Finite Group Databasebeta

← #4381093

Subgroups — #4381093

Order 512.

3718 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#539C211yesmaximal, characteristic
2C2×#2817C211yesmaximal, characteristic, complemented
3C2×#2583C211yesmaximal, characteristic, complemented
4C2×#2620C211yesmaximal, characteristic
5#16657C214yesmaximal, complemented
6#15940C214yesmaximal, complemented
7#15850C212yesmaximal, complemented
8#15849C212yesmaximal, complemented
9#15464C212yesmaximal, complemented
10#15276C214yesmaximal, complemented
11#15262C212yesmaximal, complemented
12#14731C214yesmaximal, complemented
13#14290C211yesmaximal, characteristic
14#14138C211yesmaximal, characteristic, complemented
15#13870C211yesmaximal, characteristic, complemented
16C2×#539—28no—
17C2×#539—28no—
18C2×#539—28no—
19C22×#126C2212yes—
20C22×#125C2212yes—
21C22×#124C2211yescharacteristic
22C22×#123C2212yes—
23C22×#123C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2212yes—
26C22×C22⋊Q8C2211yescharacteristic
27C22×#121C2211yescharacteristic
28C22×C4×D4C2212yes—
29C22×#117C2211yescharacteristic
30C23×C4⋊C4C2211yescharacteristic
31#2817—28no—
32#2620—28no—
33#2583—28no—
34C2×#400C2212yes—
35C2×#400C2212yes—
36C2×#398C2212yes—
37C2×#397C2214yes—
38C2×#396C2214yes—
39C2×#395C2212yes—
40C2×#395C2212yes—
41C2×#395C2212yes—
42C2×#394C2214yes—
43C2×#391C2211yescharacteristic
44C2×#391C2212yes—
45C2×#390C2211yescharacteristic
46C2×#389C2214yes—
47C2×#385C2212yes—
48C2×#385C2212yes—
49C2×#385C2214yes—
50C2×#384C2211yescharacteristic
51C2×#381C2211yescharacteristic
52C2×#381C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54#1445C2212yes—
55#1444C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1442C2214yes—
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#1437C2214yes—
72#1436C2212yes—
73#1436C2212yes—
74#1436C2212yes—
75#1436C2212yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2214yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89#539—2128no—
90C2×#126—28no—
91C2×#126—28no—
92C2×#126—28no—
93C2×#125—28no—
94C2×#125—28no—
95C2×#125—28no—
96C2×#124—24no—
97C2×#124—44no—
98C2×#123—28no—
99C2×#123—28no—
100C2×#123—28no—
101C2×#123—28no—
102C2×#123—28no—
103C2×#123—28no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—24no—
106C2×C22⋊Q8—24no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—28no—
111C2×C22⋊Q8—28no—
112C2×C22⋊Q8D418yes—
113C2×#121—24no—
114C2×#121—24no—
115C2×#121D418yes—
116C2×C4×D4—28no—
117C2×C4×D4—28no—
118C2×C4×D4—28no—
119C2×#117—24no—
120C2×#117—44no—
121C22×C4⋊C4—24no—
122C22×C4⋊C4—24no—
123C22×C4⋊C4D418yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2312yes—
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2314yes—
136C22×C4⋊C4C2314yes—
137C22×C4⋊C4C2314yes—
138C22×C4⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2312yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2312yes—
147C22×C22⋊C4C2312yes—
148C22×C22⋊C4C2314yes—
149C22×C22⋊C4C2312yes—
150C22×C22⋊C4C2314yes—
151C22×C22⋊C4C2314yes—
152C22×C42C2311yescharacteristic
153C22×C42C2312yes—
154C22×C42C2311yescharacteristic
155C22×C42C2312yes—
156#400—28no—
157#400—28no—
158#398—28no—
159#397—216no—
160#396—216no—
161#395—28no—
162#395—28no—
163#395—28no—
164#394—216no—
165#391—24no—
166#391—44no—
167#390—24no—
168#389—216no—
169#385—28no—
170#385—28no—
171#385—48no—
172#384—24no—
173#381—24no—
174#381—44no—
175C4×C22⋊C4—24no—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8C2×D414yes—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—44no—
212C22×D4—44no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4C2×D418yes—
216C22×D4C2×D414yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—44no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4C2×D414yes—
232C23×C4C2×D414yes—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249#126—264no—
250#125—264no—
251#124—416no—
252#123—264no—
253#123—264no—
254C22⋊Q8—264no—
255C22⋊Q8—232no—
256C22⋊Q8—416no—
257#121—232no—
258C4×D4—264no—
259#117—416no—
260C2×C4⋊C4—232no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—28no—
290C2×C4⋊C4—28no—
291C2×C4⋊C4—28no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—44no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4C2×D418yes—
302C2×C4⋊C4C2×D418yes—
303C2×C4⋊C4C2×D414yes—
304C2×C4⋊C4C2×D418yes—
305C2×C4⋊C4C2×D414yes—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—28no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—28no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—42no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—42no—
341C2×C22⋊C4—42no—
342C2×C22⋊C4C2×D418yes—
343C2×C22⋊C4C2×D418yes—
344C2×C42—22no—
345C2×C42—24no—
346C2×C42—24no—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—22no—
350C2×C42—24no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42—42no—
354#95—24no—
355#95—24no—
356#95—24no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—28no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95—44no—
373C24—22no—
374C24—24no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—24no—
384C24—24no—
385C24C22×D414yes—
386C24C22×D412yes—
387C24C22×D412yes—
388C24C22×D412yes—
389C24C2511yescharacteristic
390C2×Q8—28no—
391C2×Q8—48no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—82no—
396C2×Q8—82no—
397C2×Q8—84no—
398C2×Q8—84no—
399C2×Q8—84no—
400C2×D4—216no—
401C2×D4—28no—
402C2×D4—416no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—416no—
408C2×D4—84no—
409C2×D4—84no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
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—216no—
446C22×C4—28no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
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—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4—82no—
480C22×C4—84no—
481C22×C4—82no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4#120116yes—
485C22×C4C22×D414yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D414yes—
489C22×C4C22×D414yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C42−1+418yes—
498C4⋊C4—216no—
499C4⋊C4—216no—
500C4⋊C4—216no—
501C4⋊C4—216no—
502C4⋊C4—216no—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—28no—
506C4⋊C4—48no—
507C4⋊C4—416no—
508C4⋊C4—44no—
509C4⋊C4—48no—
510C4⋊C4—88no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C22⋊C4—216no—
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—216no—
519C22⋊C4—48no—
520C22⋊C4—416no—
521C22⋊C4—416no—
522C22⋊C4—48no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—84no—
526C42—28no—
527C42—216no—
528C42—44no—
529C42—84no—
530C23—24no—
531C23—22no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—24no—
544C23—28no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—42no—
556C23—42no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23C2×#12014yes—
566C23C23×D411yescharacteristic
567C23C23×D411yescharacteristic
568C23C23×D411yescharacteristic
569C23C2×H(5,𝔽2)12yes—
570C23C2×2−1+414yes—
571C23#58514yes—
572C23#58512yes—
573Q8—88no—
574Q8—88no—
575D4—432no—
576D4—88no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
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—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—416no—
603C2×C4—416no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—88no—
647C2×C4—816no—
648C2×C4C2×#12014yes—
649C2×C4C2×#12014yes—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—44no—
654C22—48no—
655C22—44no—
656C22—48no—
657C22—44no—
658C22—48no—
659C22—44no—
660C22—44no—
661C22—48no—
662C22—84no—
663C22—84no—
664C22—84no—
665C22—84no—
666C22C22×#12011yescharacteristic
667C22#344912yes—
668C22#348411yescharacteristic
669C22#348511yescharacteristic
670C22#348512yes—
671C22#348612yes—
672C22#348712yes—
673C22#349612yes—
674C22#349612yes—
675C22#349912yes—
676C22#349912yes—
677C22#354114yes—
678C22#355714yes—
679C22#355914yes—
680C22#356414yes—
681C4—24no—
682C4—24no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—48no—
687C4—44no—
688C4—44no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—84no—
694C2—44no—
695C2—84no—
696C2#3514911yescharacteristic
697C2#3534711yescharacteristic
698C2#3566611yescharacteristic
699C2#4052314yes—
700C2#4635812yes—
701C2#4681812yes—
702C2#4786014yes—
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×#2817C2semidirect
3. C2×#2583C2semidirect
5. #16657C2semidirect
6. #15940C2semidirect
7. #15850C2semidirect
8. #15849C2semidirect
9. #15464C2semidirect
10. #15276C2semidirect
11. #15262C2semidirect
12. #14731C2semidirect
14. #14138C2semidirect
15. #13870C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5392
C2×#28172
C2×#25832
C2×#26202
#166572
#159402
#158502
#158492
#154642
#152762
#152622
#147312
#142902
#141382
#138702