Finite Group Databasebeta

← #6476128

Subgroups — #6476128

Order 512.

1780 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35325C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#32985C212yesmaximal
4#32983C212yesmaximal
5#29976C212yesmaximal, complemented
6#29766C212yesmaximal, complemented
7#29344C212yesmaximal, complemented
8#29310C212yesmaximal, complemented
9#29253C214yesmaximal, complemented
10#27617C212yesmaximal, complemented
11#24606C212yesmaximal, complemented
12#24602C212yesmaximal, complemented
13#24064C212yesmaximal, complemented
14#22094C214yesmaximal, complemented
15#20717C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C2×#484C2212yes—
21C2×#484C2212yes—
22C2×#485C2214yes—
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2855C2212yes—
26#2848C2212yes—
27#2628C2211yescharacteristic
28#2628C2211yescharacteristic
29#2621C2211yescharacteristic
30#2621C2211yescharacteristic
31#2611—21no—
32#2611—21no—
33#2599—21no—
34#2599—21no—
35#2583—21no—
36#2561—21no—
37#2484—21no—
38#2482—21no—
39#2323C2212yes—
40#2323C2212yes—
41#2317—21no—
42#2317—21no—
43#2317—21no—
44#2317—21no—
45#2317C2212yes—
46#2317C2212yes—
47#2317C2212yes—
48#2317C2212yes—
49C2×#389C2212yes—
50C2×#389C2212yes—
51C2×#385—21no—
52C2×#385—21no—
53C2×#384—21no—
54#2283C2211yescharacteristic
55C2×#382—21no—
56#2101C2214yescomplemented
57#2087C2214yescomplemented
58#2086C2214yescomplemented
59#2085C2214yescomplemented
60#2068C2214yescomplemented
61#2066C2214yescomplemented
62#2057C2214yescomplemented
63#2043C2214yescomplemented
64#1944C2214yescomplemented
65#1942C2214yescomplemented
66#1935C2214yescomplemented
67#1933C2214yescomplemented
68#1919C2214yescomplemented
69#1918C2214yescomplemented
70#1873C2214yes—
71#1872C2212yes—
72#1872C2212yes—
73#1872C2214yes—
74#1872C2214yes—
75#1863C2214yescomplemented
76#1863C2214yescomplemented
77#1854C2214yes—
78#1853C2212yes—
79#1852C2214yes—
80#1852C2212yes—
81#1852C2214yes—
82#1825C2214yes—
83#1824C2212yes—
84#1824C2212yes—
85#1817C2212yes—
86#1798C2212yes—
87#1798C2212yes—
88#1779C2212yes—
89#1750C2212yes—
90#1750C2212yes—
91C24×C4—21no—
92C24×C4C2311yescharacteristic
93#533—42no—
94#533—42no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#125—21no—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×C22⋊Q8C2312yes—
102C2×C4×Q8—21no—
103C2×C4×Q8C2312yes—
104C2×#117—21no—
105C2×#117—21no—
106C2×#117—28no—
107C2×#117—21no—
108C2×#117—42no—
109C2×#117—42no—
110C2×#117D411yescharacteristic
111C2×#117D411yescharacteristic
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C2×#117C2312yes—
115C2×#117C2312yes—
116C2×#117C2311yescharacteristic
117C2×#117C2311yescharacteristic
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—22no—
124C22×C4⋊C4—22no—
125C22×C4⋊C4—42no—
126C22×C4⋊C4—42no—
127C22×C4⋊C4D411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2312yes—
132C22×C4⋊C4C2312yes—
133C22×C22⋊C4—21no—
134C22×C22⋊C4D411yescharacteristic
135C22×C22⋊C4D411yescharacteristic
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#485—216no—
140#484—44no—
141#484—44no—
142#432—42no—
143#431—42no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8C2314yes—
155C2×C22⋊C8C2314yes—
156C2×C22⋊C8C2312yes—
157C2×C22⋊C8C2312yes—
158C2×#97C2312yes—
159C2×C4×C8C2312yes—
160#400—21no—
161#400—21no—
162#399—21no—
163#399—21no—
164#398—21no—
165#398—21no—
166#398—21no—
167#398—21no—
168#397—21no—
169#397—21no—
170#397—21no—
171#397—21no—
172#395—21no—
173#395—21no—
174#393—21no—
175#393—21no—
176#391C2312yes—
177#391C2312yes—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—22no—
183#389—22no—
184#389C2314yes—
185#389C2314yes—
186#389C2314yes—
187#389C2314yes—
188#389C2312yes—
189#389C2312yes—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#385—22no—
199#385—22no—
200#385—22no—
201#385—22no—
202#385—22no—
203#385—22no—
204#385—22no—
205#385—22no—
206#385—22no—
207#385—22no—
208#385—22no—
209#385—22no—
210#384—22no—
211#384—22no—
212#384—22no—
213#384—22no—
214#383—21no—
215#382—22no—
216#382—22no—
217#382—22no—
218#382—22no—
219#382C2311yescharacteristic
220#382C2311yescharacteristic
221#381—22no—
222#381—21no—
223#380C2312yes—
224#380C2312yes—
225#380C2312yes—
226#379—22no—
227C4×C4⋊C4C2312yes—
228C4×C22⋊C4—21no—
229#376—21no—
230C2×#95—22no—
231C2×#95—22no—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2314yes—
237#340C2314yes—
238#340C2314yes—
239#340C2314yes—
240#336C2314yes—
241#336C2314yes—
242C25C2×D411yescharacteristic
243C22×Q8—21no—
244C22×Q8C2412yes—
245C23×C4—24no—
246C23×C4—22no—
247C23×C4—22no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—22no—
253C23×C4—44no—
254C23×C4—84no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C23×C4C2411yescharacteristic
264C23×C4C2411yescharacteristic
265C22×C8C2414yes—
266C22×C8C2414yes—
267C4⋊Q8—41no—
268C4⋊Q8—41no—
269C4⋊Q8—41no—
270C4⋊Q8—41no—
271#125—44no—
272C22⋊Q8—28no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—28no—
276C22⋊Q8—44no—
277C4×Q8—42no—
278C4×Q8—42no—
279C4×Q8—44no—
280#117—28no—
281#117—28no—
282#117—24no—
283#117—28no—
284#117—42no—
285#117—42no—
286#117—42no—
287#117—42no—
288#117—41no—
289#117—41no—
290#117—41no—
291#117—41no—
292#117—41no—
293#117—41no—
294#117—41no—
295#117—41no—
296#117—44no—
297#117#4118yes—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—28no—
300C2×C4⋊C4—24no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—42no—
319C2×C4⋊C4—42no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—44no—
326C2×C4⋊C4—44no—
327C2×C4⋊C4—84no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D411yescharacteristic
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2414yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2414yes—
342C2×C22⋊C4—28no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C22⋊C4C2×D411yescharacteristic
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C22⋊C4C2×D411yescharacteristic
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C22⋊C4C2×D411yescharacteristic
361C2×C42—28no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378#107—28no—
379#107—44no—
380#107—44no—
381#106—28no—
382C4⋊C8—42no—
383C4⋊C8—42no—
384#103—28no—
385#103—28no—
386#103—44no—
387#103—44no—
388C22⋊C8—28no—
389C22⋊C8—24no—
390C22⋊C8—24no—
391C22⋊C8—28no—
392#97—42no—
393C4×C8—42no—
394#95—24no—
395#95—24no—
396#95—24no—
397#95—24no—
398#95—22no—
399#95—21no—
400#95—21no—
401#95—21no—
402#95—21no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24—24no—
416C24—42no—
417C24C22×D411yescharacteristic
418C24C22×D411yescharacteristic
419C24C22×D411yescharacteristic
420C2×Q8—24no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—24no—
424C2×Q8—42no—
425C2×Q8—82no—
426C22×C4—216no—
427C22×C4—216no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—28no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—816no—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C2×#4114yes—
482C22×C4H(5,𝔽2)12yes—
483C22×C4H(5,𝔽2)12yes—
484C22×C42−1+414yes—
485C22×C4C2511yescharacteristic
486C2×C8—24no—
487C2×C8—24no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C4⋊C4—28no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—41no—
507C4⋊C4—41no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4C2×#4118yes—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—41no—
528C22⋊C4—41no—
529C22⋊C4—41no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—41no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C42—28no—
542C42—28no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—21no—
548C42—21no—
549C42—21no—
550C42—21no—
551C42—21no—
552C42—21no—
553C42—21no—
554C42—21no—
555C42—44no—
556C42—82no—
557C23—22no—
558C23—21no—
559C23—21no—
560C23—28no—
561C23—22no—
562C23—22no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—22no—
568C23—22no—
569C23—48no—
570C23—48no—
571C23—44no—
572C23—44no—
573C23#52512yes—
574C23#53412yes—
575C23#53612yes—
576C23#57814yes—
577C23C23×D411yescharacteristic
578Q8—44no—
579Q8—44no—
580Q8—84no—
581C2×C4—28no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—416no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—82no—
634C2×C4—82no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4—88no—
638C2×C4#52512yes—
639C2×C4#53412yes—
640C2×C4#53612yes—
641C2×C4C22×#4112yes—
642C2×C4C2×H(5,𝔽2)11yescharacteristic
643C2×C4C2×H(5,𝔽2)11yescharacteristic
644C2×C4C2×2−1+412yes—
645C8—44no—
646C8—44no—
647C22—22no—
648C22—24no—
649C22—24no—
650C22—22no—
651C22—24no—
652C22—24no—
653C22—22no—
654C22—22no—
655C22—416no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—42no—
661C22—42no—
662C22#320214yes—
663C22C2×#52511yescharacteristic
664C22C2×#53411yescharacteristic
665C22C2×#53611yescharacteristic
666C22C2×#57812yes—
667C22#358512yes—
668C4—22no—
669C4—22no—
670C4—22no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—48no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4—82no—
684C4—82no—
685C4#349212yes—
686C4#349212yes—
687C2—24no—
688C2—42no—
689C2—42no—
690C2#3505711yescharacteristic
691C2#6058912yes—
692C2C2×#320211yescharacteristic
693C2#6075311yescharacteristic
694C2#6087412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #29976C2semidirect
6. #29766C2semidirect
7. #29344C2semidirect
8. #29310C2semidirect
9. #29253C2semidirect
10. #27617C2semidirect
11. #24606C2semidirect
12. #24602C2semidirect
13. #24064C2semidirect
14. #22094C2semidirect
56. #2101C22semidirect
57. #2087C22semidirect
58. #2086C22semidirect
59. #2085C22semidirect
60. #2068C22semidirect
61. #2066C22semidirect
62. #2057C22semidirect
63. #2043C22semidirect
64. #1944C22semidirect
65. #1942C22semidirect
66. #1935C22semidirect
67. #1933C22semidirect
68. #1919C22semidirect
69. #1918C22semidirect
75. #1863C22semidirect
76. #1863C22semidirect

Maximal subgroups

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

subgroupindex
#353252
C2×#23172
#329852
#329832
#299762
#297662
#293442
#293102
#292532
#276172
#246062
#246022
#240642
#220942
#207172