Finite Group Databasebeta

← #6497755

Subgroups — #6497755

Order 512.

1335 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#35291C211yesmaximal, characteristic
2#35283C211yesmaximal, characteristic
3#33293C211yesmaximal, characteristic, complemented
4#33287C211yesmaximal, characteristic, complemented
5#33212C211yesmaximal, characteristic
6#33049C211yesmaximal, characteristic
7#30305C211yesmaximal, characteristic, complemented
8#30281C211yesmaximal, characteristic, complemented
9#29877C211yesmaximal, characteristic
10#29845C211yesmaximal, characteristic
11#28963C214yesmaximal, complemented
12#28863C214yesmaximal
13#28070C214yesmaximal
14#28042C214yesmaximal, complemented
15#22248C211yesmaximal, characteristic, complemented
16#22218C211yesmaximal, characteristic, complemented
17C4×#498C211yesmaximal, characteristic, complemented
18C4×#487C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic
20C2×#500C2211yescharacteristic
21C2×#498C2211yescharacteristic
22C2×#487C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26#2868C2211yescharacteristic
27#2862C2211yescharacteristic
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2857C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2647—21no—
35#2645—21no—
36#2628—21no—
37#2625—21no—
38#2514—21no—
39#2511—21no—
40#2511C2212yes—
41#2509C2212yes—
42#2508C2212yes—
43#2495C2212yes—
44#2494C2212yes—
45#2492—21no—
46#2492C2212yes—
47#2489—21no—
48#2453—21no—
49#2451—21no—
50#2451C2212yes—
51#2423—21no—
52#2423C2212yes—
53#2421—21no—
54#2331—21no—
55#2323—21no—
56#2320—21no—
57#2314—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2097C2214yes—
64#2097C2214yes—
65#2088C2214yes—
66#2084C2214yes—
67#2077C2214yes—
68#2066C2214yes—
69#2058C2214yes—
70#2035C2214yes—
71#1993C2214yes—
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1944C2214yes—
81#1941C2214yes—
82#1935C2214yes—
83#1935C2214yes—
84#1935C2214yes—
85#1933C2214yes—
86#1927C2214yes—
87#1921C2214yes—
88#1921C2214yes—
89#1894C2214yes—
90#1865C2214yes—
91#1844C2211yescharacteristic
92#1844C2211yescharacteristic
93#1836C2214yes—
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1809C2214yes—
97C4×#106C2214yes—
98#1771C2212yes—
99#1771C2212yes—
100C4×#103C2214yes—
101#1753C2212yes—
102#1753C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×M16C2312yes—
108C2×C4⋊Q8—21no—
109C2×C4⋊Q8—21no—
110C2×C4⋊Q8C2311yescharacteristic
111C2×C4⋊Q8C2311yescharacteristic
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×#125—21no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8C2312yes—
124C2×#117D411yescharacteristic
125C2×#117D411yescharacteristic
126C22×C42C2311yescharacteristic
127#500—24no—
128#498—24no—
129#487—24no—
130#487—24no—
131#431—24no—
132C2×#107C2314yes—
133C2×#106C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136C2×C4⋊C8C2312yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×C22⋊C8C2312yes—
142C4×M16—24no—
143C2×#97C2312yes—
144C2×#97C2312yes—
145C2×#97C2312yes—
146C2×C4×C8C2311yescharacteristic
147C2×C4×C8C2311yescharacteristic
148C2×C4×C8C2312yes—
149C2×C4×C8C2312yes—
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152#401—22no—
153#400—22no—
154#400—22no—
155#400—22no—
156#400—22no—
157#400—22no—
158#400—22no—
159#400—22no—
160#399—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#397—22no—
165#397—22no—
166#395—22no—
167#393—22no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—21no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391—21no—
180#391—22no—
181#391—22no—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#389—22no—
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#389—21no—
197#389—21no—
198#389—21no—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#389C2312yes—
206#389C2312yes—
207#388—22no—
208#388—22no—
209#388—22no—
210#388—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#384—21no—
219#384—21no—
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#384C2311yescharacteristic
223#384C2311yescharacteristic
224#384C2312yes—
225#383—22no—
226#382—21no—
227#382—21no—
228#382—21no—
229#382—22no—
230#382C2312yes—
231#382C2312yes—
232#380—21no—
233#380—21no—
234#380—21no—
235#380—21no—
236#380—21no—
237#380C2312yes—
238#380C2312yes—
239#380C2312yes—
240C4×C4⋊C4—21no—
241C4×C4⋊C4—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2311yescharacteristic
251C4×C4⋊C4C2311yescharacteristic
252C4×C22⋊C4D411yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4C2311yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256#376D411yescharacteristic
257#376D411yescharacteristic
258C43C2311yescharacteristic
259C43C2311yescharacteristic
260#340C2314yes—
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272C22×Q8—21no—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—22no—
276C22×Q8C2412yes—
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2411yescharacteristic
280C2×M16—24no—
281C2×M16—24no—
282C2×M16—42no—
283C22×C8C2412yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C22×C8C2412yes—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292C4⋊Q8—22no—
293C4⋊Q8—22no—
294C4⋊Q8—22no—
295C4⋊Q8—44no—
296C4⋊Q8—44no—
297C4⋊Q8—82no—
298#125—82no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301C22⋊Q8—82no—
302C22⋊Q8—82no—
303C4×Q8—44no—
304C4×Q8—44no—
305C4×Q8—82no—
306C4×Q8—82no—
307#117—24no—
308#117—42no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
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⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4C2×D411yescharacteristic
351C2×C22⋊C4C2×D411yescharacteristic
352C2×C22⋊C4C2×D411yescharacteristic
353C2×C22⋊C4C2×D411yescharacteristic
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2411yescharacteristic
371C2×C42C2411yescharacteristic
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—28no—
377#106—28no—
378#106—44no—
379#106—44no—
380C4⋊C8—24no—
381#103—28no—
382#103—28no—
383#103—28no—
384#103—44no—
385C22⋊C8—24no—
386#97—24no—
387#97—24no—
388#97—42no—
389C4×C8—24no—
390C4×C8—22no—
391C4×C8—22no—
392C4×C8—42no—
393C4×C8#4114yes—
394C4×C8#4114yes—
395#95—22no—
396#95—22no—
397#95—22no—
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#95—22no—
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D411yescharacteristic
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426#95C2×D411yescharacteristic
427#95C2×D411yescharacteristic
428#95C2×D411yescharacteristic
429C24C22×D411yescharacteristic
430C2×Q8—22no—
431C2×Q8—22no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—44no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×Q8—82no—
445C2×Q8—82no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4C22×D411yescharacteristic
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D411yescharacteristic
468C22×C4C22×D411yescharacteristic
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×C4C2511yescharacteristic
478M16—48no—
479C2×C8—22no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8C2×#4114yes—
489C2×C8C2×#4114yes—
490C2×C82−1+414yes—
491C4⋊C4—22no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—24no—
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—24no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—42no—
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—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—82no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C22⋊C4—22no—
532C22⋊C4—22no—
533C22⋊C4—41no—
534C22⋊C4—41no—
535C22⋊C4—44no—
536C22⋊C4—82no—
537C22⋊C4—82no—
538C22⋊C4—82no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—22no—
543C42—22no—
544C42—22no—
545C42—22no—
546C42—22no—
547C42—22no—
548C42—21no—
549C42—21no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—44no—
555C42—82no—
556C42—82no—
557C42—82no—
558C42#13714yes—
559C42C2×#4112yes—
560C42C2×#4112yes—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23C23×D411yescharacteristic
569Q8—44no—
570Q8—44no—
571Q8—84no—
572Q8—84no—
573Q8—84no—
574Q8—164no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
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—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
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#53212yes—
635C2×C4#53212yes—
636C2×C4C2×#13712yes—
637C2×C4C2×#13712yes—
638C2×C4C2×#13712yes—
639C2×C4#57512yes—
640C2×C4#57712yes—
641C2×C4#57712yes—
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4111yescharacteristic
644C2×C4C2×2−1+412yes—
645C2×C4#58512yes—
646C8—24no—
647C8—24no—
648C8—24no—
649C8—42no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22C2×#53211yescharacteristic
658C22#349911yescharacteristic
659C22#350011yescharacteristic
660C22C22×#13711yescharacteristic
661C22C2×#57511yescharacteristic
662C22C2×#57711yescharacteristic
663C22C2×#57711yescharacteristic
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—82no—
683C4#317214yes—
684C4#317314yes—
685C4#346112yes—
686C4#349512yes—
687C2—81no—
688C2#3531811yescharacteristic
689C2C2×#317211yescharacteristic
690C2C2×#317311yescharacteristic
691C2#6247811yescharacteristic
692C2#6248311yescharacteristic
693C2#6254011yescharacteristic
694C2#6254411yescharacteristic
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. #33293C2semidirect
4. #33287C2semidirect
7. #30305C2semidirect
8. #30281C2semidirect
11. #28963C2semidirect
14. #28042C2semidirect
15. #22248C2semidirect
16. #22218C2semidirect
17. C4×#498C2semidirect
18. C4×#487C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352832
#332932
#332872
#332122
#330492
#303052
#302812
#298772
#298452
#289632
#288632
#280702
#280422
#222482
#222182
C4×#4982
C4×#4872
#204722