, complemented
9#15687C214yesmaximal, complemented
10#15394C214yesmaximal, complemented
11#15174C212yesmaximal, complemented
12#15081C212yesmaximal, complemented
13#14841C212yesmaximal, complemented
14#14764C212yesmaximal, complemented
15#14664C211yesmaximal, characteristic, complemented
16#14356C211yesmaximal, characteristic, complemented
17#14345C211yesmaximal, characteristic, complemented
18C22×#125C2211yescharacteristic
19C22×#121C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2633—28no—
23#2562—28no—
24#2535—28no—
25C2×#400C2211yescharacteristic
26C2×#400C2211yescharacteristic
27C2×#399C2212yes—
28C2×#398C2211yescharacteristic
29C2×#398C2211yescharacteristic
30C2×#397C2212yes—
31C2×#397C2212yes—
32C2×#394C2211yescharacteristic
33C2×#394C2211yescharacteristic
34C2×#394C2211yescharacteristic
35C2×#394C2211yescharacteristic
36C2×#394C2214yes—
37C2×#392C2212yes—
38#2357—28no—
39C2×#390C2212yes—
40C2×#388C2214yes—
41C2×#388C2212yes—
42C2×#388C2212yes—
43C2×#388C2214yes—
44C2×#386C2214yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#383C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#381C2212yes—
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2212yes—
57#1445C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1444C2212yes—
61#1443C2214yes—
62#1442C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1438C2212yes—
78#1438C2214yes—
79#1438C2214yes—
80#1438C2214yes—
81#1437C2214yes—
82#1436C2214yes—
83C4⋊#95C2212yes—
84#1434C2212yes—
85#1434C2214yes—
86C22⋊#95C2214yes—
87C23×D4C2311yescharacteristic
88C23×D4C2314yes—
89C23×D4C2311yescharacteristic
90C24×C4C2312yes—
91C24×C4C2312yes—
92C2×#125—24no—
93C2×#125—24no—
94C2×#125—24no—
95C2×#121—28no—
96C2×#121—48no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4D418yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C42C2314yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#400—24no—
137#400—24no—
138#399—28no—
139#398—24no—
140#398—24no—
141#397—28no—
142#397—28no—
143#394—24no—
144#394—24no—
145#394—24no—
146#394—24no—
147#394—48no—
148#392—28no—
149#390—28no—
150#388—216no—
151#388—28no—
152#388—28no—
153#388—48no—
154#386—216no—
155#385—28no—
156#385—28no—
157#383—24no—
158#382—24no—
159#382—24no—
160#381—28no—
161#380—28no—
162#380—28no—
163#380—28no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4D418yescomplemented
166C4×C22⋊C4—28no—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C25C2412yes—
195C25C2412yes—
196C22×D4—24no—
197C22×D4—28no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—42no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4C2×D414yes—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D418yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238#125—416no—
239#121—432no—
240C4×D4—232no—
241C4×D4—416no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D418yes—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
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⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4C2×D418yes—
323C2×C42—28no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42C2×D418yes—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330C2×C42#4114yes—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—22no—
343#95—22no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95C2×D414yes—
357#95C2×D414yes—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—22no—
365C24—24no—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—22no—
377C24—22no—
378C24C22×D414yes—
379C24H(5,𝔽2)14yes—
380C24C2511yescharacteristic
381C2×D4—28no—
382C2×D4—28no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—48no—
390C2×D4—416no—
391C2×D4—416no—
392C2×D4—44no—
393C2×D4—88no—
394C2×D4—88no—
395C2×D4—88no—
396C2×D4—88no—
397C2×D4—88no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—216no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4C22×D414yes—
475C22×C4C22×D414yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C4⋊C4—28no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—88no—
500C22⋊C4—216no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—88no—
512C22⋊C4—84no—
513C22⋊C4—84no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C42—48no—
517C42—88no—
518C42#121116yes—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—28no—
531C23—42no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23C23×D411yescharacteristic
566C23C23×D411yescharacteristic
567C23C22×#4111yescharacteristic
568C23C2×H(5,𝔽2)12yes—
569C23#58512yes—
570C23#58512yes—
571C23#58514yes—
572C23#58512yes—
573D4—416no—
574D4—416no—
575D4—832no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
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—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—816no—
652C2×C4—816no—
653C2×C4C2×#12114yes—
654C2×C4C2×#12114yes—
655C2×C4C2×#12114yes—
656C22—48no—
657C22—44no—
658C22—44no—
659C22—48no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—84no—
669C22—84no—
670C22—84no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—84no—
675C22—84no—
676C22—82no—
677C22—82no—
678C22C22×#12111yescharacteristic
679C22#344912yes—
680C22#344912yes—
681C22#348411yescharacteristic
682C22#348511yescharacteristic
683C22#348511yescharacteristic
684C22#348711yescharacteristic
685C22#348812yes—
686C22#349312yes—
687C22#349412yes—
688C22#350012yes—
689C22#351012yes—
690C22#353314yes—
691C22#356014yes—
692C22#356614yes—
693C22#356714yes—
694C4—24no—
695C4—24no—
696C4—24no—
697C4—44no—
698C4—44no—
699C4—44no—
700C4—84no—
701C4—82no—
702C4—84no—
703C4—81no—
704C4—81no—
705C4—82no—
706C4—81no—
707C4—81no—
708C4—82no—
709C2—84no—
710C2—84no—
711C2#3515011yescharacteristic
712C2#3532711yescharacteristic
713C2#3551411yescharacteristic
714C2#4050212yes—
715C2#4138212yes—
716C2#4142012yes—
717C2#4198712yes—
718C2#5120014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2535C2
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×#2562C2semidirect
3. C2×#2633C2semidirect
4. C2×#2357C2semidirect
5. #17022C2semidirect
6. #16819C2semidirect
7. #16394C2semidirect
8. #16259C2semidirect
9. #15687C2semidirect
10. #15394C2semidirect
11. #15174C2semidirect
12. #15081C2semidirect
13. #14841C2semidirect
14. #14764C2semidirect
15. #14664C2semidirect
16. #14356C2semidirect
17. #14345C2semidirect
165. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#25352
C2×#26332
C2×#23572
#170222
#168192
#163942
#162592
#156872
#153942
#151742
#150812
#148412
#147642
#146642
#143562
#143452