Finite Group Databasebeta

← #626268

Subgroups — #626268

Order 512.

3816 conjugacy classes of subgroups.

The subgroup lattice has 786 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2412C212yesmaximal, complemented
2C2×#2617C211yesmaximal, characteristic
3C22×#388C211yesmaximal, characteristic, complemented
4#15578C214yesmaximal, complemented
5#15567C214yesmaximal, complemented
6#15486C214yesmaximal, complemented
7#15473C214yesmaximal, complemented
8#15019C211yesmaximal, characteristic, complemented
9#14926C211yesmaximal, characteristic, complemented
10#14821C211yesmaximal, characteristic, complemented
11#14719C211yesmaximal, characteristic, complemented
12#14706C212yesmaximal, complemented
13#14259C212yesmaximal, complemented
14#14134C212yesmaximal, complemented
15#13794C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×#121C2211yescharacteristic
18C22×#120C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25#2617—28no—
26#2412—216no—
27C2×#398C2212yescomplemented
28C2×#397C2214yes—
29C2×#397C2212yes—
30C2×#394C2214yes—
31C2×#394C2212yes—
32C2×#394C2212yes—
33C2×#394C2212yescomplemented
34C2×#392C2212yes—
35C2×#388—28no—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#386C2214yes—
41C2×#386C2212yes—
42C2×#386C2214yes—
43C2×#386C2212yes—
44C2×#385C2212yes—
45C2×#381C2212yes—
46C2×#380C2214yes—
47C2×#380C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1445C2212yescomplemented
51#1444C2212yescomplemented
52#1443C2214yescomplemented
53#1443C2214yescomplemented
54#1442C2212yescomplemented
55#1442C2214yescomplemented
56#1441C2212yescomplemented
57#1441C2214yescomplemented
58#1441C2212yescomplemented
59#1440C2214yes—
60#1440C2214yescomplemented
61#1439C2214yescomplemented
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1438C2214yes—
65#1438C2212yescomplemented
66#1437C2214yescomplemented
67#1437C2214yescomplemented
68#1436C2214yescomplemented
69#1436C2214yescomplemented
70C22⋊#95C2212yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2211yescharacteristic
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C22⋊#95C2211yescharacteristic
78C26C2311yescharacteristic
79C23×D4C2312yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C23×D4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2314yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#123—28no—
91C2×#123—28no—
92C2×#123—44no—
93C2×#121—24no—
94C2×#121—44no—
95C2×#120—28no—
96C2×#120—28no—
97C2×#120—28no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—28no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—28no—
112C22×C22⋊C4—44no—
113C22×C22⋊C4—44no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4D418yes—
116C22×C22⋊C4D418yescomplemented
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C42—24no—
135C22×C42—44no—
136C22×C42C2314yes—
137#398—44no—
138#397—28no—
139#397—216no—
140#394—28no—
141#394—28no—
142#394—216no—
143#394—44no—
144#392—28no—
145#388—216no—
146#388—216no—
147#386—28no—
148#386—28no—
149#386—216no—
150#386—28no—
151#386—48no—
152#386—44no—
153#385—28no—
154#381—28no—
155#380—24no—
156#380—216no—
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×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C25—24no—
182C25—24no—
183C25—42no—
184C25—42no—
185C25—42no—
186C25—42no—
187C25C2×D414yes—
188C25C2×D414yes—
189C25#4114yes—
190C25C2412yes—
191C25C2412yes—
192C25C2411yescharacteristic
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—42no—
208C22×D4—44no—
209C22×D4—44no—
210C22×D4—44no—
211C22×D4—44no—
212C22×D4—42no—
213C22×D4—42no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—24no—
228C23×C4—28no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—44no—
238C23×C4—44no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—44no—
242C23×C4—44no—
243C23×C4C2×D414yes—
244C23×C4C2×D414yes—
245C23×C4C2×D414yes—
246C23×C4C2×D414yes—
247C23×C4C2412yes—
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2412yes—
252C23×C4C2412yes—
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C23×C4C2414yes—
258C23×C4C2412yes—
259#123—432no—
260#121—416no—
261#120—432no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—416no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C22⋊C4—232no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
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—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—432no—
311C2×C22⋊C4—416no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—416no—
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⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4C2×D4116yes—
331C2×C22⋊C4C2×D418yes—
332C2×C22⋊C4C2×D418yes—
333C2×C22⋊C4#4118yes—
334C2×C42—28no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42—416no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—42no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—42no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—42no—
359#95—42no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—28no—
373C24—28no—
374C24—216no—
375C24—216no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—24no—
387C24—44no—
388C24—48no—
389C24—44no—
390C24—44no—
391C24—44no—
392C24—48no—
393C24—48no—
394C24—48no—
395C24—88no—
396C24—88no—
397C24—88no—
398C24—88no—
399C24#121116yes—
400C24C22×D414yes—
401C24C22×D414yes—
402C24C22×D412yes—
403C24C22×D412yes—
404C24C2×#4114yes—
405C24C2×#4112yes—
406C24C2×#4114yes—
407C24H(5,𝔽2)14yes—
408C24C2511yescharacteristic
409C2×D4—44no—
410C2×D4—416no—
411C2×D4—416no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—82no—
422C2×D4—82no—
423C2×D4—84no—
424C2×D4—84no—
425C2×D4—84no—
426C2×D4—84no—
427C2×D4—88no—
428C2×D4—88no—
429C2×D4—88no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—216no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—28no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—48no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—416no—
501C22×C4—416no—
502C22×C4—44no—
503C22×C4—48no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—416no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—48no—
511C22×C4—44no—
512C22×C4—416no—
513C22×C4—44no—
514C22×C4—44no—
515C22×C4—88no—
516C22×C4—88no—
517C22×C4—88no—
518C22×C4—88no—
519C22×C4—88no—
520C22×C4—88no—
521C22×C4—88no—
522C22×C4C22×D414yes—
523C22×C4C22×D414yes—
524C22×C4C22×D414yes—
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C2×#4118yes—
528C22×C4H(5,𝔽2)14yes—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—416no—
532C4⋊C4—44no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—88no—
536C22⋊C4—232no—
537C22⋊C4—216no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—416no—
541C22⋊C4—416no—
542C22⋊C4—416no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C22⋊C4—88no—
551C22⋊C4—88no—
552C22⋊C4—88no—
553C42—416no—
554C23—24no—
555C23—24no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—28no—
563C23—28no—
564C23—28no—
565C23—28no—
566C23—24no—
567C23—24no—
568C23—28no—
569C23—28no—
570C23—24no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—28no—
579C23—28no—
580C23—232no—
581C23—416no—
582C23—416no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—44no—
595C23—44no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—88no—
615C23—88no—
616C23—832no—
617C23—88no—
618C23—832no—
619C23—88no—
620C23C2×#12118yes—
621C23C2×#12114yes—
622C23#53214yes—
623C23#53414yes—
624C23#54618yes—
625C23#55318yes—
626C23C23×D411yescharacteristic
627C23C23×D411yescharacteristic
628C23C22×#4111yescharacteristic
629C23C22×#4112yes—
630C23C2×H(5,𝔽2)12yes—
631C23C2×H(5,𝔽2)12yes—
632C23#58512yes—
633C23#58512yes—
634C23#58512yes—
635D4—88no—
636D4—816no—
637D4—816no—
638D4—816no—
639C2×C4—24no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—28no—
658C2×C4—416no—
659C2×C4—416no—
660C2×C4—416no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
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—416no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—44no—
709C2×C4—88no—
710C2×C4—832no—
711C2×C4—88no—
712C22—24no—
713C22—24no—
714C22—24no—
715C22—216no—
716C22—24no—
717C22—24no—
718C22—216no—
719C22—24no—
720C22—28no—
721C22—28no—
722C22—28no—
723C22—44no—
724C22—44no—
725C22—48no—
726C22—48no—
727C22—44no—
728C22—48no—
729C22—48no—
730C22—44no—
731C22—832no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—82no—
736C22—84no—
737C22—84no—
738C22—82no—
739C22—82no—
740C22—84no—
741C22—84no—
742C22—82no—
743C22#254318yes—
744C22C22×#12111yescharacteristic
745C22C2×#53211yescharacteristic
746C22C2×#53411yescharacteristic
747C22#344912yes—
748C22#345912yes—
749C22C2×#54612yes—
750C22C2×#55312yes—
751C22#348411yescharacteristic
752C22#348511yescharacteristic
753C22#348511yescharacteristic
754C22#348511yescharacteristic
755C22#349112yes—
756C22#349612yes—
757C22#351414yes—
758C22#352114yes—
759C22#353314yes—
760C22#353414yes—
761C4—44no—
762C4—44no—
763C4—44no—
764C4—48no—
765C4—44no—
766C4—42no—
767C4—42no—
768C4—82no—
769C4—82no—
770C4—82no—
771C4—82no—
772C4—84no—
773C2—28no—
774C2—48no—
775C2—162no—
776C2C2×#254311yescharacteristic
777C2#3498411yescharacteristic
778C2#3508911yescharacteristic
779C2#3546611yescharacteristic
780C2#3546711yescharacteristic
781C2#3695511yescharacteristic
782C2#3711711yescharacteristic
783C2#3808614yes—
784C2#3905014yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2617C2
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×#2412C2semidirect
3. C22×#388C2semidirect
4. #15578C2semidirect
5. #15567C2semidirect
6. #15486C2semidirect
7. #15473C2semidirect
8. #15019C2semidirect
9. #14926C2semidirect
10. #14821C2semidirect
11. #14719C2semidirect
12. #14706C2semidirect
13. #14259C2semidirect
14. #14134C2semidirect
15. #13794C2semidirect
27. C2×#398C22semidirect
33. C2×#394C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
52. #1443C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
60. #1440C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
65. #1438C22semidirect
66. #1437C22semidirect
67. #1437C22semidirect
68. #1436C22semidirect
69. #1436C22semidirect
116. C22×C22⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24122
C2×#26172
C22×#3882
#155782
#155672
#154862
#154732
#150192
#149262
#148212
#147192
#147062
#142592
#141342
#137942