Finite Group Databasebeta

← #6463835

Subgroups — #6463835

Order 512.

2128 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3080C211yesmaximal, characteristic, complemented
2C22×#478C211yesmaximal, characteristic, complemented
3#34217C211yesmaximal, characteristic, complemented
4C4×#535C211yesmaximal, characteristic
5#28215C214yesmaximal, complemented
6#28214C214yesmaximal, complemented
7#28208C214yesmaximal, complemented
8#28207C214yesmaximal, complemented
9#23661C211yesmaximal, characteristic, complemented
10#23660C211yesmaximal, characteristic, complemented
11#23511C211yesmaximal, characteristic
12#23511C211yesmaximal, characteristic
13#21068C211yesmaximal, characteristic, complemented
14#21067C211yesmaximal, characteristic, complemented
15#21063C211yesmaximal, characteristic, complemented
16#21063C211yesmaximal, characteristic, complemented
17#20755C211yesmaximal, characteristic, complemented
18#20646C211yesmaximal, characteristic, complemented
19#20645C211yesmaximal, characteristic, complemented
20C2×#535C2211yescharacteristic
21C22×#125C2211yescharacteristic
22C2×C4×#41C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#3080—28no—
25C2×#479C2212yescomplemented
26C2×#479C2212yescomplemented
27C2×#478—28no—
28C2×#478C2214yes—
29C2×#478C2214yes—
30C2×#478C2212yescomplemented
31C2×#478C2212yescomplemented
32C2×#428C2211yescharacteristic
33C2×#428C2211yescharacteristic
34C22×#107C2211yescharacteristic
35C22×#106C2211yescharacteristic
36C2×#423C2211yescharacteristic
37C22×C4⋊C8C2211yescharacteristic
38C2×#417C2211yescharacteristic
39C2×#417C2211yescharacteristic
40C2×#418C2211yescharacteristic
41C2×#418C2211yescharacteristic
42C2×#416C2211yescharacteristic
43C2×#416C2211yescharacteristic
44C22×#102C2211yescharacteristic
45C22×#102C2211yescharacteristic
46#2370—22no—
47#2370—22no—
48#2345—22no—
49#2345—22no—
50#2330C2214yescomplemented
51#2328C2212yes—
52#2326C2212yes—
53#2324—22no—
54#2321—22no—
55#2317C2211yescharacteristic
56#2316C2211yescharacteristic
57#2316C2211yescharacteristic
58#2311C2211yescharacteristic
59#2309C2211yescharacteristic
60C4×#127C2212yes—
61C4×#125C2214yescomplemented
62C4×#124C2212yes—
63C4×#121—22no—
64C4×#120—22no—
65#2285C2211yescharacteristic
66#2274C2211yescharacteristic
67#2274C2211yescharacteristic
68C4×#117C2211yescharacteristic
69#2085C2214yescomplemented
70#2085C2214yescomplemented
71#2084C2214yescomplemented
72#2083C2214yescomplemented
73#2042C2214yes—
74#2042C2214yes—
75#2041C2214yes—
76#2041C2214yes—
77#1928C2214yescomplemented
78#1927C2214yescomplemented
79#1926C2214yes—
80#1926C2214yes—
81#1926C2214yescomplemented
82#1925C2214yes—
83#1925C2214yes—
84#1925C2214yescomplemented
85#1832C2214yescomplemented
86#1831C2214yescomplemented
87#1805C2212yescomplemented
88#1804C2214yes—
89#1804C2212yescomplemented
90#1803C2212yescomplemented
91#1802C2214yes—
92#1802C2212yescomplemented
93#1736C2211yescharacteristic
94#1736C2211yescharacteristic
95#1735C2211yescharacteristic
96#1735C2211yescharacteristic
97#1729C2211yescharacteristic
98#1729C2211yescharacteristic
99#1729C2211yescharacteristic
100#1729C2211yescharacteristic
101C22×#41C2311yescharacteristic
102C23×D4C2311yescharacteristic
103C24×C4—21no—
104C22×M16C2311yescharacteristic
105C22×M16C2311yescharacteristic
106C23×C8C2311yescharacteristic
107C23×C8C2311yescharacteristic
108#535—24no—
109#535—24no—
110C2×#127C2312yes—
111C2×#125—21no—
112C2×#125—21no—
113C2×#125—24no—
114C2×#125D418yescomplemented
115C2×#125C2312yes—
116C2×#125C2312yes—
117C2×#124C2312yes—
118C2×#121—22no—
119C2×#120—22no—
120C4×#41—24no—
121C4×#41—24no—
122C2×C4×Q8C2311yescharacteristic
123C2×C4×Q8C2311yescharacteristic
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4—21no—
128C2×C4×D4—24no—
129C2×C4×D4—24no—
130C2×C4×D4—24no—
131C2×C4×D4C2312yes—
132C2×C4×D4C2312yes—
133C2×C4×D4C2311yescharacteristic
134C2×C4×D4C2311yescharacteristic
135C2×#117—21no—
136C2×#117C2311yescharacteristic
137C2×#117C2311yescharacteristic
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C2×#117C2311yescharacteristic
141C22×C4⋊C4—21no—
142C22×C4⋊C4—21no—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2311yescharacteristic
146C22×C22⋊C4—21no—
147C22×C42—21no—
148C22×C42C2311yescharacteristic
149#479—28no—
150#479—28no—
151#478—216no—
152#478—216no—
153#478—28no—
154#478—28no—
155#428—24no—
156#428—24no—
157C2×#107—24no—
158C2×#107C2314yes—
159C2×#107C2312yes—
160C2×#107C2312yes—
161C2×#106—24no—
162C2×#106C2314yes—
163C2×#106C2312yes—
164C2×#106C2312yes—
165#423—24no—
166C2×C4⋊C8—24no—
167C2×C4⋊C8C2312yes—
168C2×C4⋊C8C2312yes—
169C2×C4⋊C8C2312yes—
170C2×C4⋊C8C2312yes—
171#418—24no—
172#418—24no—
173#417—24no—
174#417—24no—
175#416—24no—
176#416—24no—
177C2×#103C2312yes—
178C2×#103C2312yes—
179C2×#103C2312yes—
180C2×#103C2312yes—
181C2×#102—24no—
182C2×#102—24no—
183C2×#102C2314yes—
184C2×#102C2314yes—
185C2×#102C2312yes—
186C2×#102C2312yes—
187C2×#102C2314yes—
188C2×#102C2314yes—
189C2×#102C2312yes—
190C2×#102C2312yes—
191#391C2312yes—
192#390—22no—
193#390—24no—
194#390C2312yes—
195#390C2312yes—
196#390C2312yes—
197#389—22no—
198#389—22no—
199#389—22no—
200#389—24no—
201#389—24no—
202#389C2314yes—
203#389C2314yes—
204#388—24no—
205#388—24no—
206#387—24no—
207#387—24no—
208#387—24no—
209#387—24no—
210#386—22no—
211#386—24no—
212#386—24no—
213#385—22no—
214#385—24no—
215#384C2312yes—
216#384C2312yes—
217#382C2312yes—
218#382C2312yes—
219#380—21no—
220#380—21no—
221#380—22no—
222#380—22no—
223#380—22no—
224#379—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—24no—
228C4×C4⋊C4—24no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4D412yes—
232C4×C4⋊C4D412yes—
233C4×C4⋊C4C2314yes—
234C4×C4⋊C4C2314yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—24no—
238C4×C22⋊C4D412yes—
239C4×C22⋊C4D412yes—
240#376C2312yes—
241C43C2312yes—
242#340C2314yes—
243#340C2314yes—
244#340C2312yes—
245#340C2312yes—
246#340C2312yes—
247#340C2312yes—
248#336C2314yes—
249#336C2314yes—
250#336C2314yes—
251#336C2314yes—
252C25—21no—
253C2×#41—22no—
254C2×#41—22no—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—22no—
258C2×#41#4114yes—
259C22×Q8C2411yescharacteristic
260C22×D4—21no—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—24no—
264C22×D4—22no—
265C22×D4—22no—
266C22×D4—22no—
267C22×D4—22no—
268C22×D4#4114yes—
269C22×D4C2412yes—
270C22×D4C2411yescharacteristic
271C23×C4—21no—
272C23×C4—21no—
273C23×C4—21no—
274C23×C4—21no—
275C23×C4—21no—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4—21no—
279C23×C4—44no—
280C23×C4—84no—
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2411yescharacteristic
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16—22no—
288C2×M16—22no—
289C2×M16—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8—22no—
295C22×C8—22no—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C22×C8C2412yes—
299C22×C8C2412yes—
300#127—24no—
301#127—24no—
302#125—28no—
303#125—28no—
304#125—24no—
305#125—24no—
306#125C2×D418yes—
307#125C2×D418yes—
308#124—24no—
309#124—24no—
310#121—216no—
311#120—216no—
312C4×Q8—22no—
313C4×Q8—22no—
314C4×Q8—22no—
315C4×Q8—22no—
316C4×D4—24no—
317C4×D4—24no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—22no—
323C4×D4—22no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—44no—
328C4×D4—88no—
329#117—22no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—22no—
335#117—22no—
336#117—22no—
337#117—22no—
338#117—22no—
339#117—44no—
340C2×C4⋊C4—28no—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—21no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—24no—
359C2×C4⋊C4—24no—
360C2×C4⋊C4C2×D412yes—
361C2×C4⋊C4C2×D412yes—
362C2×C4⋊C4C2×D414yes—
363C2×C4⋊C4C2×D414yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2411yescharacteristic
366C2×C4⋊C4C2411yescharacteristic
367C2×C4⋊C4C2412yes—
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C22⋊C4—24no—
372C2×C22⋊C4—21no—
373C2×C22⋊C4—21no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—21no—
378C2×C22⋊C4—21no—
379C2×C22⋊C4—24no—
380C2×C22⋊C4—24no—
381C2×C22⋊C4—24no—
382C2×C22⋊C4—44no—
383C2×C22⋊C4C2×D412yes—
384C2×C22⋊C4C2×D412yes—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42—21no—
388C2×C42—21no—
389C2×C42—21no—
390C2×C42—24no—
391C2×C42—24no—
392C2×C42—44no—
393C2×C42C2×D414yes—
394C2×C42C2×D414yes—
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D414yes—
398C2×C42#4114yes—
399C2×C42C2412yes—
400C2×C42C2412yes—
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403#107—28no—
404#107—24no—
405#107—24no—
406#106—28no—
407#106—24no—
408#106—24no—
409C4⋊C8—24no—
410C4⋊C8—24no—
411C4⋊C8—24no—
412C4⋊C8—24no—
413#103—24no—
414#103—24no—
415#103—24no—
416#103—24no—
417#102—24no—
418#102—24no—
419#102—28no—
420#102—28no—
421#102—28no—
422#102—28no—
423#102—24no—
424#102—24no—
425#95—24no—
426#95—24no—
427#95—24no—
428#95—24no—
429#95—22no—
430#95—22no—
431#95—24no—
432#95—24no—
433#95C2×D414yes—
434#95C2×D414yes—
435C24—22no—
436C24—21no—
437C24—24no—
438C24—42no—
439C24—42no—
440C24—82no—
441C24—82no—
442C24C22×D411yescharacteristic
443#41—28no—
444#41—28no—
445#41—44no—
446#41—44no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×#4112yes—
453C2×D4—24no—
454C2×D4—216no—
455C2×D4—24no—
456C2×D4—28no—
457C2×D4—28no—
458C2×D4—22no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4—21no—
462C2×D4—22no—
463C2×D4—21no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—44no—
473C2×D4—44no—
474C2×D4—84no—
475C2×D4—84no—
476C2×D4C2×#4114yes—
477C2×D4C2×#4112yes—
478C22×C4—21no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—21no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—44no—
492C22×C4—48no—
493C22×C4—42no—
494C22×C4—42no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—82no—
501C22×C4—84no—
502C22×C4—82no—
503C22×C4—82no—
504C22×C4—82no—
505C22×C4—816no—
506C22×C4#12114yes—
507C22×C4#12114yes—
508C22×C4C22×D414yes—
509C22×C4C22×D414yes—
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D412yes—
516C22×C4C2×#4112yes—
517C22×C4C2×#4112yes—
518C22×C4C2×#4112yes—
519C22×C42−1+412yes—
520C22×C42−1+412yes—
521C22×C4C2511yescharacteristic
522M16—44no—
523M16—44no—
524C2×C8—22no—
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—44no—
537C2×C8—44no—
538C4⋊C4—28no—
539C4⋊C4—28no—
540C4⋊C4—28no—
541C4⋊C4—28no—
542C4⋊C4—24no—
543C4⋊C4—24no—
544C4⋊C4—24no—
545C4⋊C4—24no—
546C4⋊C4—22no—
547C4⋊C4—22no—
548C4⋊C4—22no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—22no—
552C4⋊C4—22no—
553C4⋊C4—22no—
554C4⋊C4—22no—
555C4⋊C4—22no—
556C4⋊C4—21no—
557C4⋊C4—21no—
558C4⋊C4—42no—
559C4⋊C4—42no—
560C4⋊C4—42no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—84no—
564C4⋊C4—84no—
565C4⋊C4—84no—
566C4⋊C4C22×D414yes—
567C4⋊C4C22×D414yes—
568C22⋊C4—216no—
569C22⋊C4—24no—
570C22⋊C4—24no—
571C22⋊C4—42no—
572C22⋊C4—42no—
573C22⋊C4—42no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C22⋊C4—42no—
577C22⋊C4—42no—
578C22⋊C4—84no—
579C22⋊C4—84no—
580C22⋊C4—84no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—44no—
586C42—44no—
587C42—42no—
588C42—42no—
589C42—42no—
590C42—42no—
591C42—42no—
592C42—84no—
593C42—84no—
594C42#12114yes—
595C42#12114yes—
596C42C22×D412yes—
597C42C22×D412yes—
598C42C2×#4112yes—
599C42C2×#4112yes—
600C23—22no—
601C23—22no—
602C23—24no—
603C23—22no—
604C23—44no—
605C23—42no—
606C23—42no—
607C23—44no—
608C23—41no—
609C23—42no—
610C23—41no—
611C23—42no—
612C23—48no—
613C23—88no—
614C23—84no—
615C23—84no—
616C23—88no—
617C23C2×#12112yes—
618C23#52512yes—
619C23#53612yes—
620C23C23×D411yescharacteristic
621Q8—22no—
622Q8—22no—
623Q8—41no—
624Q8—41no—
625D4—24no—
626D4—22no—
627D4—24no—
628D4—22no—
629D4—48no—
630D4—42no—
631D4—41no—
632D4—42no—
633D4—41no—
634D4—48no—
635D4—816no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—41no—
654C2×C4—41no—
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—41no—
664C2×C4—41no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—84no—
669C2×C4—88no—
670C2×C4—84no—
671C2×C4—88no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4C2×#12112yes—
676C2×C4C2×#12112yes—
677C2×C4C2×#12112yes—
678C2×C4C2×#12112yes—
679C2×C4C2×#12112yes—
680C2×C4#52512yes—
681C2×C4#53612yes—
682C2×C4#54814yes—
683C2×C4#54814yes—
684C2×C4#55214yes—
685C2×C4#55214yes—
686C2×C4C23×D411yescharacteristic
687C2×C4C22×#4111yescharacteristic
688C2×C4C22×#4111yescharacteristic
689C2×C4C22×#4111yescharacteristic
690C2×C4C2×2−1+411yescharacteristic
691C2×C4C2×2−1+411yescharacteristic
692C8—42no—
693C8—42no—
694C8—42no—
695C8—42no—
696C22—22no—
697C22—22no—
698C22—24no—
699C22—24no—
700C22—42no—
701C22—44no—
702C22—44no—
703C22—48no—
704C22—816no—
705C22—81no—
706C22—82no—
707C22—82no—
708C22—81no—
709C22—82no—
710C22—84no—
711C22#253614yes—
712C22C22×#12111yescharacteristic
713C22C2×#52511yescharacteristic
714C22C2×#53611yescharacteristic
715C22#358412yes—
716C22#358512yes—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—44no—
722C4—44no—
723C4—42no—
724C4—44no—
725C4—44no—
726C4—81no—
727C4—81no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4#253614yes—
732C4C2×#54811yescharacteristic
733C4C2×#54811yescharacteristic
734C4C2×#55211yescharacteristic
735C4C2×#55211yescharacteristic
736C2—24no—
737C2—82no—
738C2—84no—
739C2C2×#253611yescharacteristic
740C2#6042211yescharacteristic
741C2#6042611yescharacteristic
742C2#6060112yes—
743C2#6089012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
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×#3080C2semidirect
2. C22×#478C2semidirect
3. #34217C2semidirect
5. #28215C2semidirect
6. #28214C2semidirect
7. #28208C2semidirect
8. #28207C2semidirect
9. #23661C2semidirect
10. #23660C2semidirect
13. #21068C2semidirect
14. #21067C2semidirect
15. #21063C2semidirect
16. #21063C2semidirect
17. #20755C2semidirect
18. #20646C2semidirect
19. #20645C2semidirect
25. C2×#479C22semidirect
26. C2×#479C22semidirect
30. C2×#478C22semidirect
31. C2×#478C22semidirect
50. #2330C22semidirect
61. C4×#125C22semidirect
69. #2085C22semidirect
70. #2085C22semidirect
71. #2084C22semidirect
72. #2083C22semidirect
77. #1928C22semidirect
78. #1927C22semidirect
81. #1926C22semidirect
84. #1925C22semidirect
85. #1832C22semidirect
86. #1831C22semidirect
87. #1805C22semidirect
89. #1804C22semidirect
90. #1803C22semidirect
92. #1802C22semidirect
114. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#30802
C22×#4782
#342172
C4×#5352
#282152
#282142
#282082
#282072
#236612
#236602
#235112
#210682
#210672
#210632
#207552
#206462
#206452