Finite Group Databasebeta

← #6539663

Subgroups — #6539663

Order 512.

2125 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3331C212yesmaximal, complemented
2#36127C211yesmaximal, characteristic
3#34332C211yesmaximal, characteristic, complemented
4#29799C211yesmaximal, characteristic, complemented
5#29756C212yesmaximal, complemented
6#28748C211yesmaximal, characteristic, complemented
7#28581C212yesmaximal, complemented
8#28202C211yesmaximal, characteristic, complemented
9#25273C212yesmaximal, complemented
10#25100C214yesmaximal, complemented
11#24729C212yesmaximal, complemented
12#24491C214yesmaximal, complemented
13#24216C212yesmaximal, complemented
14#23678C211yesmaximal, characteristic, complemented
15#23508C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22431C212yesmaximal, complemented
18C2×#554C2211yescharacteristic
19C2×#547C2212yes—
20C2×#532C2212yes—
21#3331—216no—
22C2×#487C2212yes—
23C2×#483C2214yes—
24C2×#476C2212yes—
25C2×#470C2214yes—
26C2×#464C2214yes—
27C2×#462C2212yes—
28C2×#452C2214yes—
29C2×C4×SD16C2212yes—
30C2×#436C2212yes—
31C2×#417C2214yes—
32C2×#417C2214yes—
33#2868C2211yescharacteristic
34#2857C2212yes—
35#2547C2212yes—
36#2537—21no—
37#2537—21no—
38#2432C2214yes—
39#2422—22no—
40#2422—22no—
41#2370—21no—
42#2369C2211yescharacteristic
43#2360—22no—
44#2360C2212yes—
45Q8×C4⋊C4C2211yescharacteristic
46D4×C4⋊C4—21no—
47D4×C4⋊C4—21no—
48D4×C4⋊C4—21no—
49D4×C4⋊C4C2211yescharacteristic
50#2345C2212yes—
51#2330—21no—
52#2329—22no—
53C4×C4⋊Q8C2211yescharacteristic
54C4×#127C2211yescharacteristic
55C4×C22⋊Q8C2212yes—
56#2295C2211yescharacteristic
57C2×#389C2212yes—
58#2275—21no—
59C42×D4C2211yescharacteristic
60#2084C2212yes—
61#2084C2211yescharacteristic
62#2072C2211yescharacteristic
63#2066C2214yes—
64#2066C2214yes—
65#2065C2212yes—
66#2055C2212yes—
67#2043C2212yes—
68#2043C2212yes—
69#2043C2212yes—
70#2043C2212yes—
71#2042C2214yes—
72#2042C2214yes—
73#2041C2212yes—
74#2041C2211yescharacteristic
75#2016C2212yes—
76#2016C2212yes—
77#2014C2211yescharacteristic
78#1983C2214yes—
79#1981C2214yes—
80#1981C2212yes—
81#1956C2212yes—
82#1939C2212yes—
83#1927C2214yes—
84#1926C2214yes—
85#1894C2214yes—
86#1878C2214yes—
87#1872C2214yes—
88#1851C2214yes—
89#1831C2211yescharacteristic
90#1802C2212yes—
91#1797C2212yes—
92#1788C2214yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#102C2212yes—
95#1728C2212yes—
96C22×#41C2314yes—
97C22×Q16C2314yes—
98C22×SD16C2312yes—
99C22×M16C2314yes—
100#554—24no—
101#554D418yescomplemented
102#547—28no—
103#547—28no—
104#532—48no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×C4⋊Q8C2312yes—
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#125—21no—
111C2×#124C2314yes—
112C2×#123—24no—
113C2×C22⋊Q8—22no—
114C2×C22⋊Q8C2314yes—
115C2×C22⋊Q8C2312yes—
116C2×#121—22no—
117C2×#121—22no—
118C2×#121—22no—
119C2×C4×Q8—21no—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2312yes—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—22no—
127C2×C4×D4—22no—
128C2×C4×D4D411yescharacteristic
129C2×C4×D4D411yescharacteristic
130C2×C4×D4C2312yes—
131C2×C4×D4C2311yescharacteristic
132C22×C4⋊C4—22no—
133C22×C4⋊C4—22no—
134C22×C4⋊C4C2312yes—
135C22×C4⋊C4C2312yes—
136C22×C42C2312yes—
137#487—28no—
138#483—216no—
139#476—28no—
140#470—216no—
141#464—48no—
142#462—28no—
143#452—216no—
144C4×SD16—28no—
145#436—28no—
146C2×#106C2312yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2314yes—
150C2×C4⋊C8C2312yes—
151#417—216no—
152#417—48no—
153C2×#103C2312yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164C2×#102C2314yes—
165C2×#102C2312yes—
166C2×C22⋊C8C2314yes—
167C2×C4×C8C2312yes—
168#397—24no—
169#394—24no—
170#394—24no—
171#391—21no—
172#391—22no—
173#391C2311yescharacteristic
174#391C2314yes—
175#391C2311yescharacteristic
176#390—21no—
177#390—21no—
178#390C2312yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—22no—
183#389—22no—
184#389—22no—
185#389—22no—
186#389—22no—
187#389—22no—
188#389C2312yes—
189#389C2311yescharacteristic
190#389C2312yes—
191#389C2312yes—
192#389C2311yescharacteristic
193#387—21no—
194#387—22no—
195#387—21no—
196#387—22no—
197#387—22no—
198#387—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—24no—
203#385—24no—
204#384—21no—
205#384—21no—
206#384—21no—
207#384—21no—
208#384—21no—
209#384C2311yescharacteristic
210#384C2311yescharacteristic
211#380—22no—
212#380—22no—
213#380—22no—
214#380—22no—
215#380—22no—
216#380C2312yes—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4D411yescharacteristic
224C4×C4⋊C4C2312yes—
225C4×C4⋊C4C2312yes—
226C4×C4⋊C4C2311yescharacteristic
227C4×C4⋊C4C2311yescharacteristic
228C4×C22⋊C4—22no—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233C4×C22⋊C4C2312yes—
234#376—21no—
235C2×#95—22no—
236C43C2311yescharacteristic
237#340C2312yes—
238#340C2314yes—
239#340C2312yes—
240#336C2311yescharacteristic
241#336C2312yes—
242#336C2311yescharacteristic
243#336C2312yes—
244#336C2314yes—
245C2×#41—28no—
246C2×#41—28no—
247C2×#41—28no—
248C2×#41—44no—
249C2×#41—44no—
250C2×#41—44no—
251C22×Q8—21no—
252C22×Q8C2414yes—
253C22×Q8C2411yescharacteristic
254C22×D4—24no—
255C22×D4—21no—
256C22×D4—21no—
257C22×D4C2×D411yescharacteristic
258C22×D4C2412yes—
259C23×C4—22no—
260C23×C4—22no—
261C23×C4—22no—
262C23×C4—24no—
263C23×C4C2×D412yes—
264C23×C4C2×D412yes—
265C23×C4C2412yes—
266C2×Q16—28no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×SD16—24no—
270C2×SD16—24no—
271C2×SD16—42no—
272C2×M16—28no—
273C2×M16—44no—
274C2×M16—44no—
275C22×C8C2412yes—
276C22×C8C2414yes—
277C22×C8C2412yes—
278C4⋊Q8—24no—
279C4⋊Q8—22no—
280C4⋊Q8—24no—
281C4⋊Q8—44no—
282C4⋊Q8—44no—
283C4⋊Q8C2×D414yes—
284#127—24no—
285#125—28no—
286#124—28no—
287#124—44no—
288#123—416no—
289C22⋊Q8—24no—
290C22⋊Q8—216no—
291C22⋊Q8—28no—
292C22⋊Q8—28no—
293C22⋊Q8C2×D418yes—
294#121—48no—
295#121—48no—
296#121—48no—
297C4×Q8—22no—
298C4×Q8—44no—
299C4×Q8—82no—
300C4×Q8C2×D414yes—
301C4×D4—28no—
302C4×D4—24no—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—42no—
306C4×D4—44no—
307C4×D4—44no—
308C4×D4—48no—
309C4×D4—48no—
310C4×D4—44no—
311C4×D4C2×D414yes—
312C4×D4#4114yes—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—44no—
337C2×C4⋊C4—48no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4C2×D411yescharacteristic
340C2×C4⋊C4C2×D411yescharacteristic
341C2×C4⋊C4C2×D418yes—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C4⋊C4C2411yescharacteristic
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4C2×D412yes—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—44no—
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2412yes—
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#106—24no—
377#106—28no—
378C4⋊C8—24no—
379C4⋊C8—42no—
380C4⋊C8—44no—
381#103—24no—
382#103—28no—
383#103—28no—
384#103—28no—
385#103—44no—
386#103—44no—
387#103—44no—
388#103—44no—
389#102—24no—
390#102—28no—
391#102—24no—
392#102—28no—
393#102—42no—
394C22⋊C8—28no—
395C4×C8—24no—
396#95—21no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—21no—
404#95—22no—
405#95—21no—
406#95—22no—
407#95—22no—
408#95—21no—
409#95—24no—
410#95—24no—
411#95C2×D412yes—
412#95C2×D412yes—
413#95C2×D412yes—
414C24—22no—
415C24C22×D412yes—
416#41—416no—
417#41—416no—
418#41—88no—
419#41—88no—
420C2×Q8—24no—
421C2×Q8—21no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—22no—
425C2×Q8—24no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—81no—
431C2×Q8—81no—
432C2×Q8C22×D412yes—
433C2×D4—24no—
434C2×D4—24no—
435C2×D4—22no—
436C2×D4—22no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—42no—
443C2×D4—42no—
444C2×D4—48no—
445C2×D4—48no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—84no—
449C2×D4#12114yes—
450C2×D4H(5,𝔽2)14yes—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—84no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—82no—
478C22×C4—82no—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D412yes—
484C22×C4C22×D414yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C2×#4114yes—
490C22×C4C2511yescharacteristic
491Q16—88no—
492SD16—48no—
493M16—88no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—24no—
502C2×C8—22no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—28no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—21no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—42no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—81no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4#12114yes—
535C4⋊C4C22×D412yes—
536C4⋊C4C22×D414yes—
537C22⋊C4—28no—
538C22⋊C4—24no—
539C22⋊C4—42no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—48no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4#12118yes—
548C42—24no—
549C42—22no—
550C42—22no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—42no—
555C42—42no—
556C42—41no—
557C42—42no—
558C42—82no—
559C42—82no—
560C42—82no—
561C42#13714yes—
562C42C22×D412yes—
563C42C2×#4112yes—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—24no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23C2×#12114yes—
578C23C23×D411yescharacteristic
579Q8—24no—
580Q8—42no—
581Q8—44no—
582Q8—44no—
583Q8—44no—
584Q8—44no—
585Q8—84no—
586D4—24no—
587D4—24no—
588D4—28no—
589D4—48no—
590D4—48no—
591D4—42no—
592D4—42no—
593D4—816no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—84no—
640C2×C4—84no—
641C2×C4—816no—
642C2×C4—88no—
643C2×C4—88no—
644C2×C4—164no—
645C2×C4C2×#12112yes—
646C2×C4C2×#12114yes—
647C2×C4C2×#12112yes—
648C2×C4C2×#13712yes—
649C2×C4C2×#13712yes—
650C2×C4C2×#13712yes—
651C2×C4#57512yes—
652C2×C4#57712yes—
653C2×C4C23×D411yescharacteristic
654C2×C4C22×#4111yescharacteristic
655C2×C4C2×H(5,𝔽2)12yes—
656C2×C4#58512yes—
657C8—42no—
658C8—44no—
659C8—42no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—44no—
668C22—44no—
669C22—44no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22C22×#12111yescharacteristic
675C22#348411yescharacteristic
676C22#350011yescharacteristic
677C22C22×#13711yescharacteristic
678C22C2×#57511yescharacteristic
679C22C2×#57711yescharacteristic
680C22#358611yescharacteristic
681C4—22no—
682C4—22no—
683C4—24no—
684C4—22no—
685C4—22no—
686C4—42no—
687C4—44no—
688C4—41no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—81no—
693C4—84no—
694C4—82no—
695C4—81no—
696C4—82no—
697C4#303714yes—
698C4#348812yes—
699C4#351312yes—
700C2—44no—
701C2—84no—
702C2#3572811yescharacteristic
703C2C2×#303711yescharacteristic
704C2#6042911yescharacteristic
705C2#6158611yescharacteristic
706C2#6160411yescharacteristic
707C2#6254211yescharacteristic
708C2#6259111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3331C2semidirect
3. #34332C2semidirect
4. #29799C2semidirect
5. #29756C2semidirect
6. #28748C2semidirect
7. #28581C2semidirect
8. #28202C2semidirect
9. #25273C2semidirect
10. #25100C2semidirect
11. #24729C2semidirect
12. #24491C2semidirect
13. #24216C2semidirect
14. #23678C2semidirect
15. #23508C2semidirect
16. #22700C2semidirect
17. #22431C2semidirect
101. #554D4semidirect

Maximal subgroups

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

subgroupindex
C2×#33312
#361272
#343322
#297992
#297562
#287482
#285812
#282022
#252732
#251002
#247292
#244912
#242162
#236782
#235082
#227002
#224312