Finite Group Databasebeta

← #6457103

Subgroups — #6457103

Order 512.

1862 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2#34739C211yesmaximal, characteristic
3#27722C214yesmaximal, complemented
4#27716C214yesmaximal, complemented
5#27708C214yesmaximal, complemented
6#27698C214yesmaximal, complemented
7#23746C211yesmaximal, characteristic, complemented
8#23745C211yesmaximal, characteristic, complemented
9#23629C211yesmaximal, characteristic, complemented
10#23455C211yesmaximal, characteristic, complemented
11#21322C211yesmaximal, characteristic, complemented
12#21094C211yesmaximal, characteristic, complemented
13#21094C211yesmaximal, characteristic, complemented
14#20972C211yesmaximal, characteristic
15#20911C211yesmaximal, characteristic
16#20765C211yesmaximal, characteristic, complemented
17#20701C211yesmaximal, characteristic
18C22⋊C4×M16C211yesmaximal, characteristic
19#20347C211yesmaximal, characteristic
20C23×M16C2211yescharacteristic
21C2×#533C2211yescharacteristic
22C22×C22⋊Q8C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C22×#117C2211yescharacteristic
25C2×#419C2211yescharacteristic
26C2×#419C2211yescharacteristic
27C2×#419C2211yescharacteristic
28C2×#419C2211yescharacteristic
29C2×#407C2211yescharacteristic
30C2×C4×M16C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2499—22no—
33#2499—22no—
34#2499C2212yes—
35#2499C2212yes—
36#2499C2212yes—
37#2499C2212yes—
38#2442C2212yes—
39#2441C2212yes—
40#2440C2212yes—
41#2439C2212yes—
42#2438—22no—
43#2437—22no—
44#2396—22no—
45C2×#397—22no—
46C4×C22⋊Q8—22no—
47#2296C2211yescharacteristic
48#2296C2211yescharacteristic
49C2×#391C2211yescharacteristic
50C2×#391C2211yescharacteristic
51C2×#387—22no—
52#2285C2211yescharacteristic
53C2×#384C2211yescharacteristic
54C2×#382C2211yescharacteristic
55#2280C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57#2101C2214yescomplemented
58#2101C2214yescomplemented
59#2100C2214yescomplemented
60#2099C2214yescomplemented
61#2074C2214yescomplemented
62#2072C2214yescomplemented
63#2026C2214yescomplemented
64#2024C2214yescomplemented
65#1992C2214yescomplemented
66#1990C2214yescomplemented
67#1935C2214yescomplemented
68#1935C2214yescomplemented
69#1933C2214yescomplemented
70#1933C2214yescomplemented
71#1894C2214yes—
72#1892C2214yes—
73#1873C2214yes—
74#1872C2214yes—
75#1836C2214yescomplemented
76#1835C2214yescomplemented
77#1825C2214yes—
78#1825C2214yes—
79#1753C2212yes—
80#1753C2212yes—
81#1752C2212yes—
82C8×C22⋊C4C2212yes—
83#1736C2211yescharacteristic
84#1736C2211yescharacteristic
85#1736C2211yescharacteristic
86#1736C2211yescharacteristic
87#1736C2214yes—
88#1736C2211yescharacteristic
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1736C2211yescharacteristic
96#1735C2214yes—
97#1728C2211yescharacteristic
98#1728C2211yescharacteristic
99C23×Q8C2311yescharacteristic
100C24×C4C2311yescharacteristic
101C22×M16—24no—
102C22×M16—24no—
103C22×M16—42no—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C22×M16C2312yes—
108C22×M16C2311yescharacteristic
109C23×C8C2312yes—
110#533—24no—
111#533—42no—
112C2×C4⋊Q8C2312yes—
113C2×#125C2312yes—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8—22no—
117C2×C22⋊Q8—22no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8—24no—
120C2×C22⋊Q8C2312yes—
121C2×C22⋊Q8C2312yes—
122C2×C4×Q8—22no—
123C2×#117—21no—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117—21no—
127C2×#117—24no—
128C2×#117—24no—
129C2×#117—24no—
130C2×#117—42no—
131C2×#117C2312yes—
132C2×#117C2312yes—
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2311yescharacteristic
136C2×#117C2312yes—
137C2×#117C2311yescharacteristic
138C22×C4⋊C4—22no—
139C22×C4⋊C4—21no—
140C22×C4⋊C4C2311yescharacteristic
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2311yescharacteristic
143C22×C4⋊C4C2312yes—
144C22×C22⋊C4—21no—
145C22×C22⋊C4—21no—
146C22×C22⋊C4—21no—
147C22×C22⋊C4D412yes—
148C22×C42—21no—
149C22×C42—21no—
150C22×C42C2311yescharacteristic
151C22×C42C2311yescharacteristic
152#419—24no—
153#419—24no—
154#419—42no—
155#419—42no—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160#407—24no—
161C2×C22⋊C8C2312yes—
162C2×C22⋊C8C2312yes—
163C4×M16—24no—
164C4×M16—42no—
165C2×#97C2312yes—
166C2×#97C2312yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2312yes—
169#400—22no—
170#399—22no—
171#398—22no—
172#398—22no—
173#397—24no—
174#397—24no—
175#397—24no—
176#397—24no—
177#396—22no—
178#396—22no—
179#395—22no—
180#393—22no—
181#391—22no—
182#391—22no—
183#391—22no—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#389—22no—
189#389C2312yes—
190#389C2312yes—
191#389C2312yes—
192#389C2312yes—
193#387—22no—
194#387—22no—
195#387—24no—
196#387—24no—
197#387—24no—
198#387—24no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#384—22no—
204#384C2312yes—
205#384C2312yes—
206#384C2312yes—
207#382—22no—
208#382C2312yes—
209#382C2312yes—
210#382C2312yes—
211#381—21no—
212#380—22no—
213#380—22no—
214#380C2312yes—
215#380C2312yes—
216#379—21no—
217C4×C4⋊C4—22no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C22⋊C4—21no—
221C4×C22⋊C4—21no—
222C4×C22⋊C4D412yes—
223C4×C22⋊C4D412yes—
224C4×C22⋊C4C2311yescharacteristic
225C4×C22⋊C4C2311yescharacteristic
226C4×C22⋊C4C2311yescharacteristic
227C4×C22⋊C4C2311yescharacteristic
228C2×#95—22no—
229C2×#95—22no—
230C2×#95—22no—
231C2×#95D412yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#340C2314yes—
244#336C2312yes—
245#336C2312yes—
246#336C2312yes—
247#336C2312yes—
248C25C2×D411yescharacteristic
249C22×Q8—21no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8C2412yes—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C23×C4—21no—
266C23×C4—22no—
267C23×C4C2×D412yes—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D412yes—
272C23×C4#4114yes—
273C23×C4#4114yes—
274C23×C4C2411yescharacteristic
275C23×C4C2411yescharacteristic
276C23×C4C2411yescharacteristic
277C2×M16—22no—
278C2×M16—22no—
279C2×M16—24no—
280C2×M16—22no—
281C2×M16—22no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—24no—
290C2×M16—42no—
291C2×M16—416no—
292C2×M16#4114yes—
293C22×C8—24no—
294C22×C8—24no—
295C22×C8—42no—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C4⋊Q8—24no—
301C4⋊Q8—42no—
302#125—24no—
303#125—42no—
304C22⋊Q8—24no—
305C22⋊Q8—24no—
306C22⋊Q8—24no—
307C22⋊Q8—24no—
308C22⋊Q8—48no—
309C22⋊Q8—48no—
310C22⋊Q8—44no—
311C22⋊Q8—44no—
312C4×Q8—48no—
313#117—22no—
314#117—24no—
315#117—24no—
316#117—24no—
317#117—24no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—24no—
325#117—44no—
326#117—44no—
327#117—42no—
328#117—44no—
329#117—44no—
330#117#4114yes—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—42no—
356C2×C4⋊C4—48no—
357C2×C4⋊C4—44no—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—44no—
377C2×C22⋊C4—44no—
378C2×C22⋊C4—44no—
379C2×C22⋊C4—44no—
380C2×C22⋊C4C2×D412yes—
381C2×C22⋊C4C2×D412yes—
382C2×C22⋊C4C2×D412yes—
383C2×C22⋊C4C2×D412yes—
384C2×C42—21no—
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—21no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—21no—
394C2×C42—21no—
395C2×C42—44no—
396C2×C42—44no—
397C2×C42C2×D412yes—
398C2×C42C2×D412yes—
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#103—28no—
404#103—28no—
405#103—44no—
406#103—44no—
407C22⋊C8—24no—
408C22⋊C8—24no—
409#97—24no—
410#97—42no—
411C4×C8—24no—
412C4×C8—42no—
413#95—24no—
414#95—24no—
415#95—24no—
416#95—24no—
417#95—24no—
418#95—24no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95C2×D412yes—
428#95C2×D412yes—
429#95C2×D412yes—
430#95C2×D412yes—
431C24—22no—
432C24—22no—
433C24—42no—
434C24—42no—
435C24#12414yes—
436C24C22×D411yescharacteristic
437C24C22×D411yescharacteristic
438C24C22×D411yescharacteristic
439C2×Q8—22no—
440C2×Q8—22no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—44no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—44no—
450C2×Q8—44no—
451C2×Q8—44no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—82no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—82no—
499C22×C4#12414yes—
500C22×C4#12414yes—
501C22×C4#12414yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4C2×#4112yes—
516C22×C4C2×#4112yes—
517C22×C4H(5,𝔽2)12yes—
518C22×C42−1+412yes—
519C22×C4C2511yescharacteristic
520M16—28no—
521M16—48no—
522M16—44no—
523M16—44no—
524M16—44no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8—22no—
530C2×C8—22no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—22no—
534C2×C8—22no—
535C2×C8—22no—
536C2×C8—48no—
537C2×C8C2×#4114yes—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—22no—
550C4⋊C4—22no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—48no—
556C4⋊C4—44no—
557C4⋊C4—44no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—44no—
562C4⋊C4—44no—
563C4⋊C4—48no—
564C4⋊C4—44no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4C2×#4114yes—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—24no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—44no—
577C22⋊C4—44no—
578C22⋊C4—44no—
579C22⋊C4—44no—
580C22⋊C4—42no—
581C22⋊C4—42no—
582C22⋊C4—42no—
583C22⋊C4—82no—
584C42—24no—
585C42—24no—
586C42—24no—
587C42—21no—
588C42—21no—
589C42—21no—
590C42—21no—
591C42—21no—
592C42—21no—
593C42—21no—
594C42—21no—
595C42—44no—
596C42—44no—
597C42—44no—
598C42—42no—
599C42—42no—
600C42—42no—
601C23—22no—
602C23—22no—
603C23—22no—
604C23—22no—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—22no—
610C23—22no—
611C23—28no—
612C23—44no—
613C23—44no—
614C23—84no—
615C23—84no—
616C23C2×#12412yes—
617C23C2×#12412yes—
618C23C2×#12412yes—
619C23#54612yes—
620C23#54712yes—
621C23C23×D411yescharacteristic
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—42no—
626Q8—88no—
627Q8—84no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—28no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—22no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—48no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—84no—
677C2×C4—88no—
678C2×C4—84no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4C2×#12412yes—
683C2×C4C2×#12412yes—
684C2×C4C2×#12412yes—
685C2×C4#52914yes—
686C2×C4#53914yes—
687C2×C4#54612yes—
688C2×C4#54712yes—
689C2×C4#55114yes—
690C2×C4#55214yes—
691C2×C4#57514yes—
692C2×C4C22×#4111yescharacteristic
693C2×C4C22×#4111yescharacteristic
694C2×C4C22×#4111yescharacteristic
695C2×C4C22×#4111yescharacteristic
696C2×C4C2×H(5,𝔽2)11yescharacteristic
697C2×C4C2×2−1+411yescharacteristic
698C8—24no—
699C8—42no—
700C8—42no—
701C8—42no—
702C22—24no—
703C22—22no—
704C22—22no—
705C22—24no—
706C22—22no—
707C22—22no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—42no—
713C22—42no—
714C22—42no—
715C22—42no—
716C22—88no—
717C22#248814yes—
718C22C22×#12411yescharacteristic
719C22C2×#54611yescharacteristic
720C22C2×#54711yescharacteristic
721C22C2×#57512yes—
722C22#358612yes—
723C4—22no—
724C4—22no—
725C4—24no—
726C4—42no—
727C4—42no—
728C4—42no—
729C4—44no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4#248814yes—
738C4C2×#52911yescharacteristic
739C4C2×#53911yescharacteristic
740C4C2×#55111yescharacteristic
741C4C2×#55211yescharacteristic
742C2—24no—
743C2—42no—
744C2—42no—
745C2C2×#248811yescharacteristic
746C2#6061811yescharacteristic
747C2#6062011yescharacteristic
748C2#6116312yes—
749C2#6125712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27722C2semidirect
4. #27716C2semidirect
5. #27708C2semidirect
6. #27698C2semidirect
7. #23746C2semidirect
8. #23745C2semidirect
9. #23629C2semidirect
10. #23455C2semidirect
11. #21322C2semidirect
12. #21094C2semidirect
13. #21094C2semidirect
16. #20765C2semidirect
57. #2101C22semidirect
58. #2101C22semidirect
59. #2100C22semidirect
60. #2099C22semidirect
61. #2074C22semidirect
62. #2072C22semidirect
63. #2026C22semidirect
64. #2024C22semidirect
65. #1992C22semidirect
66. #1990C22semidirect
67. #1935C22semidirect
68. #1935C22semidirect
69. #1933C22semidirect
70. #1933C22semidirect
75. #1836C22semidirect
76. #1835C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
#347392
#277222
#277162
#277082
#276982
#237462
#237452
#236292
#234552
#213222
#210942
#209722
#209112
#207652
#207012
C22⋊C4×M162
#203472