Finite Group Databasebeta

← #6752319

Subgroups — #6752319

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36210C211yesmaximal, characteristic
2#36194C211yesmaximal, characteristic
3#33049C211yesmaximal, characteristic, complemented
4#32214C214yesmaximal, complemented
5#32023C214yesmaximal
6#30305C211yesmaximal, characteristic, complemented
7#29845C211yesmaximal, characteristic, complemented
8#29218C211yesmaximal, characteristic, complemented
9#29191C211yesmaximal, characteristic, complemented
10#27975C211yesmaximal, characteristic, complemented
11#27889C211yesmaximal, characteristic, complemented
12#27875C211yesmaximal, characteristic, complemented
13#27875C211yesmaximal, characteristic, complemented
14#27380C214yesmaximal, complemented
15#27356C214yesmaximal, complemented
16#22221C211yesmaximal, characteristic, complemented
17#21984C211yesmaximal, characteristic, complemented
18#21102C211yesmaximal, characteristic
19#20824C211yesmaximal, characteristic, complemented
20C2×#487C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×C4×M16C2211yescharacteristic
23#2868C2211yescharacteristic
24#2859C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27#2857C2211yescharacteristic
28#2857C2211yescharacteristic
29#2571—21no—
30#2563—21no—
31#2562—21no—
32#2538C2212yes—
33#2514—21no—
34#2509C2212yes—
35#2497—21no—
36#2497—21no—
37#2495—21no—
38#2494—21no—
39#2494C2212yes—
40#2492C2212yes—
41#2492C2212yes—
42#2485C2212yes—
43#2439—21no—
44#2439—21no—
45#2421—21no—
46#2415—21no—
47#2370—21no—
48#2369—21no—
49#2362—21no—
50#2355C2212yes—
51#2354—21no—
52#2347C2212yes—
53C4×C4⋊Q8C2211yescharacteristic
54C4×C4⋊Q8C2211yescharacteristic
55#2285C2211yescharacteristic
56#2285C2211yescharacteristic
57#2285C2211yescharacteristic
58#2285C2211yescharacteristic
59C42×D4C2211yescharacteristic
60#2100C2214yes—
61#2084C2214yes—
62#2077C2214yes—
63#2077C2214yes—
64#2074C2214yes—
65#2066C2214yes—
66#2066C2214yes—
67#2058C2214yes—
68#2035C2214yes—
69#2035C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1987C2214yes—
74#1956C2211yescharacteristic
75#1956C2211yescharacteristic
76#1956C2211yescharacteristic
77#1949C2214yes—
78#1935C2214yes—
79#1928C2214yes—
80#1927C2214yes—
81#1924C2212yes—
82#1920C2214yes—
83#1920C2214yes—
84#1917C2212yes—
85#1894C2214yes—
86#1873C2214yes—
87#1870C2212yes—
88#1865C2214yes—
89#1863C2214yes—
90#1854C2214yes—
91#1851C2212yes—
92#1845C2211yescharacteristic
93#1844C2211yescharacteristic
94#1844C2211yescharacteristic
95#1844C2211yescharacteristic
96#1834C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1834C2211yescharacteristic
100#1834C2211yescharacteristic
101#1809C2214yes—
102#1776C2214yes—
103C4×C4⋊C8C2211yescharacteristic
104C4×C22⋊C8C2211yescharacteristic
105C4×C22⋊C8C2211yescharacteristic
106#1744C2211yescharacteristic
107C22×M16C2312yes—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2312yes—
113C2×#125—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2312yes—
118C2×C4×Q8—21no—
119C2×C4×Q8C2312yes—
120C2×C4×D4D411yescharacteristic
121C2×C4×D4D411yescharacteristic
122C2×C4×D4C2311yescharacteristic
123C2×#117—21no—
124C2×#117—21no—
125C2×#117—21no—
126C2×#117C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#487—24no—
130#431—24no—
131C2×#107C2314yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2312yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8C2312yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C4×M16—24no—
145C2×#97C2312yes—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150#401—22no—
151#400—22no—
152#400—22no—
153#400—22no—
154#400—22no—
155#397—22no—
156#397—22no—
157#396—22no—
158#396—22no—
159#394—22no—
160#393—22no—
161#393—22no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—22no—
166#391—22no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#390—22no—
171#389—22no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389—21no—
176#389—21no—
177#389—21no—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#389C2312yes—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#388—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#387—22no—
199#387—22no—
200#384—21no—
201#384—21no—
202#384C2311yescharacteristic
203#384C2311yescharacteristic
204#384C2312yes—
205#384C2311yescharacteristic
206#384C2311yescharacteristic
207#384C2311yescharacteristic
208#384C2311yescharacteristic
209#383—22no—
210#382—22no—
211#382—21no—
212#382—21no—
213#382—21no—
214#382—21no—
215#382C2312yes—
216#380—21no—
217#380—21no—
218#380—21no—
219#380—21no—
220#380—21no—
221#380—21no—
222#380—21no—
223#380—21no—
224#380—21no—
225#380—21no—
226#380—21no—
227#380C2312yes—
228#380C2312yes—
229#380C2312yes—
230#380C2312yes—
231#380C2312yes—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—22no—
242C4×C4⋊C4D411yescharacteristic
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4C2312yes—
245C4×C4⋊C4C2312yes—
246C4×C4⋊C4C2312yes—
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2311yescharacteristic
249C4×C22⋊C4D411yescharacteristic
250C4×C22⋊C4D411yescharacteristic
251C4×C22⋊C4D411yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4C2311yescharacteristic
254C4×C22⋊C4C2311yescharacteristic
255C43C2311yescharacteristic
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#336C2312yes—
265#336C2312yes—
266#336C2312yes—
267#336C2312yes—
268#336C2312yes—
269C22×Q8—21no—
270C22×Q8—22no—
271C22×Q8C2412yes—
272C22×D4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C23×C4C2411yescharacteristic
279C2×M16—24no—
280C2×M16—42no—
281C2×M16—42no—
282C22×C8C2412yes—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—44no—
291C4⋊Q8—44no—
292C4⋊Q8—82no—
293#125—44no—
294C22⋊Q8—44no—
295C22⋊Q8—44no—
296C22⋊Q8—44no—
297C22⋊Q8—82no—
298C4×Q8—44no—
299C4×Q8—82no—
300C4×D4—24no—
301C4×D4—42no—
302C4×D4—42no—
303#117—44no—
304#117—82no—
305#117—82no—
306#117—82no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
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—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
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⋊C4C2×D411yescharacteristic
352C2×C22⋊C4C2×D411yescharacteristic
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C22⋊C4C2×D411yescharacteristic
355C2×C22⋊C4C2×D411yescharacteristic
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—44no—
381#106—44no—
382#106—44no—
383#106—44no—
384C4⋊C8—24no—
385C4⋊C8—42no—
386#103—28no—
387#103—44no—
388#103—44no—
389#103—44no—
390C22⋊C8—24no—
391C22⋊C8—42no—
392C22⋊C8—42no—
393#97—24no—
394C4×C8—22no—
395C4×C8—22no—
396C4×C8—22no—
397C4×C8#4114yes—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
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—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95C2×D411yescharacteristic
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426C24C22×D411yescharacteristic
427C24C22×D411yescharacteristic
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—44no—
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—82no—
437C2×Q8—82no—
438C2×D4—22no—
439C2×D4—41no—
440C2×D4—41no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
482M16—48no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—22no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8C2×#4114yes—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C22⋊C4—22no—
533C22⋊C4—22no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C22⋊C4—41no—
537C22⋊C4—41no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C22⋊C4—82no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—22no—
558C42—22no—
559C42—44no—
560C42—44no—
561C42—82no—
562C42—82no—
563C42—82no—
564C42—82no—
565C42#13714yes—
566C42C2×#4112yes—
567C422−1+412yes—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23C23×D411yescharacteristic
583Q8—44no—
584Q8—44no—
585Q8—84no—
586Q8—164no—
587D4—44no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
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—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4#54712yes—
649C2×C4#54712yes—
650C2×C4C2×#13712yes—
651C2×C4C2×#13712yes—
652C2×C4C2×#13712yes—
653C2×C4#57512yes—
654C2×C4#57712yes—
655C2×C4#57712yes—
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×2−1+411yescharacteristic
658C2×C4#58512yes—
659C2×C4#58512yes—
660C8—24no—
661C8—42no—
662C8—42no—
663C8—42no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22#344911yescharacteristic
679C22C2×#54711yescharacteristic
680C22#350011yescharacteristic
681C22C22×#13711yescharacteristic
682C22C2×#57511yescharacteristic
683C22C2×#57711yescharacteristic
684C22C2×#57711yescharacteristic
685C4—22no—
686C4—22no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—41no—
693C4—41no—
694C4—41no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4#335014yes—
704C4#335614yes—
705C4#349312yes—
706C4#356512yes—
707C2—81no—
708C2—81no—
709C2#3787111yescharacteristic
710C2#6096611yescharacteristic
711C2#6101511yescharacteristic
712C2C2×#335011yescharacteristic
713C2C2×#335611yescharacteristic
714C2#6254011yescharacteristic
715C2#6254411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
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
3. #33049C2semidirect
4. #32214C2semidirect
6. #30305C2semidirect
7. #29845C2semidirect
8. #29218C2semidirect
9. #29191C2semidirect
10. #27975C2semidirect
11. #27889C2semidirect
12. #27875C2semidirect
13. #27875C2semidirect
14. #27380C2semidirect
15. #27356C2semidirect
16. #22221C2semidirect
17. #21984C2semidirect
19. #20824C2semidirect

Maximal subgroups

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

subgroupindex
#362102
#361942
#330492
#322142
#320232
#303052
#298452
#292182
#291912
#279752
#278892
#278752
#273802
#273562
#222212
#219842
#211022
#208242