Finite Group Databasebeta

← #6529761

Subgroups — #6529761

Order 512.

1870 conjugacy classes of subgroups.

The subgroup lattice has 734 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic, complemented
2C2×#3079C211yesmaximal, characteristic
3#34979C211yesmaximal, characteristic
4#34216C211yesmaximal, characteristic, complemented
5#28283C212yesmaximal, complemented
6#28281C212yesmaximal, complemented
7#28205C212yesmaximal, complemented
8#28203C212yesmaximal, complemented
9#25454C212yesmaximal, complemented
10#25451C212yesmaximal, complemented
11#25278C212yesmaximal, complemented
12#25273C212yesmaximal, complemented
13#24590C214yesmaximal, complemented
14#24491C214yesmaximal
15#21974C211yesmaximal, characteristic, complemented
16#21894C211yesmaximal, characteristic, complemented
17#20737C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21#3229—28no—
22C2×#498C2212yes—
23C2×#487C2212yes—
24C2×#483C2214yes—
25#3079—28no—
26C2×#477C2212yes—
27C2×#475C2212yes—
28C2×#475C2212yes—
29C2×#475C2212yes—
30C2×#476C2212yes—
31C2×#474C2212yes—
32C2×#431C2211yescharacteristic
33C2×#428C2212yes—
34C2×#423C2211yescharacteristic
35C2×#417C2212yes—
36#2611—22no—
37#2568C2211yescharacteristic
38#2547C2211yescharacteristic
39#2546C2211yescharacteristic
40#2542C2211yescharacteristic
41#2499C2212yes—
42#2432C2212yes—
43#2415—21no—
44#2415—21no—
45#2395—22no—
46#2369C2212yes—
47#2369C2212yes—
48#2365—21no—
49#2365—21no—
50#2345C2214yes—
51Q8×C22⋊C4—22no—
52#2331—22no—
53#2316—21no—
54C4×C4⋊Q8C2212yes—
55C4×C4⋊Q8C2212yes—
56C4×C4⋊Q8C2212yes—
57C4×#124C2212yes—
58C4×C22⋊Q8—21no—
59C4×C22⋊Q8—21no—
60#2285C2211yescharacteristic
61#2274—21no—
62C4×#117C2211yescharacteristic
63#2085C2214yes—
64#2084C2214yes—
65#2068C2214yes—
66#2066C2214yes—
67#2066C2214yes—
68#2065C2214yes—
69#2043C2214yes—
70#2042C2214yes—
71#1988C2212yes—
72#1983C2214yes—
73#1981C2214yes—
74#1981C2212yes—
75#1949C2214yes—
76#1948C2214yes—
77#1939C2212yes—
78#1939C2212yes—
79#1928C2214yes—
80#1927C2214yes—
81#1927C2214yes—
82#1926C2214yes—
83#1894C2214yes—
84#1851C2214yes—
85#1842C2212yes—
86#1831C2212yes—
87#1825C2212yes—
88#1803C2211yescharacteristic
89#1803C2211yescharacteristic
90#1803C2212yes—
91#1803C2212yes—
92#1802C2211yescharacteristic
93#1802C2211yescharacteristic
94#1788C2212yes—
95C4×#103C2212yes—
96C4×#102C2212yes—
97C22×#41C2311yescharacteristic
98C23×Q8—21no—
99C22×M16C2312yes—
100#533—24no—
101#533—24no—
102C2×C4⋊Q8—21no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—24no—
105C2×C4⋊Q8D418yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2312yes—
110C2×C4⋊Q8C2312yes—
111C2×#125—21no—
112C2×#124C2312yes—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2312yes—
117C4×#41—24no—
118C4×#41—24no—
119C2×C4×Q8—21no—
120C2×C4×Q8—22no—
121C2×C4×Q8—22no—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2312yes—
124C2×C4×Q8C2311yescharacteristic
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2311yescharacteristic
128C2×C4×D4C2311yescharacteristic
129C2×#117—21no—
130C2×#117—21no—
131C2×#117—21no—
132C2×#117D411yescharacteristic
133C2×#117D411yescharacteristic
134C2×#117C2311yescharacteristic
135C22×C4⋊C4—21no—
136C22×C4⋊C4C2312yes—
137C22×C42—21no—
138C22×C42C2311yescharacteristic
139#498—28no—
140#487—28no—
141#483—216no—
142#477—28no—
143#476—28no—
144#475—28no—
145#475—28no—
146#475—28no—
147#474—28no—
148#431—24no—
149#428—28no—
150C2×#107C2314yes—
151C2×#106C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154#423—24no—
155C2×C4⋊C8C2311yescharacteristic
156C2×C4⋊C8C2311yescharacteristic
157C2×C4⋊C8C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2314yes—
161#417—28no—
162C2×#103C2314yes—
163C2×#103C2314yes—
164C2×#103C2314yes—
165C2×#103C2312yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#102C2312yes—
170C2×#102C2314yes—
171C2×#102C2312yes—
172C2×C22⋊C8C2312yes—
173C2×C4×C8C2312yes—
174#400—22no—
175#399—22no—
176#398—22no—
177#397—22no—
178#397—22no—
179#397—22no—
180#394—22no—
181#394—22no—
182#391—21no—
183#391—21no—
184#391—22no—
185#391—22no—
186#391—24no—
187#391—24no—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#390C2312yes—
192#389—22no—
193#389—22no—
194#389—21no—
195#389—21no—
196#389—22no—
197#389—22no—
198#389—22no—
199#389—22no—
200#389C2314yes—
201#389C2314yes—
202#388—22no—
203#388—22no—
204#387—24no—
205#387—24no—
206#387—22no—
207#387—22no—
208#385—22no—
209#385—22no—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#384C2312yes—
213#382—21no—
214#382C2312yes—
215#380—21no—
216#380—21no—
217#380—21no—
218#380—24no—
219#380C2312yes—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—22no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4C2314yes—
230C4×C4⋊C4C2314yes—
231C4×C4⋊C4C2312yes—
232C4×C4⋊C4C2311yescharacteristic
233C4×C4⋊C4C2311yescharacteristic
234C4×C22⋊C4—24no—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4D412yes—
238C4×C22⋊C4C2312yes—
239#376D412yes—
240C43C2312yes—
241#340C2312yes—
242#340C2312yes—
243#340C2314yes—
244#340C2312yes—
245#340C2314yes—
246#340C2312yes—
247#336C2314yes—
248#336C2314yes—
249C2×#41—22no—
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—22no—
253C2×#41—41no—
254C2×#41—41no—
255C22×Q8—28no—
256C22×Q8—22no—
257C22×Q8—22no—
258C22×Q8—21no—
259C22×Q8—82no—
260C22×Q8—82no—
261C22×Q8C2412yes—
262C22×Q8C2411yescharacteristic
263C22×D4—21no—
264C22×D4C2411yescharacteristic
265C23×C4—22no—
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—21no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2411yescharacteristic
272C2×M16—24no—
273C2×M16—42no—
274C2×M16—42no—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C4⋊Q8—24no—
279C4⋊Q8—28no—
280C4⋊Q8—28no—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—42no—
288C4⋊Q8C2×D418yes—
289C4⋊Q8C2×D418yes—
290#125—44no—
291#124—24no—
292#124—42no—
293C22⋊Q8—24no—
294C22⋊Q8—24no—
295C22⋊Q8—44no—
296C22⋊Q8—44no—
297C22⋊Q8—44no—
298C4×Q8—22no—
299C4×Q8—22no—
300C4×Q8—24no—
301C4×Q8—22no—
302C4×Q8—22no—
303C4×Q8—24no—
304C4×Q8—44no—
305C4×Q8—84no—
306C4×Q8—84no—
307C4×D4—22no—
308C4×D4—22no—
309C4×D4—22no—
310C4×D4—22no—
311C4×D4—44no—
312C4×D4—44no—
313#117—24no—
314#117—22no—
315#117—22no—
316#117—42no—
317#117—44no—
318#117—44no—
319#117—44no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—24no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D418yes—
346C2×C4⋊C4C2×D411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2414yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—21no—
368C2×C42—21no—
369C2×C42—24no—
370C2×C42—24no—
371C2×C42—44no—
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2×D412yes—
375C2×C42C2×D412yes—
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D414yes—
379C2×C42#4114yes—
380C2×C42C2412yes—
381C2×C42C2412yes—
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384#107—28no—
385#106—28no—
386#106—28no—
387#106—28no—
388C4⋊C8—24no—
389C4⋊C8—24no—
390C4⋊C8—22no—
391C4⋊C8—22no—
392C4⋊C8—44no—
393C4⋊C8—44no—
394#103—24no—
395#103—28no—
396#103—28no—
397#103—28no—
398#103—44no—
399#103—42no—
400#103—44no—
401#102—24no—
402#102—28no—
403#102—42no—
404C22⋊C8—24no—
405C4×C8—24no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—24no—
419#95—24no—
420#95C2×D412yes—
421#95C2×D412yes—
422#95C2×D412yes—
423#95C2×D412yes—
424C24—21no—
425C24C22×D411yescharacteristic
426#41—28no—
427#41—44no—
428#41—44no—
429#41—44no—
430C2×Q8—28no—
431C2×Q8—28no—
432C2×Q8—21no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—22no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—21no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—44no—
447C2×Q8—44no—
448C2×Q8—82no—
449C2×Q8—82no—
450C2×Q8—816no—
451C2×Q82−1+412yes—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D42−1+412yes—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—24no—
471C22×C4—28no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—82no—
488C22×C4—82no—
489C22×C4#12114yes—
490C22×C4#12114yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C2×#4112yes—
502C22×C4C2511yescharacteristic
503M16—48no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—24no—
512C2×C8—24no—
513C4⋊C4—28no—
514C4⋊C4—28no—
515C4⋊C4—28no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—21no—
521C4⋊C4—22no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—44no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C4⋊C4#13714yes—
548C4⋊C4C22×D418yes—
549C22⋊C4—24no—
550C22⋊C4—42no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—42no—
556C22⋊C4—44no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—44no—
560C42—24no—
561C42—24no—
562C42—24no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—42no—
568C42—42no—
569C42—42no—
570C42—42no—
571C42—44no—
572C42—44no—
573C42—84no—
574C42—84no—
575C42#12114yes—
576C42#12114yes—
577C42C22×D412yes—
578C42C22×D412yes—
579C42C2×#4112yes—
580C42C2×#4112yes—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—22no—
588C23—41no—
589C23—41no—
590C23—41no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23C2×#12112yes—
595C23C23×D411yescharacteristic
596Q8—22no—
597Q8—22no—
598Q8—41no—
599Q8—41no—
600Q8—416no—
601Q8—42no—
602Q8—42no—
603Q8—42no—
604Q8—42no—
605Q8—88no—
606Q8—84no—
607D4—22no—
608D4—22no—
609D4—48no—
610D4—41no—
611D4—41no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—48no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—41no—
659C2×C4—41no—
660C2×C4—88no—
661C2×C4—84no—
662C2×C4—84no—
663C2×C4C2×#12112yes—
664C2×C4C2×#12112yes—
665C2×C4C2×#12112yes—
666C2×C4C2×#12112yes—
667C2×C4C2×#12112yes—
668C2×C4C2×#13714yes—
669C2×C4C2×#13712yes—
670C2×C4#57512yes—
671C2×C4#57712yes—
672C2×C4#57712yes—
673C2×C4C23×D411yescharacteristic
674C2×C4C22×#4111yescharacteristic
675C2×C4C2×2−1+411yescharacteristic
676C2×C4C2×2−1+411yescharacteristic
677C2×C4#58511yescharacteristic
678C2×C4#58511yescharacteristic
679C8—42no—
680C8—42no—
681C8—44no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—41no—
686C22—41no—
687C22—41no—
688C22—41no—
689C22—42no—
690C22—42no—
691C22—42no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22C22×#12111yescharacteristic
697C22#348911yescharacteristic
698C22#348911yescharacteristic
699C22C22×#13711yescharacteristic
700C22C2×#57511yescharacteristic
701C22C2×#57711yescharacteristic
702C22C2×#57711yescharacteristic
703C4—22no—
704C4—22no—
705C4—24no—
706C4—22no—
707C4—48no—
708C4—42no—
709C4—42no—
710C4—42no—
711C4—42no—
712C4—81no—
713C4—82no—
714C4—82no—
715C4—82no—
716C4—81no—
717C4—82no—
718C4#303714yes—
719C4#304214yes—
720C4#348611yescharacteristic
721C4#348711yescharacteristic
722C4#349511yescharacteristic
723C4#349511yescharacteristic
724C2—42no—
725C2—82no—
726C2#3566111yescharacteristic
727C2C2×#303711yescharacteristic
728C2C2×#304211yescharacteristic
729C2#6213111yescharacteristic
730C2#6213411yescharacteristic
731C2#6214511yescharacteristic
732C2#6214711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3229C2semidirect
4. #34216C2semidirect
5. #28283C2semidirect
6. #28281C2semidirect
7. #28205C2semidirect
8. #28203C2semidirect
9. #25454C2semidirect
10. #25451C2semidirect
11. #25278C2semidirect
12. #25273C2semidirect
13. #24590C2semidirect
15. #21974C2semidirect
16. #21894C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
C2×#30792
#349792
#342162
#282832
#282812
#282052
#282032
#254542
#254512
#252782
#252732
#245902
#244912
#219742
#218942
#207372