Finite Group Databasebeta

← #6498607

Subgroups — #6498607

Order 512.

1902 conjugacy classes of subgroups.

The subgroup lattice has 769 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3252C212yesmaximal, complemented
2#36631C211yesmaximal, characteristic
3C4×#533C211yesmaximal, characteristic, complemented
4#33029C214yesmaximal, complemented
5#29871C212yesmaximal, complemented
6#29801C212yesmaximal, complemented
7#28582C214yesmaximal, complemented
8#28581C212yesmaximal, complemented
9#28580C212yesmaximal
10#28206C211yesmaximal, characteristic, complemented
11#28204C212yesmaximal, complemented
12#28202C211yesmaximal, characteristic, complemented
13#28161C211yesmaximal, characteristic, complemented
14#28159C212yesmaximal, complemented
15#28157C211yesmaximal, characteristic, complemented
16#21890C212yesmaximal, complemented
17#20752C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2212yes—
20#3252—216no—
21C2×#479C2214yes—
22C2×#477C2212yes—
23C2×#476C2212yes—
24C2×#470C2214yes—
25C2×#468C2214yes—
26C2×#464C2214yes—
27C2×#462C2212yes—
28C2×#461C2212yes—
29C2×#423C2212yes—
30#2861C2212yes—
31#2860—21no—
32#2860—21no—
33#2859C2211yescharacteristic
34#2858C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2854—21no—
38#2854—21no—
39#2850C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125—21no—
47C4×#125—21no—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50C4×C22⋊Q8—21no—
51C4×C22⋊Q8C2212yes—
52C4×C22⋊Q8C2212yes—
53C4×#121—21no—
54C4×#121—21no—
55#2296—21no—
56#2295—21no—
57#2295C2212yes—
58#2293—21no—
59#2293—21no—
60#2285C2211yescharacteristic
61C42×Q8—21no—
62C42×Q8C2212yes—
63C4×#117—21no—
64C2×C43C2211yescharacteristic
65#2085C2214yes—
66#2084C2212yes—
67#2083C2212yes—
68#2074C2214yes—
69#2072C2212yes—
70#2072C2212yes—
71#2043C2212yes—
72#2042C2214yes—
73#2041C2212yes—
74#2016C2212yes—
75#2015C2214yes—
76#2014C2212yes—
77#1989C2214yes—
78#1983C2214yes—
79#1928C2214yes—
80#1927C2214yes—
81#1926C2214yes—
82#1925C2214yes—
83#1895C2214yes—
84#1894C2214yes—
85#1880C2214yes—
86#1880C2214yes—
87#1878C2214yes—
88#1878C2214yes—
89#1831C2211yescharacteristic
90#1831C2212yes—
91#1831C2212yes—
92#1831C2211yescharacteristic
93#1831C2212yes—
94#1803C2212yes—
95#1803C2212yes—
96#1802C2212yes—
97#1802C2212yes—
98#1728C2212yes—
99#1728C2212yes—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102C22×M16C2314yes—
103#533—24no—
104#533D418yescomplemented
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127C2311yescharacteristic
109#529—28no—
110#529—28no—
111C2×#126—22no—
112C2×#125—21no—
113C2×#125C2312yes—
114C2×#124C2312yes—
115C2×#124C2312yes—
116C2×#123—22no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8C2312yes—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121C2312yes—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×D4—22no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117—22no—
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#479—216no—
145#477—28no—
146#476—28no—
147#470—216no—
148#468—216no—
149#464—216no—
150#462—28no—
151#461—28no—
152C2×#107C2314yes—
153C2×#106C2314yes—
154#423—28no—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2314yes—
157C2×C4⋊C8C2312yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#103C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166#391—21no—
167#391—21no—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390C2312yes—
181#390C2312yes—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389C2312yes—
197#389C2312yes—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#386—22no—
211#386—22no—
212#385—22no—
213#385—22no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#382—21no—
219#382C2312yes—
220#380—21no—
221#380—21no—
222#380—21no—
223#380C2312yes—
224C4×C4⋊C4—21no—
225C4×C4⋊C4—21no—
226C4×C4⋊C4—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4C2312yes—
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2312yes—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—22no—
246#376—22no—
247C43—22no—
248C43D411yescharacteristic
249C43D411yescharacteristic
250C43D411yescharacteristic
251C43D411yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#336C2312yes—
261#336C2312yes—
262#336C2314yes—
263#336C2312yes—
264#336C2314yes—
265#336C2312yes—
266C22×Q8—21no—
267C22×Q8C2412yes—
268C22×Q8C2412yes—
269C22×D4—22no—
270C22×D4—21no—
271C22×D4—21no—
272C22×D4C2412yes—
273C23×C4—22no—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C2×Q16—28no—
282C2×Q16—44no—
283C2×Q16—44no—
284C2×SD16—28no—
285C2×SD16—44no—
286C2×SD16—44no—
287C2×M16—28no—
288C2×M16—44no—
289C2×M16—44no—
290C22×C8C2414yes—
291C22×C8C2414yes—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—42no—
295C4⋊Q8C2×D414yes—
296C4⋊Q8C2×D414yes—
297#127—42no—
298#126—48no—
299#125—28no—
300#125—24no—
301#125—24no—
302#124—24no—
303#124—24no—
304#124—42no—
305#124—42no—
306#123—48no—
307C22⋊Q8—28no—
308C22⋊Q8—24no—
309C22⋊Q8—24no—
310C22⋊Q8—24no—
311C22⋊Q8C2×D418yes—
312#121—24no—
313#121—24no—
314#121—44no—
315#121—44no—
316C4×Q8—28no—
317C4×Q8—24no—
318C4×Q8—24no—
319C4×Q8—24no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8C2×D418yes—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—44no—
328C4×D4—48no—
329C4×D4—44no—
330#117—28no—
331#117—44no—
332#117—48no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
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—21no—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C42—22no—
375C2×C42—24no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—42no—
383C2×C42—42no—
384C2×C42—42no—
385C2×C42—42no—
386C2×C42—42no—
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2×D411yescharacteristic
405C2×C42C2×D411yescharacteristic
406C2×C42C2×D411yescharacteristic
407C2×C42C2×D411yescharacteristic
408C2×C42C2×D414yes—
409C2×C42C2×D411yescharacteristic
410C2×C42#4114yes—
411C2×C42C2411yescharacteristic
412C2×C42C2411yescharacteristic
413C2×C42C2411yescharacteristic
414C2×C42C2411yescharacteristic
415C2×C42C2411yescharacteristic
416C2×C42C2411yescharacteristic
417#107—28no—
418#106—28no—
419C4⋊C8—24no—
420C4⋊C8—24no—
421C4⋊C8—28no—
422#103—28no—
423#103—28no—
424#103—28no—
425#103—44no—
426#103—44no—
427#102—28no—
428#102—44no—
429#102—44no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442#95—22no—
443#95—22no—
444#95—22no—
445#95—22no—
446C24—22no—
447C24C22×D411yescharacteristic
448C2×Q8—24no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8C22×D414yes—
461C2×D4—22no—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—42no—
468C2×D4—42no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—44no—
474C2×D4—42no—
475C2×D4—42no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—44no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4#12114yes—
505C22×C4#12114yes—
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D412yes—
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C2×#4112yes—
521C22×C4C2511yescharacteristic
522Q16—88no—
523SD16—88no—
524M16—88no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—24no—
530C2×C8—24no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—28no—
535C4⋊C4—28no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—22no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—44no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—44no—
554C4⋊C4—44no—
555C4⋊C4—44no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—42no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4C22×D414yes—
562C22⋊C4—28no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C42—28no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—41no—
590C42—41no—
591C42—41no—
592C42—41no—
593C42—41no—
594C42—41no—
595C42—41no—
596C42—41no—
597C42—41no—
598C42—41no—
599C42—44no—
600C42—44no—
601C42—44no—
602C42—44no—
603C42—44no—
604C42—44no—
605C42#12114yes—
606C42#12114yes—
607C42#13714yes—
608C42C22×D412yes—
609C42C22×D412yes—
610C42C2×#4112yes—
611C42C2×#4112yes—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—21no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23—42no—
625C23C2×#12112yes—
626C23C23×D411yescharacteristic
627Q8—28no—
628Q8—48no—
629Q8—44no—
630Q8—42no—
631Q8—42no—
632Q8—42no—
633Q8—42no—
634D4—42no—
635D4—42no—
636D4—42no—
637D4—42no—
638D4—84no—
639D4—84no—
640D4—88no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—24no—
658C2×C4—24no—
659C2×C4—28no—
660C2×C4—24no—
661C2×C4—24no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—21no—
671C2×C4—21no—
672C2×C4—21no—
673C2×C4—21no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—88no—
699C2×C4C2×#12112yes—
700C2×C4C2×#12112yes—
701C2×C4C2×#12112yes—
702C2×C4C2×#12112yes—
703C2×C4C2×#12112yes—
704C2×C4C2×#13712yes—
705C2×C4C2×#13712yes—
706C2×C4C2×#13712yes—
707C2×C4#57512yes—
708C2×C4#57512yes—
709C2×C4#57512yes—
710C2×C4C23×D411yescharacteristic
711C2×C4C22×#4111yescharacteristic
712C2×C4#58512yes—
713C2×C4#58512yes—
714C8—44no—
715C8—44no—
716C22—22no—
717C22—22no—
718C22—22no—
719C22—41no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—82no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22C22×#12111yescharacteristic
731C22#344911yescharacteristic
732C22#344911yescharacteristic
733C22C22×#13711yescharacteristic
734C22C2×#57511yescharacteristic
735C22C2×#57511yescharacteristic
736C22C2×#57511yescharacteristic
737C4—22no—
738C4—22no—
739C4—22no—
740C4—22no—
741C4—22no—
742C4—22no—
743C4—22no—
744C4—22no—
745C4—44no—
746C4—41no—
747C4—41no—
748C4—41no—
749C4—41no—
750C4—44no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4#303714yes—
757C4#350012yes—
758C4#351212yes—
759C2—42no—
760C2—162no—
761C2#3532811yescharacteristic
762C2C2×#303711yescharacteristic
763C2#6041511yescharacteristic
764C2#6095011yescharacteristic
765C2#6095311yescharacteristic
766C2#6096611yescharacteristic
767C2#6097011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3252C2semidirect
3. C4×#533C2semidirect
4. #33029C2semidirect
5. #29871C2semidirect
6. #29801C2semidirect
7. #28582C2semidirect
8. #28581C2semidirect
10. #28206C2semidirect
11. #28204C2semidirect
12. #28202C2semidirect
13. #28161C2semidirect
14. #28159C2semidirect
15. #28157C2semidirect
16. #21890C2semidirect
17. #20752C2semidirect
104. #533D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32522
#366312
C4×#5332
#330292
#298712
#298012
#285822
#285812
#285802
#282062
#282042
#282022
#281612
#281592
#281572
#218902
#207522