Finite Group Databasebeta

← #518259

Subgroups — #518259

Order 512.

2766 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic
2C2×#2362C211yesmaximal, characteristic
3C2×#2274C211yesmaximal, characteristic
4#14307C212yesmaximal, complemented
5#14258C211yesmaximal, characteristic
6#14250C212yesmaximal, complemented
7#14239C212yesmaximal, complemented
8#14104C214yesmaximal
9#14103C214yesmaximal
10#14030C212yesmaximal, complemented
11#14021C212yesmaximal, complemented
12#13924C211yesmaximal, characteristic
13C4×#400C212yesmaximal, complemented
14#13844C211yesmaximal, characteristic
15#13817C212yesmaximal, complemented
16#13804C212yesmaximal, complemented
17#13801C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2503C4116yescomplemented
24C2×#400C2212yes—
25C2×#400C2212yes—
26#2362—28no—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#389C2211yescharacteristic
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#387C2212yes—
34C2×#382C2212yes—
35C2×#381C2214yes—
36C2×#380C2211yescharacteristic
37#2274—28no—
38C2×C4×C4⋊C4C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2214yes—
48C2×C4×C22⋊C4C2211yescharacteristic
49#1445C2212yes—
50#1445C2212yes—
51#1444C2211yescharacteristic
52#1444C2211yescharacteristic
53#1443C2214yes—
54#1443C2214yes—
55#1442C2211yescharacteristic
56#1442C2211yescharacteristic
57#1442C2214yes—
58#1442C2214yes—
59#1441C2212yes—
60#1441C2212yes—
61#1440C2214yes—
62#1439C2212yes—
63#1439C2212yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75#1434C2214yes—
76#1434C2214yes—
77#1434C2214yes—
78#1434C2214yes—
79#1434C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2214yes—
84C4×#95C2214yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8C2×C418yes—
92C2×C4×Q8—24no—
93C2×C4×Q8—28no—
94C2×#117—24no—
95C2×#117—28no—
96C2×#117—28no—
97C2×#117D418yes—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4C2×C418yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2314yes—
130C22×C42C2314yes—
131#400C2×C4116yes—
132#400C2×C4116yes—
133#391—24no—
134#391—28no—
135#391C2×C4116yes—
136#389—24no—
137#388—28no—
138#388—28no—
139#387—28no—
140#382C2×C4116yes—
141#381C2×C4132yes—
142#380—24no—
143C4×C4⋊C4—24no—
144C4×C4⋊C4—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—28no—
150C4×C4⋊C4—28no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—216no—
153C4×C22⋊C4C2×C418yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—24no—
191C22×Q8C22×C414yes—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4C22×C414yes—
199C23×C4C22×C418yes—
200C23×C4C2×D414yes—
201C23×C4#4114yes—
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2414yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C22⋊Q8—232no—
219C4×Q8—416no—
220#117—232no—
221#117—232no—
222C2×C4⋊C4—232no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4C22×C418yes—
251C2×C4⋊C4C22×C418yes—
252C2×C4⋊C4C22×C414yes—
253C2×C4⋊C4C22×C418yes—
254C2×C4⋊C4C2×D414yes—
255C2×C4⋊C4C2×D414yes—
256C2×C4⋊C4#4118yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—42no—
274C2×C22⋊C4C22×C418yes—
275C2×C22⋊C4C22×C418yes—
276C2×C22⋊C4C2×D418yes—
277C2×C42—24no—
278C2×C42—28no—
279C2×C42—24no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—22no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—24no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—28no—
291C2×C42—28no—
292C2×C42—28no—
293C2×C42—42no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C22×C418yes—
301C2×C42C2×D418yes—
302C2×C42#4118yes—
303#95—22no—
304#95—28no—
305#95—28no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—24no—
318#95—44no—
319#95—42no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95C22×C414yes—
330#95C22×C414yes—
331#95C22×C4116yes—
332#95C22×C4116yes—
333#95C22×C4116yes—
334#95C22×C4116yes—
335C24—22no—
336C24—22no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24C23×C412yes—
344C24C22×D412yes—
345C24C2×#4112yes—
346C24C2511yescharacteristic
347C2×Q8—28no—
348C2×Q8—216no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—48no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—216no—
358C22×C4—28no—
359C22×C4—28no—
360C22×C4—216no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—216no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—416no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—88no—
435C22×C4C4×D4116yes—
436C22×C4C23×C414yes—
437C22×C4C23×C414yes—
438C22×C4C23×C414yes—
439C22×C4C23×C414yes—
440C22×C4C23×C418yes—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C42−1+412yes—
455C22×C42−1+412yes—
456C4⋊C4—28no—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4#123116yes—
469C22⋊C4—216no—
470C22⋊C4—216no—
471C22⋊C4—216no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—84no—
476C42—216no—
477C42—216no—
478C42—216no—
479C42—48no—
480C42—48no—
481C42—44no—
482C42—44no—
483C42—88no—
484C42—88no—
485C42—88no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—24no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23C2×C4×D414yes—
509C23C24×C411yescharacteristic
510C23C23×D411yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4112yes—
513C23C22×#4112yes—
514C23C2×2−1+411yescharacteristic
515C23C2×2−1+411yescharacteristic
516C23#58512yes—
517C23#58512yes—
518C23#58511yescharacteristic
519C23#58511yescharacteristic
520Q8—416no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—28no—
529C2×C4—232no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—44no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—816no—
610C2×C4C2×C4×D414yes—
611C2×C4C2×C4×D414yes—
612C2×C4C4×#4118yes—
613C2×C4C2×#12318yes—
614C2×C4C2×#12314yes—
615C2×C4#52514yes—
616C2×C4#52918yes—
617C22—24no—
618C22—24no—
619C22—24no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—44no—
626C22—42no—
627C22—44no—
628C22C22×C4×D411yescharacteristic
629C22C2×C4×#4112yes—
630C22#342812yes—
631C22C4×2−1+411yescharacteristic
632C22C4×2−1+411yescharacteristic
633C22C22×#12311yescharacteristic
634C22C2×#52511yescharacteristic
635C22C2×#52912yes—
636C22#348911yescharacteristic
637C22#348911yescharacteristic
638C22#349312yes—
639C22#349512yes—
640C22#350011yescharacteristic
641C22#350011yescharacteristic
642C22#350712yes—
643C22#350911yescharacteristic
644C22#351012yes—
645C22#351011yescharacteristic
646C22#351212yes—
647C22#351512yes—
648C22#351612yes—
649C22#352112yes—
650C22#352412yes—
651C4—28no—
652C4—24no—
653C4—44no—
654C4—44no—
655C4—42no—
656C4—48no—
657C4—44no—
658C4—44no—
659C4—44no—
660C4—44no—
661C4—42no—
662C4—44no—
663C4—81no—
664C4—84no—
665C4—81no—
666C4C4×#12318yes—
667C2—44no—
668C2C2×C4×#12311yescharacteristic
669C2#3421111yescharacteristic
670C2#3426811yescharacteristic
671C2#3430012yes—
672C2#3431812yes—
673C2#3575911yescharacteristic
674C2#3581812yes—
675C2#3598311yescharacteristic
676C2#3603712yes—
677C2#3606711yescharacteristic
678C2#3608011yescharacteristic
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC25C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #14307C2semidirect
6. #14250C2semidirect
7. #14239C2semidirect
10. #14030C2semidirect
11. #14021C2semidirect
13. C4×#400C2semidirect
15. #13817C2semidirect
16. #13804C2semidirect
23. #2503C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#23622
C2×#22742
#143072
#142582
#142502
#142392
#141042
#141032
#140302
#140212
#139242
C4×#4002
#138442
#138172
#138042
#138012