Finite Group Databasebeta

← #636506

Subgroups — #636506

Order 512.

2770 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C211yesmaximal, characteristic
2C22×#384C211yesmaximal, characteristic
3#16570C214yesmaximal, complemented
4#16523C214yesmaximal, complemented
5#15231C212yesmaximal, complemented
6#15228C212yesmaximal, complemented
7#15146C212yesmaximal, complemented
8#15137C212yesmaximal, complemented
9#14822C212yesmaximal
10#14728C212yesmaximal
11#14685C211yesmaximal, characteristic
12#14339C212yesmaximal
13#14263C212yesmaximal
14#14130C211yesmaximal, characteristic
15#13907C211yesmaximal, characteristic
16#13881C211yesmaximal, characteristic
17C4×#381C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24#2442—28no—
25C2×#400C2212yes—
26C2×#400C2214yes—
27C2×#398C2212yes—
28C2×#398C2214yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#384—28no—
36C2×#384C2212yes—
37C2×#384C2212yes—
38C2×#384C2214yes—
39C2×#382C2211yescharacteristic
40C2×#382C2212yes—
41C2×#381C2211yescharacteristic
42C2×#381C2212yes—
43C2×#381C2211yescharacteristic
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×#380C2211yescharacteristic
47C2×#380C2212yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#1444C2214yes—
51#1444C2212yes—
52#1444C2214yes—
53#1444C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73#1434C2214yes—
74#1434C2212yes—
75#1434C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2211yescharacteristic
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×#125—24no—
91C2×#125—44no—
92C22×C4⋊C4—28no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—28no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4—48no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4D418yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C42—24no—
126C22×C42—24no—
127C22×C42Q818yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133#400—28no—
134#400—48no—
135#398—28no—
136#398—48no—
137#389—24no—
138#389—24no—
139#389—44no—
140#389—44no—
141#385—28no—
142#385—44no—
143#384—28no—
144#384—28no—
145#384—216no—
146#382—24no—
147#382—44no—
148#381—24no—
149#381—24no—
150#381—28no—
151#380—28no—
152#380—24no—
153#380—24no—
154#380—44no—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—24no—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C25C2411yescharacteristic
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C2×D414yes—
209C23×C4C2×D418yes—
210C23×C4C2×Q814yes—
211C23×C4C2×Q814yes—
212C23×C4C2×Q814yes—
213C23×C4#4114yes—
214C23×C4#4114yes—
215C23×C4#4114yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232#125—416no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—416no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—432no—
268C2×C4⋊C4—432no—
269C2×C4⋊C4—416no—
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⋊C4—44no—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D418yes—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C42—232no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42C2×Q818yes—
312C2×C42C2×Q818yes—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24C22×D412yes—
349C24C22×Q812yes—
350C24C2×#4112yes—
351C24C2×#4112yes—
352C24C2×#4112yes—
353C24H(5,𝔽2)12yes—
354C242−1+412yes—
355C24C2511yescharacteristic
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—28no—
378C22×C4—216no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—216no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4C22⋊Q8116yes—
458C22×C4#125116yescomplemented
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×Q812yes—
464C22×C4C22×Q814yes—
465C22×C4C22×Q814yes—
466C22×C4C22×Q812yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4112yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C4H(5,𝔽2)14yes—
473C22×C42−1+414yes—
474C22×C42−1+414yes—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—48no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C42—216no—
500C42—216no—
501C42—44no—
502C42—44no—
503C42—48no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23C2×C22⋊Q814yes—
521C23C2×#12514yes—
522C23#54414yes—
523C23#54614yes—
524C23#54814yes—
525C23#55214yes—
526C23#55314yes—
527C23C23×D411yescharacteristic
528C23C23×Q811yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4111yescharacteristic
532C23C2×H(5,𝔽2)12yes—
533C23C2×H(5,𝔽2)11yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23C2×2−1+412yes—
536C23C2×2−1+411yescharacteristic
537C23C2×2−1+412yes—
538C2×C4—232no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—232no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—232no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—816no—
626C2×C4—816no—
627C2×C4C2×C22⋊Q814yes—
628C2×C4C2×C22⋊Q814yes—
629C2×C4C2×#12518yes—
630C2×C4#53414yes—
631C2×C4#53614yes—
632C2×C4#53918yes—
633C2×C4#54418yes—
634C2×C4#54418yes—
635C22—24no—
636C22—24no—
637C22—24no—
638C22—24no—
639C22—28no—
640C22—28no—
641C22—24no—
642C22—28no—
643C22—28no—
644C22—24no—
645C22#244218yes—
646C22C22×C22⋊Q811yescharacteristic
647C22C22×#12511yescharacteristic
648C22C2×#53411yescharacteristic
649C22C2×#53611yescharacteristic
650C22C2×#53912yes—
651C22C2×#54411yescharacteristic
652C22C2×#54412yes—
653C22C2×#54412yes—
654C22C2×#54611yescharacteristic
655C22C2×#54811yescharacteristic
656C22C2×#55211yescharacteristic
657C22C2×#55311yescharacteristic
658C22#348411yescharacteristic
659C22#348611yescharacteristic
660C22#349212yes—
661C22#350812yes—
662C22#350912yes—
663C22#351112yes—
664C22#351112yes—
665C22#354312yes—
666C22#354412yes—
667C22#355212yes—
668C22#355212yes—
669C4—28no—
670C4—28no—
671C4—24no—
672C4—44no—
673C4—44no—
674C4—81no—
675C4—84no—
676C4—81no—
677C4—82no—
678C4—81no—
679C4—81no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4#262718yes—
685C2—28no—
686C2C2×#262711yescharacteristic
687C2C2×#244211yescharacteristic
688C2#3522411yescharacteristic
689C2#3528612yes—
690C2#3762312yes—
691C2#3946411yescharacteristic
692C2#3946511yescharacteristic
693C2#3947612yes—
694C2#3972011yescharacteristic
695C2#3972111yescharacteristic
696C2#3973212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #16570C2semidirect
4. #16523C2semidirect
5. #15231C2semidirect
6. #15228C2semidirect
7. #15146C2semidirect
8. #15137C2semidirect
458. C22×C4#125semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C22×#3842
#165702
#165232
#152312
#152282
#151462
#151372
#148222
#147282
#146852
#143392
#142632
#141302
#139072
#138812
C4×#3812