Finite Group Databasebeta

← #6453085

Subgroups — #6453085

Order 512.

2213 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×#3092C211yesmaximal, characteristic
2C22×#428C211yesmaximal, characteristic, complemented
3#34810C211yesmaximal, characteristic
4C2×#2345C211yesmaximal, characteristic, complemented
5#24591C214yesmaximal, complemented
6#24590C214yesmaximal, complemented
7#24583C212yesmaximal, complemented
8#24582C212yesmaximal, complemented
9#24579C212yesmaximal, complemented
10#24578C212yesmaximal, complemented
11#23657C212yesmaximal, complemented
12#23617C212yesmaximal, complemented
13#21170C212yesmaximal, complemented
14#21059C212yesmaximal, complemented
15#20971C211yesmaximal, characteristic
16#20837C211yesmaximal, characteristic, complemented
17#20702C211yesmaximal, characteristic, complemented
18C23×M16C2211yescharacteristic
19C2×#525C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#3092—28no—
23C2×#484C2212yes—
24C2×#481C2212yes—
25C2×#483C2212yes—
26C2×#480C2212yes—
27C2×#428—28no—
28C2×#428C2214yes—
29C2×#428C2212yes—
30C2×#428C2212yes—
31C2×#428C2211yescharacteristic
32C2×#428C2211yescharacteristic
33C22×#107C2212yes—
34C22×#106C2212yes—
35C2×#423C2212yes—
36C2×#417C2211yescharacteristic
37C2×#417C2212yes—
38C2×#417C2211yescharacteristic
39C2×#407C2211yescharacteristic
40#2504C2211yescharacteristic
41#2503C2211yescharacteristic
42#2502C2211yescharacteristic
43#2501C2211yescharacteristic
44#2500C2212yes—
45#2499C2212yes—
46#2498—22no—
47#2412—21no—
48#2412—21no—
49#2395—22no—
50#2345—22no—
51#2345—21no—
52#2345—21no—
53#2345C2212yes—
54#2345C2212yes—
55#2345C2214yes—
56C4×#123—21no—
57C4×#123—21no—
58#2297C2212yes—
59C2×#389C2212yes—
60C2×#387—22no—
61#2285C2212yes—
62C2×#380—21no—
63C2×#380C2211yescharacteristic
64C2×C4×C4⋊C4C2212yes—
65C2×C4×C22⋊C4—21no—
66C2×C4×C22⋊C4C2211yescharacteristic
67#2085C2214yescomplemented
68#2085C2214yescomplemented
69#2084C2214yescomplemented
70#2083C2214yescomplemented
71#2068C2214yescomplemented
72#2067C2214yescomplemented
73#2066C2214yescomplemented
74#2065C2214yescomplemented
75#1949C2214yescomplemented
76#1949C2214yescomplemented
77#1948C2214yescomplemented
78#1947C2214yescomplemented
79#1928C2214yescomplemented
80#1927C2214yescomplemented
81#1926C2214yescomplemented
82#1925C2214yescomplemented
83#1895C2212yes—
84#1894C2212yes—
85#1893C2212yes—
86#1892C2212yes—
87#1851C2214yes—
88#1850C2212yes—
89#1850C2212yes—
90#1825C2214yes—
91#1824C2212yes—
92#1823C2212yes—
93#1736C2212yes—
94#1736C2212yes—
95#1736C2212yes—
96#1728C2212yes—
97C22×#41C2311yescharacteristic
98C24×C4—21no—
99C24×C4C2311yescharacteristic
100C22×M16—24no—
101C22×M16—28no—
102C22×M16C2312yes—
103C22×M16C2311yescharacteristic
104C22×M16C2311yescharacteristic
105C22×M16C2312yes—
106C23×C8C2312yes—
107#525—24no—
108#525—24no—
109C2×#123—21no—
110C2×#123—21no—
111C2×#123—21no—
112C2×C22⋊Q8C2311yescharacteristic
113C2×C22⋊Q8C2311yescharacteristic
114C2×#121C2311yescharacteristic
115C2×#121C2311yescharacteristic
116C2×#120—21no—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×#117—21no—
120C2×#117—21no—
121C2×#117—24no—
122C2×#117—24no—
123C2×#117C2312yes—
124C2×#117C2312yes—
125C2×#117C2311yescharacteristic
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—24no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4D418yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C22⋊C4—21no—
140C22×C22⋊C4—21no—
141C22×C22⋊C4D411yescharacteristic
142C22×C22⋊C4D411yescharacteristic
143C22×C42—21no—
144C22×C42—22no—
145C22×C42C2312yes—
146#484—28no—
147#483—28no—
148#481—28no—
149#480—28no—
150#428—24no—
151#428—24no—
152#428—28no—
153#428—28no—
154#428—216no—
155C2×#107—28no—
156C2×#107C2312yes—
157C2×#107C2312yes—
158C2×#107C2314yes—
159C2×#107C2314yes—
160C2×#106—28no—
161C2×#106C2312yes—
162C2×#106C2312yes—
163C2×#106C2314yes—
164C2×#106C2314yes—
165#423—28no—
166C2×C4⋊C8C2314yes—
167C2×C4⋊C8C2314yes—
168#417—28no—
169#417—24no—
170#417—24no—
171C2×#103C2312yes—
172C2×#103C2312yes—
173C2×#103C2314yes—
174C2×#102C2312yes—
175C2×#102C2312yes—
176C2×#102C2314yes—
177#407—24no—
178C2×C22⋊C8C2312yes—
179C2×C22⋊C8C2312yes—
180#400—22no—
181#399—22no—
182#398—22no—
183#397—22no—
184#397—22no—
185#397—22no—
186#394—22no—
187#394—22no—
188#391C2312yes—
189#391C2312yes—
190#390C2312yes—
191#390C2312yes—
192#389—22no—
193#389—22no—
194#389C2314yes—
195#389C2314yes—
196#388—22no—
197#388—22no—
198#387—24no—
199#387—24no—
200#387—24no—
201#387—24no—
202#387—22no—
203#387—22no—
204#386—22no—
205#386—22no—
206#386—21no—
207#386—21no—
208#385—21no—
209#385—21no—
210#384C2312yes—
211#384C2312yes—
212#382C2312yes—
213#382C2312yes—
214#381—21no—
215#381—21no—
216#380—21no—
217#380—21no—
218#380—22no—
219#380—22no—
220#380—22no—
221#380—22no—
222#380—24no—
223#380C2312yes—
224#380C2312yes—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4C2314yes—
230C4×C4⋊C4C2314yes—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—24no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4D412yes—
237C4×C22⋊C4D411yescharacteristic
238C4×C22⋊C4D411yescharacteristic
239C4×C22⋊C4C2311yescharacteristic
240C4×C22⋊C4C2311yescharacteristic
241C4×C22⋊C4C2312yes—
242C2×#95—22no—
243C2×#95—22no—
244C2×#95—21no—
245C2×#95D412yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251#340C2314yes—
252#336C2314yes—
253#336C2314yes—
254C25C2×D411yescharacteristic
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—24no—
258C2×#41—24no—
259C22×Q8C2411yescharacteristic
260C22×D4—21no—
261C22×D4—21no—
262C22×D4—21no—
263C22×D4C2411yescharacteristic
264C23×C4—24no—
265C23×C4—24no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—28no—
269C23×C4—22no—
270C23×C4—22no—
271C23×C4—22no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—84no—
276C23×C4C2×D412yes—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D414yes—
281C23×C4C2×D412yes—
282C23×C4#4114yes—
283C23×C4C2411yescharacteristic
284C23×C4C2411yescharacteristic
285C23×C4C2411yescharacteristic
286C2×M16—232no—
287C2×M16—24no—
288C2×M16—24no—
289C2×M16—24no—
290C2×M16—22no—
291C2×M16—22no—
292C2×M16—24no—
293C2×M16—24no—
294C2×M16—28no—
295C2×M16#4118yes—
296C22×C8—24no—
297C22×C8—28no—
298C22×C8C2412yes—
299C22×C8C2412yes—
300C22×C8C2414yes—
301#123—44no—
302#123—44no—
303#123—44no—
304C22⋊Q8—22no—
305C22⋊Q8—22no—
306C22⋊Q8—22no—
307C22⋊Q8—22no—
308#121—22no—
309#121—22no—
310#121—22no—
311#121—22no—
312#120—44no—
313C4×D4—44no—
314C4×D4—44no—
315#117—24no—
316#117—24no—
317#117—22no—
318#117—24no—
319#117—24no—
320#117—22no—
321#117—44no—
322#117—44no—
323C2×C4⋊C4—28no—
324C2×C4⋊C4—28no—
325C2×C4⋊C4—28no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—24no—
333C2×C4⋊C4—24no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—24no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—24no—
348C2×C4⋊C4—24no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—84no—
352C2×C4⋊C4C2×D414yes—
353C2×C4⋊C4C2×D414yes—
354C2×C4⋊C4C2×D418yes—
355C2×C4⋊C4C2×D418yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2414yes—
359C2×C4⋊C4C2411yescharacteristic
360C2×C4⋊C4C2411yescharacteristic
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C22⋊C4—24no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—21no—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—21no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—44no—
379C2×C22⋊C4—44no—
380C2×C22⋊C4C2×D411yescharacteristic
381C2×C22⋊C4C2×D411yescharacteristic
382C2×C22⋊C4C2×D412yes—
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D412yes—
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C42—24no—
387C2×C42—24no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42—21no—
391C2×C42—24no—
392C2×C42—24no—
393C2×C42—44no—
394C2×C42—84no—
395C2×C42C2×D412yes—
396C2×C42C2×D412yes—
397C2×C42C2412yes—
398C2×C42C2412yes—
399#107—28no—
400#107—28no—
401#107—24no—
402#107—24no—
403#106—28no—
404#106—28no—
405#106—24no—
406#106—24no—
407C4⋊C8—28no—
408C4⋊C8—28no—
409#103—28no—
410#103—24no—
411#103—24no—
412#102—28no—
413#102—24no—
414#102—24no—
415C22⋊C8—24no—
416C22⋊C8—24no—
417#95—24no—
418#95—24no—
419#95—24no—
420#95—24no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95C2×D412yes—
432#95C2×D412yes—
433#95C2×D412yes—
434#95C2×D412yes—
435C24—24no—
436C24—22no—
437C24—22no—
438C24—21no—
439C24—42no—
440C24#12114yes—
441C24C22×D411yescharacteristic
442C24C22×D411yescharacteristic
443C24C22×D411yescharacteristic
444#41—44no—
445#41—44no—
446#41—48no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—21no—
450C2×Q8—22no—
451C2×D4—21no—
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—22no—
455C2×D4—42no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—44no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D4—82no—
462C22×C4—24no—
463C22×C4—28no—
464C22×C4—28no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—216no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—28no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—21no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—44no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—44no—
501C22×C4—82no—
502C22×C4—84no—
503C22×C4—84no—
504C22×C4—816no—
505C22×C4#12114yes—
506C22×C4#12114yes—
507C22×C4#12114yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D412yes—
514C22×C4C22×D412yes—
515C22×C4C22×D412yes—
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D412yes—
518C22×C4C2×#4112yes—
519C22×C4C2×#4112yes—
520C22×C4C2×#4112yes—
521C22×C4C2×#4114yes—
522C22×C42−1+414yes—
523C22×C4C2511yescharacteristic
524M16—216no—
525M16—216no—
526M16—44no—
527M16—44no—
528C2×C8—216no—
529C2×C8—24no—
530C2×C8—24no—
531C2×C8—22no—
532C2×C8—22no—
533C2×C8—24no—
534C2×C8—24no—
535C2×C8C2×#4118yes—
536C4⋊C4—28no—
537C4⋊C4—28no—
538C4⋊C4—24no—
539C4⋊C4—24no—
540C4⋊C4—22no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—21no—
544C4⋊C4—22no—
545C4⋊C4—21no—
546C4⋊C4—24no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—24no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—44no—
558C4⋊C4—44no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—84no—
562C4⋊C4—84no—
563C4⋊C4C22×D414yes—
564C4⋊C4C22×D414yes—
565C22⋊C4—22no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—24no—
569C22⋊C4—42no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—42no—
575C22⋊C4—42no—
576C22⋊C4—42no—
577C22⋊C4—44no—
578C22⋊C4—42no—
579C22⋊C4—42no—
580C22⋊C4—44no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4#13714yes—
584C42—24no—
585C42—24no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—44no—
591C42—44no—
592C42—42no—
593C42—84no—
594C42—84no—
595C23—22no—
596C23—22no—
597C23—22no—
598C23—21no—
599C23—21no—
600C23—22no—
601C23—28no—
602C23—22no—
603C23—28no—
604C23—21no—
605C23—21no—
606C23—44no—
607C23—48no—
608C23—48no—
609C23—42no—
610C23—41no—
611C23—42no—
612C23—41no—
613C23—41no—
614C23C2×#12112yes—
615C23C2×#12112yes—
616C23C2×#12112yes—
617C23#54714yes—
618C23C2×#13712yes—
619C23#57512yes—
620C23C23×D411yescharacteristic
621Q8—41no—
622Q8—41no—
623Q8—42no—
624D4—41no—
625D4—41no—
626D4—42no—
627D4—84no—
628D4—84no—
629D4—84no—
630C2×C4—22no—
631C2×C4—28no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—22no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—48no—
655C2×C4—48no—
656C2×C4—48no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—41no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—41no—
675C2×C4—84no—
676C2×C4—88no—
677C2×C4—88no—
678C2×C4—88no—
679C2×C4—84no—
680C2×C4—84no—
681C2×C4—84no—
682C2×C4C2×#12112yes—
683C2×C4C2×#12112yes—
684C2×C4C2×#12112yes—
685C2×C4#52514yes—
686C2×C4#53614yes—
687C2×C4#54714yes—
688C2×C4#55218yes—
689C2×C4C2×#13714yes—
690C2×C4C23×D411yescharacteristic
691C2×C4C22×#4112yes—
692C2×C4C22×#4111yescharacteristic
693C2×C4C2×2−1+412yes—
694C8—28no—
695C8—42no—
696C8—42no—
697C22—22no—
698C22—22no—
699C22—24no—
700C22—24no—
701C22—24no—
702C22—24no—
703C22—22no—
704C22—22no—
705C22—22no—
706C22—22no—
707C22—22no—
708C22—416no—
709C22—42no—
710C22—42no—
711C22—81no—
712C22—82no—
713C22—82no—
714C22—81no—
715C22—81no—
716C22#241314yes—
717C22#303714yes—
718C22C22×#12111yescharacteristic
719C22C2×#54712yes—
720C22C22×#13711yescharacteristic
721C22C2×#57511yescharacteristic
722C22#358512yes—
723C4—24no—
724C4—22no—
725C4—22no—
726C4—44no—
727C4—44no—
728C4—42no—
729C4—42no—
730C4—42no—
731C4—82no—
732C4—81no—
733C4—84no—
734C4—81no—
735C4—82no—
736C4—81no—
737C4#241314yes—
738C4C2×#52511yescharacteristic
739C4C2×#53611yescharacteristic
740C4C2×#55212yes—
741C2—24no—
742C2—24no—
743C2—42no—
744C2—161no—
745C2C2×#241311yescharacteristic
746C2C2×#303711yescharacteristic
747C2#6042311yescharacteristic
748C2#6125412yes—
749C2#6125512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
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
2. C22×#428C2semidirect
4. C2×#2345C2semidirect
5. #24591C2semidirect
6. #24590C2semidirect
7. #24583C2semidirect
8. #24582C2semidirect
9. #24579C2semidirect
10. #24578C2semidirect
11. #23657C2semidirect
12. #23617C2semidirect
13. #21170C2semidirect
14. #21059C2semidirect
16. #20837C2semidirect
17. #20702C2semidirect
67. #2085C22semidirect
68. #2085C22semidirect
69. #2084C22semidirect
70. #2083C22semidirect
71. #2068C22semidirect
72. #2067C22semidirect
73. #2066C22semidirect
74. #2065C22semidirect
75. #1949C22semidirect
76. #1949C22semidirect
77. #1948C22semidirect
78. #1947C22semidirect
79. #1928C22semidirect
80. #1927C22semidirect
81. #1926C22semidirect
82. #1925C22semidirect

Maximal subgroups

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

subgroupindex
C2×#30922
C22×#4282
#348102
C2×#23452
#245912
#245902
#245832
#245822
#245792
#245782
#236572
#236172
#211702
#210592
#209712
#208372
#207022