Finite Group Databasebeta

← #4309456

Subgroups — #4309456

Order 512.

2701 conjugacy classes of subgroups.

The subgroup lattice has 732 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2852C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic, complemented
3#17006C212yesmaximal, complemented
4#16988C212yesmaximal, complemented
5#16981C212yesmaximal, complemented
6#16980C212yesmaximal, complemented
7#16962C212yesmaximal, complemented
8#16674C212yesmaximal, complemented
9#16567C212yesmaximal, complemented
10#16564C212yesmaximal, complemented
11#15073C211yesmaximal, characteristic, complemented
12#15069C211yesmaximal, characteristic
13#14935C212yesmaximal, complemented
14#14927C212yesmaximal
15#14843C211yesmaximal, characteristic, complemented
16#14777C211yesmaximal, characteristic
17#14682C211yesmaximal, characteristic, complemented
18#14263C212yesmaximal, complemented
19#14256C212yesmaximal
20C22×#126C2211yescharacteristic
21C22×#123C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24C23×C4⋊C4C2211yescharacteristic
25C23×C42C2211yescharacteristic
26#2852—28no—
27C2×#400C2212yes—
28C2×#399C2212yes—
29C2×#389—28no—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#388C2211yescharacteristic
35C2×#388C2211yescharacteristic
36C2×#388C2211yescharacteristic
37C2×#387C2211yescharacteristic
38C2×#387C2211yescharacteristic
39C2×#387C2211yescharacteristic
40C2×#386C2211yescharacteristic
41C2×#386C2211yescharacteristic
42C2×#386C2211yescharacteristic
43C2×#386C2212yes—
44C2×#385C2211yescharacteristic
45C2×#385C2211yescharacteristic
46C2×#385C2211yescharacteristic
47C2×#385C2212yes—
48C2×#383C2211yescharacteristic
49C2×#382C2212yes—
50C2×#381C2212yes—
51C2×#381C2212yes—
52C2×#381C2212yes—
53C2×#380C2212yes—
54C22×#95C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1444C2214yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2212yes—
66#1442C2212yes—
67#1442C2212yes—
68#1442C2214yes—
69#1441C2214yes—
70#1441C2212yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2212yes—
76#1441C2212yes—
77#1441C2214yes—
78#1440C2214yes—
79#1439C2214yes—
80#1439C2212yes—
81#1439C2214yes—
82#1439C2212yes—
83#1439C2212yes—
84#1439C2214yes—
85C4⋊#95C2212yes—
86#1434C2212yes—
87#1434C2212yes—
88C22⋊#95C2212yes—
89C22⋊#95C2212yes—
90C22⋊#95C2212yes—
91C22⋊#95C2211yescharacteristic
92C22⋊#95C2211yescharacteristic
93C23×D4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C24×C4C2311yescharacteristic
97C24×C4C2312yes—
98C24×C4C2311yescharacteristic
99C2×#126—24no—
100C2×#126—44no—
101C2×#123—28no—
102C2×#123—44no—
103C2×#123—44no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—44no—
109C22×C4⋊C4—44no—
110C22×C4⋊C4—44no—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2314yes—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2311yescharacteristic
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2311yescharacteristic
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2314yes—
139C22×C42—44no—
140C22×C42D418yes—
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144C22×C42C2311yescharacteristic
145#400—44no—
146#399—44no—
147#389—28no—
148#389—28no—
149#389—28no—
150#389—28no—
151#388—24no—
152#388—24no—
153#388—24no—
154#387—24no—
155#387—24no—
156#387—24no—
157#386—24no—
158#386—24no—
159#386—24no—
160#386—44no—
161#385—24no—
162#385—24no—
163#385—24no—
164#385—44no—
165#383—24no—
166#382—44no—
167#381—28no—
168#381—44no—
169#381—44no—
170#380—44no—
171C2×#95—28no—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C25C2411yescharacteristic
206C25C2411yescharacteristic
207C22×D4—22no—
208C22×D4—24no—
209C22×D4—22no—
210C22×D4—24no—
211C22×D4—22no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4#4118yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2414yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2411yescharacteristic
250C23×C4C2414yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254#126—416no—
255#123—432no—
256C2×C4⋊C4—232no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—432no—
289C2×C4⋊C4—416no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
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—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—416no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335#95—28no—
336#95—28no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—28no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—24no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—21no—
377C24—22no—
378C24—22no—
379C24—21no—
380C24—22no—
381C24C22×D412yes—
382C24C2×#4114yes—
383C24H(5,𝔽2)14yes—
384C24H(5,𝔽2)12yes—
385C242−1+412yes—
386C24C2511yescharacteristic
387C2×D4—44no—
388C2×D4—44no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—48no—
392C22×C4—232no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—216no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—216no—
407C22×C4—216no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—21no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—21no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—44no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—416no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—88no—
491C22×C4—88no—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C2×#4118yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C42−1+414yes—
501C22×C42−1+414yes—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—416no—
507C4⋊C4—416no—
508C4⋊C4—416no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—44no—
515C4⋊C4—88no—
516C4⋊C4—88no—
517C22⋊C4—44no—
518C22⋊C4—416no—
519C22⋊C4—44no—
520C22⋊C4—44no—
521C22⋊C4—44no—
522C22⋊C4—44no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—88no—
530C42—28no—
531C42—28no—
532C42—44no—
533C42—44no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—28no—
539C23—28no—
540C23—24no—
541C23—24no—
542C23—28no—
543C23—28no—
544C23—24no—
545C23—24no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—41no—
552C23—41no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—44no—
560C23—44no—
561C23—42no—
562C23—42no—
563C23#52514yes—
564C23#53514yes—
565C23#53714yes—
566C23#53818yes—
567C23#53918yes—
568C23C23×D411yescharacteristic
569C23C22×#4112yes—
570C23C2×H(5,𝔽2)12yes—
571C23C2×H(5,𝔽2)12yes—
572C23C2×H(5,𝔽2)11yescharacteristic
573C23C2×2−1+412yes—
574C23C2×2−1+412yes—
575C23C2×2−1+411yescharacteristic
576C23#58512yes—
577D4—88no—
578C2×C4—232no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—28no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—432no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
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—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—816no—
665C2×C4#53414yes—
666C2×C4#53614yes—
667C2×C4#53614yes—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—24no—
672C22—24no—
673C22—216no—
674C22—216no—
675C22—24no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—81no—
682C22—84no—
683C22—84no—
684C22—81no—
685C22#263618yes—
686C22C2×#52511yescharacteristic
687C22C2×#53411yescharacteristic
688C22C2×#53511yescharacteristic
689C22C2×#53611yescharacteristic
690C22C2×#53611yescharacteristic
691C22C2×#53711yescharacteristic
692C22#344911yescharacteristic
693C22C2×#53812yes—
694C22C2×#53912yes—
695C22#349214yes—
696C22#350714yes—
697C22#353612yes—
698C22#353812yes—
699C22#354214yes—
700C22#354414yes—
701C22#355912yes—
702C22#356312yes—
703C4—24no—
704C4—24no—
705C4—24no—
706C4—48no—
707C4—42no—
708C4—42no—
709C4—42no—
710C4—81no—
711C4—81no—
712C4—81no—
713C4—81no—
714C4—81no—
715C4—84no—
716C4—81no—
717C4—84no—
718C4—84no—
719C4—81no—
720C2—28no—
721C2—161no—
722C2C2×#263611yescharacteristic
723C2#3505711yescharacteristic
724C2#3511111yescharacteristic
725C2#3682314yes—
726C2#3682414yes—
727C2#3744211yescharacteristic
728C2#3744811yescharacteristic
729C2#3754911yescharacteristic
730C2#3763211yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#389C2semidirect
3. #17006C2semidirect
4. #16988C2semidirect
5. #16981C2semidirect
6. #16980C2semidirect
7. #16962C2semidirect
8. #16674C2semidirect
9. #16567C2semidirect
10. #16564C2semidirect
11. #15073C2semidirect
13. #14935C2semidirect
15. #14843C2semidirect
17. #14682C2semidirect
18. #14263C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28522
C22×#3892
#170062
#169882
#169812
#169802
#169622
#166742
#165672
#165642
#150732
#150692
#149352
#149272
#148432
#147772
#146822
#142632
#142562