Finite Group Databasebeta

← #6493831

Subgroups — #6493831

Order 512.

1576 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35294C211yesmaximal, characteristic
2#35281C211yesmaximal, characteristic
3#33287C211yesmaximal, characteristic, complemented
4#33278C211yesmaximal, characteristic, complemented
5#33232C211yesmaximal, characteristic, complemented
6#33225C211yesmaximal, characteristic, complemented
7#30220C211yesmaximal, characteristic, complemented
8#30220C211yesmaximal, characteristic, complemented
9#30206C211yesmaximal, characteristic, complemented
10#30205C211yesmaximal, characteristic, complemented
11#27782C214yesmaximal, complemented
12#27762C214yesmaximal, complemented
13#27283C214yesmaximal
14#27277C214yesmaximal
15#21864C211yesmaximal, characteristic
16#21804C211yesmaximal, characteristic
17#21409C211yesmaximal, characteristic
18#21202C211yesmaximal, characteristic
19#20472C211yesmaximal, characteristic
20C2×#533C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C2×#491C2214yes—
23C2×#487C2214yes—
24C2×#431C2211yescharacteristic
25C2×#419C2214yes—
26C2×C4×M16C2211yescharacteristic
27#2868C2211yescharacteristic
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2862C2211yescharacteristic
32#2859C2211yescharacteristic
33#2859C2211yescharacteristic
34#2859C2211yescharacteristic
35#2660—21no—
36#2660—21no—
37#2629—21no—
38#2628—21no—
39#2570—21no—
40#2570C2212yes—
41#2567—21no—
42#2567—21no—
43#2550—21no—
44#2550C2212yes—
45#2546C2212yes—
46#2546C2212yes—
47#2502—21no—
48#2502—21no—
49#2441—21no—
50#2441C2212yes—
51#2439—21no—
52#2439C2212yes—
53#2434C2212yes—
54#2434C2212yes—
55#2331—21no—
56#2330—21no—
57#2317—21no—
58#2317—21no—
59#2285C2211yescharacteristic
60#2285C2211yescharacteristic
61C4×#117C2211yescharacteristic
62#2074C2214yes—
63#2072C2214yes—
64#2043C2214yes—
65#2043C2214yes—
66#2043C2214yes—
67#2043C2214yes—
68#2026C2214yes—
69#2024C2214yes—
70#1990C2211yescharacteristic
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1981C2214yes—
75#1960C2214yes—
76#1960C2214yes—
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1956C2214yes—
88#1956C2211yescharacteristic
89#1956C2211yescharacteristic
90#1935C2214yes—
91#1935C2214yes—
92#1933C2214yes—
93#1933C2214yes—
94#1894C2214yes—
95#1873C2214yes—
96#1805C2214yes—
97#1787C2214yes—
98#1771C2212yes—
99#1771C2212yes—
100#1763C2214yes—
101#1753C2212yes—
102#1753C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×Q8C2312yes—
108C22×M16C2312yes—
109#533—24no—
110#533—24no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8—21no—
115C2×C4⋊Q8—24no—
116C2×C4⋊Q8—24no—
117C2×C4⋊Q8C2311yescharacteristic
118C2×C4⋊Q8C2311yescharacteristic
119C2×C4⋊Q8C2311yescharacteristic
120C2×C4⋊Q8C2312yes—
121C2×C4⋊Q8C2312yes—
122C2×C4⋊Q8C2312yes—
123C2×C4⋊Q8C2312yes—
124C2×C4⋊Q8C2312yes—
125C2×C4⋊Q8C2311yescharacteristic
126C2×C4⋊Q8C2312yes—
127C2×#125—21no—
128C2×#125—21no—
129C2×#125—21no—
130C2×#125C2312yes—
131C2×C22⋊Q8—21no—
132C2×C22⋊Q8C2312yes—
133C2×C4×Q8—21no—
134C2×C4×Q8C2312yes—
135C2×#117—21no—
136C2×#117—21no—
137C2×#117D411yescharacteristic
138C2×#117D411yescharacteristic
139C2×#117C2312yes—
140C22×C4⋊C4—21no—
141C22×C4⋊C4—21no—
142C22×C42C2311yescharacteristic
143#491—48no—
144#487—48no—
145#431—24no—
146C2×C4⋊C8C2312yes—
147#419—48no—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×C22⋊C8C2312yes—
161C4×M16—24no—
162C2×#97C2312yes—
163C2×#97C2312yes—
164C2×#97C2312yes—
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2311yescharacteristic
167C2×C4×C8C2312yes—
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2312yes—
171#400—22no—
172#399—22no—
173#398—22no—
174#398—22no—
175#398—22no—
176#398—22no—
177#398—22no—
178#398—22no—
179#396—22no—
180#396—22no—
181#395—22no—
182#395—22no—
183#395—22no—
184#395—22no—
185#395—22no—
186#393—22no—
187#391—21no—
188#391—21no—
189#391—21no—
190#391—21no—
191#391—22no—
192#391—22no—
193#391C2312yes—
194#391C2312yes—
195#391C2312yes—
196#391C2312yes—
197#389—22no—
198#389—22no—
199#389—22no—
200#389—22no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—22no—
210#389—22no—
211#389C2312yes—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#385—22no—
220#385—22no—
221#385—22no—
222#385—22no—
223#384—21no—
224#384—21no—
225#384—21no—
226#384—21no—
227#384—21no—
228#384—21no—
229#384—21no—
230#384—21no—
231#384—21no—
232#384—21no—
233#384C2312yes—
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#382—21no—
239#382—21no—
240#382—21no—
241#382—21no—
242#382—21no—
243#382—21no—
244#382—21no—
245#382C2312yes—
246#382C2312yes—
247#380—21no—
248#380—21no—
249#380—21no—
250#380C2312yes—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2311yescharacteristic
258C4×C4⋊C4C2311yescharacteristic
259C4×C22⋊C4D411yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4C2311yescharacteristic
262C4×C22⋊C4C2311yescharacteristic
263#376D411yescharacteristic
264#376D411yescharacteristic
265C43C2311yescharacteristic
266C43C2311yescharacteristic
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8—22no—
278C22×Q8—24no—
279C22×Q8—24no—
280C22×Q8—44no—
281C22×Q8—44no—
282C22×Q8C2412yes—
283C22×Q8C2412yes—
284C22×Q8C2412yes—
285C23×C4—22no—
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2411yescharacteristic
289C2×M16—24no—
290C2×M16—24no—
291C2×M16—42no—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—24no—
303C4⋊Q8—24no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308C4⋊Q8—44no—
309C4⋊Q8—44no—
310C4⋊Q8—44no—
311C4⋊Q8—44no—
312C4⋊Q8—44no—
313C4⋊Q8—44no—
314C4⋊Q8—82no—
315#125—44no—
316#125—44no—
317#125—44no—
318#125—82no—
319C22⋊Q8—24no—
320C22⋊Q8—24no—
321C22⋊Q8—44no—
322C4×Q8—24no—
323C4×Q8—24no—
324C4×Q8—44no—
325#117—24no—
326#117—24no—
327#117—44no—
328#117—44no—
329#117—82no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—22no—
361C2×C4⋊C4—22no—
362C2×C4⋊C4—22no—
363C2×C4⋊C4—22no—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—22no—
366C2×C4⋊C4—44no—
367C2×C4⋊C4—44no—
368C2×C4⋊C4C2×D411yescharacteristic
369C2×C4⋊C4C2×D411yescharacteristic
370C2×C4⋊C4C2×D411yescharacteristic
371C2×C4⋊C4C2×D411yescharacteristic
372C2×C4⋊C4C2412yes—
373C2×C4⋊C4C2412yes—
374C2×C4⋊C4C2412yes—
375C2×C4⋊C4C2412yes—
376C2×C4⋊C4C2412yes—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4C2×D411yescharacteristic
381C2×C22⋊C4C2×D411yescharacteristic
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C42—22no—
385C2×C42—22no—
386C2×C42—22no—
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×C42#4114yes—
398C2×C42#4114yes—
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C4⋊C8—24no—
406#103—28no—
407#103—28no—
408#103—28no—
409#103—44no—
410#103—44no—
411#103—44no—
412#103—44no—
413#103—44no—
414#103—44no—
415#103—44no—
416#103—44no—
417#103—44no—
418C22⋊C8—24no—
419#97—24no—
420#97—24no—
421#97—42no—
422C4×C8—24no—
423C4×C8—22no—
424C4×C8—22no—
425C4×C8—22no—
426C4×C8—22no—
427C4×C8—42no—
428#95—22no—
429#95—22no—
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—
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454C24C22×D411yescharacteristic
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—24no—
462C2×Q8—24no—
463C2×Q8—24no—
464C2×Q8—24no—
465C2×Q8—24no—
466C2×Q8—24no—
467C2×Q8—416no—
468C2×Q8—42no—
469C2×Q8—44no—
470C2×Q8—44no—
471C2×Q8—42no—
472C2×Q8—42no—
473C2×Q8—82no—
474C2×Q8—82no—
475C2×Q8—82no—
476C2×Q8—82no—
477C2×Q8—88no—
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—22no—
492C22×C4—44no—
493C22×C4—82no—
494C22×C4—82no—
495C22×C4#12414yes—
496C22×C4#12414yes—
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2511yescharacteristic
513M16—48no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8H(5,𝔽2)14yes—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—24no—
536C4⋊C4—44no—
537C4⋊C4—44no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—42no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C4⋊C4—82no—
562C4⋊C4—82no—
563C4⋊C4—82no—
564C4⋊C4—82no—
565C4⋊C4—82no—
566C4⋊C4—82no—
567C4⋊C4—82no—
568C4⋊C4—82no—
569C4⋊C4—82no—
570C4⋊C4—82no—
571C4⋊C4—82no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—22no—
576C22⋊C4—44no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—44no—
590C42—82no—
591C42—82no—
592C42#12414yes—
593C42#12414yes—
594C42#13714yes—
595C42#13714yes—
596C42C2×#4112yes—
597C42C2×#4112yes—
598C42C2×#4112yes—
599C42C2×#4112yes—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—21no—
606C23C2×#12412yes—
607C23C23×D411yescharacteristic
608Q8—44no—
609Q8—44no—
610Q8—44no—
611Q8—44no—
612Q8—44no—
613Q8—44no—
614Q8—88no—
615Q8—84no—
616Q8—164no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—28no—
632C2×C4—28no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—88no—
672C2×C4C2×#12412yes—
673C2×C4C2×#12412yes—
674C2×C4C2×#12412yes—
675C2×C4C2×#12412yes—
676C2×C4C2×#12412yes—
677C2×C4C2×#13712yes—
678C2×C4C2×#13712yes—
679C2×C4C2×#13712yes—
680C2×C4C2×#13712yes—
681C2×C4C2×#13712yes—
682C2×C4C2×#13712yes—
683C2×C4#57712yes—
684C2×C4#57812yes—
685C2×C4C22×#4111yescharacteristic
686C2×C4C22×#4111yescharacteristic
687C2×C4C2×H(5,𝔽2)12yes—
688C2×C4#58512yes—
689C8—24no—
690C8—42no—
691C8—42no—
692C8—42no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22C22×#12411yescharacteristic
701C22#349811yescharacteristic
702C22#350011yescharacteristic
703C22C22×#13711yescharacteristic
704C22C22×#13711yescharacteristic
705C22C2×#57711yescharacteristic
706C22C2×#57811yescharacteristic
707C4—22no—
708C4—22no—
709C4—22no—
710C4—22no—
711C4—22no—
712C4—22no—
713C4—22no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4#313714yes—
727C4#313914yes—
728C4#346012yes—
729C4#349412yes—
730C2—42no—
731C2#3526411yescharacteristic
732C2C2×#313711yescharacteristic
733C2C2×#313911yescharacteristic
734C2#6242211yescharacteristic
735C2#6242411yescharacteristic
736C2#6253611yescharacteristic
737C2#6254011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #33287C2semidirect
4. #33278C2semidirect
5. #33232C2semidirect
6. #33225C2semidirect
7. #30220C2semidirect
8. #30220C2semidirect
9. #30206C2semidirect
10. #30205C2semidirect
11. #27782C2semidirect
12. #27762C2semidirect

Maximal subgroups

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

subgroupindex
#352942
#352812
#332872
#332782
#332322
#332252
#302202
#302062
#302052
#277822
#277622
#272832
#272772
#218642
#218042
#214092
#212022
#204722