Finite Group Databasebeta

← #7508728

Subgroups — #7508728

Order 512.

1894 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2#34809C211yesmaximal, characteristic
3#32277C214yesmaximal, complemented
4#32270C212yesmaximal, complemented
5#32268C212yesmaximal, complemented
6#32266C214yesmaximal, complemented
7#32261C212yesmaximal, complemented
8#32259C212yesmaximal, complemented
9#23642C212yesmaximal, complemented
10#23642C212yesmaximal, complemented
11#23552C212yesmaximal, complemented
12#21033C212yesmaximal, complemented
13#20898C211yesmaximal, characteristic
14#20846C212yesmaximal, complemented
15#20661C212yesmaximal, complemented
16C23×M16C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#407C2212yes—
24C2×#407C2211yescharacteristic
25#2563—21no—
26#2563—21no—
27#2563—21no—
28#2563—21no—
29#2563C2212yes—
30#2563C2212yes—
31#2563C2212yes—
32#2563C2212yes—
33#2487C2211yescharacteristic
34#2487C2211yescharacteristic
35#2486C2211yescharacteristic
36#2486C2211yescharacteristic
37#2485C2212yes—
38#2485C2212yes—
39#2484—21no—
40#2483—21no—
41#2482—21no—
42#2405—21no—
43#2395—21no—
44#2395—21no—
45C2×#393—21no—
46C2×#393—21no—
47C4×#123—21no—
48C4×#123—21no—
49#2296C2212yes—
50C2×#390—21no—
51C2×#382—21no—
52#2280C2211yescharacteristic
53#2280C2212yes—
54C2×#380C2212yes—
55C2×#380C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57#2077C2214yescomplemented
58#2077C2214yescomplemented
59#2077C2214yescomplemented
60#2077C2214yescomplemented
61#2076C2214yescomplemented
62#2076C2214yescomplemented
63#2075C2214yescomplemented
64#2075C2214yescomplemented
65#2058C2214yescomplemented
66#2058C2214yescomplemented
67#2057C2214yescomplemented
68#2057C2214yescomplemented
69#1921C2214yescomplemented
70#1920C2214yescomplemented
71#1919C2214yescomplemented
72#1918C2214yescomplemented
73#1870C2212yes—
74#1870C2212yes—
75#1869C2212yes—
76#1869C2212yes—
77#1854C2214yescomplemented
78#1854C2214yescomplemented
79#1853C2214yescomplemented
80#1852C2214yescomplemented
81#1809C2214yescomplemented
82#1809C2214yescomplemented
83#1808C2214yescomplemented
84#1807C2214yescomplemented
85#1736C2212yes—
86#1736C2212yes—
87#1736C2212yes—
88#1736C2212yes—
89#1736C2212yes—
90#1728C2212yes—
91C23×Q8—21no—
92C23×D4D411yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C22×M16—28no—
96C22×M16—42no—
97C22×M16C2312yes—
98C22×M16C2312yes—
99C22×M16C2312yes—
100C23×C8C2312yes—
101C2×C4⋊Q8—21no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—42no—
104C2×C4⋊Q8—42no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#123—21no—
108C2×#123—21no—
109C2×C4×D4—42no—
110C2×C4×D4—42no—
111C2×C4×D4D411yescharacteristic
112C2×C4×D4D411yescharacteristic
113C2×C4×D4D411yescharacteristic
114C2×C4×D4D411yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×#117—21no—
120C2×#117—21no—
121C2×#117—42no—
122C2×#117—42no—
123C2×#117C2312yes—
124C2×#117C2312yes—
125C2×#117C2312yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4C2312yes—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—21no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4—22no—
135C22×C22⋊C4D411yescharacteristic
136C22×C22⋊C4D411yescharacteristic
137C22×C42—21no—
138C22×C42—21no—
139C22×C42C2311yescharacteristic
140#428—28no—
141#428—44no—
142#428—44no—
143C2×#107C2314yes—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148C2×#106C2314yes—
149#407—28no—
150#407—42no—
151C2×C22⋊C8C2314yes—
152C2×C22⋊C8C2314yes—
153C2×C22⋊C8C2312yes—
154C2×C22⋊C8C2312yes—
155#400—21no—
156#400—21no—
157#399—21no—
158#399—21no—
159#398—21no—
160#398—21no—
161#397—21no—
162#397—21no—
163#397—22no—
164#397—22no—
165#394—22no—
166#394—22no—
167#393—22no—
168#393—22no—
169#393—22no—
170#393—22no—
171#393—22no—
172#393—22no—
173#393—22no—
174#393—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#389C2312yes—
180#389C2312yes—
181#389C2312yes—
182#389C2312yes—
183#388—22no—
184#388—22no—
185#387—22no—
186#387—22no—
187#386—21no—
188#386—21no—
189#385—22no—
190#385—22no—
191#385—21no—
192#385—21no—
193#382—22no—
194#382—22no—
195#382—22no—
196#382—22no—
197#381—21no—
198#381—22no—
199#381—21no—
200#381—21no—
201#380—22no—
202#380—22no—
203#380—22no—
204#380—22no—
205#380C2314yes—
206#380C2314yes—
207#380C2314yes—
208#380C2314yes—
209#380C2312yes—
210#380C2312yes—
211#380C2312yes—
212#380C2312yes—
213#380C2312yes—
214#380C2311yescharacteristic
215#380C2311yescharacteristic
216#379—22no—
217#379—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4C2312yes—
223C4×C4⋊C4C2312yes—
224C4×C22⋊C4—21no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—22no—
227C4×C22⋊C4—21no—
228C2×#95—21no—
229C2×#95—22no—
230C2×#95—22no—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#340C2314yes—
241#336C2314yes—
242#336C2314yes—
243C25C2×D411yescharacteristic
244C25C2×D411yescharacteristic
245C22×Q8—22no—
246C22×Q8—22no—
247C22×Q8—44no—
248C22×Q8—84no—
249C22×D4—24no—
250C22×D4—42no—
251C22×D4—42no—
252C22×D4C2×D411yescharacteristic
253C22×D4C2×D411yescharacteristic
254C22×D4C2×D411yescharacteristic
255C22×D4C2×D411yescharacteristic
256C23×C4—24no—
257C23×C4—24no—
258C23×C4—22no—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—24no—
263C23×C4—24no—
264C23×C4—22no—
265C23×C4—22no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4C2×D411yescharacteristic
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C23×C4C2411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×M16—24no—
282C2×M16—24no—
283C2×M16—24no—
284C2×M16—24no—
285C2×M16—24no—
286C2×M16—28no—
287C2×M16—42no—
288C2×M16—416no—
289C2×M16#4118yes—
290C22×C8—28no—
291C22×C8—42no—
292C22×C8C2414yes—
293C22×C8C2414yes—
294C4⋊Q8—42no—
295C4⋊Q8—42no—
296C4⋊Q8—42no—
297C4⋊Q8—42no—
298C4⋊Q8—44no—
299C4⋊Q8—44no—
300#123—44no—
301#123—44no—
302C4×D4—42no—
303C4×D4—42no—
304C4×D4—42no—
305C4×D4—42no—
306C4×D4—41no—
307C4×D4—41no—
308C4×D4—41no—
309C4×D4—41no—
310C4×D4—41no—
311C4×D4—41no—
312C4×D4—41no—
313C4×D4—41no—
314#117—24no—
315#117—24no—
316#117—24no—
317#117—24no—
318#117—42no—
319#117—42no—
320#117—44no—
321#117—44no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—24no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—41no—
337C2×C4⋊C4—41no—
338C2×C4⋊C4—41no—
339C2×C4⋊C4—41no—
340C2×C4⋊C4—42no—
341C2×C4⋊C4—44no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2414yes—
349C2×C4⋊C4C2414yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—24no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—42no—
367C2×C22⋊C4—42no—
368C2×C22⋊C4—44no—
369C2×C22⋊C4—84no—
370C2×C22⋊C4—84no—
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C22⋊C4C2×D411yescharacteristic
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—44no—
387C2×C42—44no—
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392#107—28no—
393#107—44no—
394#107—44no—
395#106—28no—
396#106—44no—
397#106—44no—
398C22⋊C8—28no—
399C22⋊C8—28no—
400C22⋊C8—42no—
401C22⋊C8—42no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—21no—
414#95—21no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419C24—22no—
420C24—22no—
421C24—22no—
422C24—22no—
423C24—24no—
424C24—24no—
425C24—42no—
426C24—42no—
427C24—41no—
428C24—41no—
429C24C22×D411yescharacteristic
430C24C22×D411yescharacteristic
431C24C22×D411yescharacteristic
432C24C22×D411yescharacteristic
433C24C22×D411yescharacteristic
434C2×Q8—44no—
435C2×Q8—44no—
436C2×Q8—84no—
437C2×Q8—84no—
438C2×Q8—816no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—41no—
445C2×D4—41no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—41no—
450C2×D4—416no—
451C2×D4#13714yes—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—24no—
490C22×C4—42no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—44no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—42no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—42no—
507C22×C4—82no—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C2×#4114yes—
518C22×C4H(5,𝔽2)12yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C42−1+414yes—
521C22×C4C2511yescharacteristic
522M16—216no—
523M16—48no—
524M16—48no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—48no—
531C2×C8C2×#4118yes—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—22no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—24no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—41no—
545C4⋊C4—41no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—44no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C22⋊C4—41no—
559C22⋊C4—41no—
560C22⋊C4—41no—
561C22⋊C4—41no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—41no—
566C22⋊C4—44no—
567C22⋊C4—42no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—42no—
572C22⋊C4—44no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—84no—
577C22⋊C4—84no—
578C22⋊C4—84no—
579C22⋊C4—84no—
580C42—22no—
581C42—22no—
582C42—21no—
583C42—21no—
584C42—21no—
585C42—21no—
586C42—44no—
587C42—44no—
588C42—44no—
589C42—44no—
590C42—44no—
591C23—22no—
592C23—21no—
593C23—21no—
594C23—22no—
595C23—22no—
596C23—21no—
597C23—21no—
598C23—22no—
599C23—22no—
600C23—21no—
601C23—21no—
602C23—22no—
603C23—22no—
604C23—21no—
605C23—21no—
606C23—22no—
607C23—22no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—44no—
613C23—44no—
614C23—48no—
615C23—48no—
616C23—44no—
617C23—44no—
618C23—84no—
619C23—84no—
620C23—82no—
621C23—82no—
622C23#53412yes—
623C23#54714yes—
624C23C2×#13712yes—
625C23C2×#13712yes—
626C23C23×D411yescharacteristic
627Q8—88no—
628Q8—88no—
629D4—44no—
630D4—44no—
631D4—44no—
632D4—44no—
633C2×C4—28no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—88no—
688C2×C4—84no—
689C2×C4—88no—
690C2×C4—84no—
691C2×C4#53412yes—
692C2×C4#54218yes—
693C2×C4#54714yes—
694C2×C4#55114yes—
695C2×C4#56314yes—
696C2×C4C2×#13712yes—
697C2×C4#57512yes—
698C2×C4C22×#4112yes—
699C2×C4C2×H(5,𝔽2)11yescharacteristic
700C2×C4C2×H(5,𝔽2)11yescharacteristic
701C2×C4C2×2−1+412yes—
702C8—28no—
703C8—44no—
704C22—24no—
705C22—24no—
706C22—22no—
707C22—22no—
708C22—24no—
709C22—24no—
710C22—22no—
711C22—22no—
712C22—42no—
713C22—42no—
714C22—41no—
715C22—41no—
716C22—41no—
717C22—41no—
718C22—42no—
719C22—42no—
720C22—42no—
721C22—42no—
722C22—41no—
723C22—41no—
724C22—41no—
725C22—41no—
726C22—41no—
727C22—41no—
728C22—88no—
729C22—88no—
730C22#271714yes—
731C22C2×#53411yescharacteristic
732C22C2×#54712yes—
733C22C22×#13711yescharacteristic
734C22C2×#57511yescharacteristic
735C22#358512yes—
736C4—22no—
737C4—22no—
738C4—41no—
739C4—41no—
740C4—41no—
741C4—41no—
742C4—41no—
743C4—41no—
744C4—82no—
745C4—82no—
746C4—84no—
747C4—82no—
748C4—82no—
749C4—84no—
750C4#271714yes—
751C4C2×#54212yes—
752C4C2×#55111yescharacteristic
753C4C2×#56311yescharacteristic
754C2—24no—
755C2—42no—
756C2—42no—
757C2—81no—
758C2—81no—
759C2C2×#271711yescharacteristic
760C2#6075111yescharacteristic
761C2#6075311yescharacteristic
762C2#6125412yes—
763C2#6125612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32277C2semidirect
4. #32270C2semidirect
5. #32268C2semidirect
6. #32266C2semidirect
7. #32261C2semidirect
8. #32259C2semidirect
9. #23642C2semidirect
10. #23642C2semidirect
11. #23552C2semidirect
12. #21033C2semidirect
14. #20846C2semidirect
15. #20661C2semidirect
57. #2077C22semidirect
58. #2077C22semidirect
59. #2077C22semidirect
60. #2077C22semidirect
61. #2076C22semidirect
62. #2076C22semidirect
63. #2075C22semidirect
64. #2075C22semidirect
65. #2058C22semidirect
66. #2058C22semidirect
67. #2057C22semidirect
68. #2057C22semidirect
69. #1921C22semidirect
70. #1920C22semidirect
71. #1919C22semidirect
72. #1918C22semidirect
77. #1854C22semidirect
78. #1854C22semidirect
79. #1853C22semidirect
80. #1852C22semidirect
81. #1809C22semidirect
82. #1809C22semidirect
83. #1808C22semidirect
84. #1807C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
#348092
#322772
#322702
#322682
#322662
#322612
#322592
#236422
#235522
#210332
#208982
#208462
#206612