Finite Group Databasebeta

← #4381105

Subgroups — #4381105

Order 512.

3542 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic, complemented
3C2×#2499C211yesmaximal, characteristic, complemented
4C2×#2624C211yesmaximal, characteristic
5#16657C214yesmaximal, complemented
6#16104C214yesmaximal, complemented
7#15983C214yesmaximal, complemented
8#15928C212yesmaximal, complemented
9#15927C212yesmaximal, complemented
10#15466C212yesmaximal, complemented
11#15465C212yesmaximal, complemented
12#14823C214yesmaximal, complemented
13#14660C211yesmaximal, characteristic, complemented
14#14290C211yesmaximal, characteristic
15#13923C211yesmaximal, characteristic, complemented
16C2×#539—28no—
17C2×#539—28no—
18C2×#539—28no—
19C22×#126C2212yes—
20C22×#125C2212yes—
21C22×#123C2212yes—
22C22×#123C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×#121C2211yescharacteristic
26C22×C4×D4C2212yes—
27C22×#117C2211yescharacteristic
28C23×C4⋊C4C2211yescharacteristic
29#2624—28no—
30#2499—28no—
31#2484—28no—
32C2×#400C2212yes—
33C2×#399C2212yes—
34C2×#398C2212yes—
35C2×#397C2214yes—
36C2×#397C2212yes—
37C2×#394C2214yes—
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#390C2212yes—
41C2×#389C2212yes—
42C2×#387C2214yes—
43C2×#387C2214yes—
44C2×#385C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382C2211yescharacteristic
47C2×#381C2211yescharacteristic
48C2×#381C2212yes—
49C2×#380C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51#1446C2212yes—
52#1445C2212yes—
53#1445C2212yes—
54#1444C2214yes—
55#1444C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2214yes—
70#1438C2212yes—
71#1437C2212yes—
72#1437C2212yes—
73#1436C2212yes—
74#1436C2214yes—
75#1436C2212yes—
76#1436C2214yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C23×Q8C2311yescharacteristic
82C23×D4C2312yes—
83C23×D4C2311yescharacteristic
84C24×C4C2312yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88#539—2128no—
89C2×#126—28no—
90C2×#126—28no—
91C2×#126—28no—
92C2×#125—28no—
93C2×#125—28no—
94C2×#125—28no—
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—28no—
98C2×#123—28no—
99C2×#123—28no—
100C2×#123—28no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8D418yes—
107C2×#121—24no—
108C2×#121—24no—
109C2×#121D418yes—
110C2×C4×D4—28no—
111C2×C4×D4—28no—
112C2×C4×D4—28no—
113C2×#117—24no—
114C2×#117—24no—
115C2×#117—24no—
116C22×C4⋊C4—24no—
117C22×C4⋊C4—24no—
118C22×C4⋊C4D418yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2314yes—
131C22×C4⋊C4C2314yes—
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2312yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2314yes—
142C22×C22⋊C4C2314yes—
143C22×C22⋊C4C2314yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2314yes—
146C22×C42C2312yes—
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149C22×C42C2311yescharacteristic
150#400—28no—
151#399—28no—
152#398—28no—
153#397—28no—
154#397—216no—
155#394—216no—
156#391—24no—
157#391—24no—
158#390—28no—
159#389—28no—
160#387—216no—
161#387—216no—
162#385—28no—
163#384—24no—
164#382—24no—
165#381—24no—
166#381—28no—
167#380—24no—
168C4×C22⋊C4—24no—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8C2×D414yes—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—42no—
204C22×D4—42no—
205C22×D4C2×D418yes—
206C22×D4C2×D414yes—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—44no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239#126—264no—
240#125—264no—
241#123—264no—
242#123—264no—
243C22⋊Q8—264no—
244C22⋊Q8—232no—
245#121—232no—
246C4×D4—264no—
247#117—416no—
248C2×C4⋊C4—232no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4C2×D418yes—
288C2×C4⋊C4C2×D414yes—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2×D414yes—
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—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4C2×D418yes—
329C2×C22⋊C4C2×D418yes—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—42no—
339C2×C42—42no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24C22×D414yes—
375C24C22×D412yes—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C2511yescharacteristic
379C2×Q8—28no—
380C2×Q8—44no—
381C2×Q8—44no—
382C2×Q8—84no—
383C2×Q8—84no—
384C2×D4—216no—
385C2×D4—28no—
386C2×D4—416no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—416no—
391C2×D4—82no—
392C2×D4—82no—
393C2×D4—84no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—28no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
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—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—88no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—84no—
468C22×C4—88no—
469C22×C4#120116yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D414yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C42−1+414yes—
482C4⋊C4—216no—
483C4⋊C4—216no—
484C4⋊C4—216no—
485C4⋊C4—216no—
486C4⋊C4—216no—
487C4⋊C4—28no—
488C4⋊C4—216no—
489C4⋊C4—28no—
490C4⋊C4—416no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—88no—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—216no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—88no—
505C22⋊C4—88no—
506C22⋊C4—84no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C42—216no—
510C42—44no—
511C42—44no—
512C42—48no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—24no—
528C23—24no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23C2×#12014yes—
550C23C23×D411yescharacteristic
551C23C23×D411yescharacteristic
552C23C23×D411yescharacteristic
553C23C2×2−1+412yes—
554C23#58514yes—
555C23#58512yes—
556C23#58512yes—
557C23#58512yes—
558Q8—88no—
559D4—432no—
560D4—88no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—88no—
631C2×C4—816no—
632C2×C4C2×#12014yes—
633C2×C4C2×#12014yes—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—44no—
638C22—44no—
639C22—48no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—44no—
645C22—48no—
646C22—48no—
647C22—84no—
648C22—84no—
649C22—84no—
650C22—84no—
651C22C22×#12011yescharacteristic
652C22#344912yes—
653C22#348512yes—
654C22#348511yescharacteristic
655C22#348511yescharacteristic
656C22#348512yes—
657C22#348512yes—
658C22#348712yes—
659C22#348912yes—
660C22#349712yes—
661C22#349911yescharacteristic
662C22#350011yescharacteristic
663C22#355714yes—
664C22#356414yes—
665C22#356614yes—
666C22#356714yes—
667C4—24no—
668C4—24no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—44no—
673C4—44no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—84no—
679C4—84no—
680C2—44no—
681C2—84no—
682C2#3514911yescharacteristic
683C2#3544611yescharacteristic
684C2#3566011yescharacteristic
685C2#4137714yes—
686C2#4681712yes—
687C2#4681812yes—
688C2#4878212yes—
689C2#4878312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2484C2semidirect
3. C2×#2499C2semidirect
5. #16657C2semidirect
6. #16104C2semidirect
7. #15983C2semidirect
8. #15928C2semidirect
9. #15927C2semidirect
10. #15466C2semidirect
11. #15465C2semidirect
12. #14823C2semidirect
13. #14660C2semidirect
15. #13923C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#24842
C2×#24992
C2×#26242
#166572
#161042
#159832
#159282
#159272
#154662
#154652
#148232
#146602
#142902
#139232