Finite Group Databasebeta

← #6469530

Subgroups — #6469530

Order 512.

1522 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#29526C214yesmaximal, complemented
4#29178C211yesmaximal, characteristic, complemented
5#29176C211yesmaximal, characteristic, complemented
6#28656C211yesmaximal, characteristic, complemented
7#28656C211yesmaximal, characteristic, complemented
8#27962C211yesmaximal, characteristic
9#27908C211yesmaximal, characteristic
10#25902C211yesmaximal, characteristic
11#25813C211yesmaximal, characteristic
12#24613C214yesmaximal
13#23788C214yesmaximal, complemented
14#21726C211yesmaximal, characteristic, complemented
15C4×#500C211yesmaximal, characteristic, complemented
16#21647C211yesmaximal, characteristic, complemented
17C4×#487C211yesmaximal, characteristic, complemented
18#21589C214yesmaximal
19C4×#433C211yesmaximal, characteristic
20C22×C4⋊Q8C2211yescharacteristic
21C2×#500C2211yescharacteristic
22C2×#500C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#485C2214yes—
26C2×#433C2212yes—
27#2862C2211yescharacteristic
28#2859C2211yescharacteristic
29#2857C2211yescharacteristic
30#2857C2211yescharacteristic
31#2645C2212yes—
32#2625C2212yes—
33#2514—21no—
34#2509—21no—
35#2508—21no—
36#2495—21no—
37#2494—21no—
38#2489—21no—
39#2453—21no—
40#2421—21no—
41#2369—21no—
42#2369—21no—
43#2365—21no—
44#2365—21no—
45#2345—21no—
46#2345—21no—
47Q8×C22⋊C4—21no—
48Q8×C22⋊C4—21no—
49#2331C2212yes—
50#2331C2212yes—
51#2323C2212yes—
52#2320C2212yes—
53#2316C2212yes—
54C4×C4⋊Q8C2211yescharacteristic
55C4×C4⋊Q8C2211yescharacteristic
56C4×C4⋊Q8C2211yescharacteristic
57C4×C4⋊Q8C2211yescharacteristic
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60#2285C2211yescharacteristic
61#2274C2212yes—
62C4×#117C2211yescharacteristic
63#2101C2214yes—
64#2097C2214yes—
65#2088C2214yes—
66#2086C2214yes—
67#2068C2214yes—
68#2068C2214yes—
69#2057C2214yes—
70#2024C2214yes—
71#1990C2211yescharacteristic
72#1990C2211yescharacteristic
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1943C2214yes—
80#1942C2214yes—
81#1935C2214yes—
82#1933C2214yes—
83#1928C2214yes—
84#1928C2214yes—
85#1920C2214yes—
86#1918C2214yes—
87#1872C2214yes—
88#1852C2214yes—
89#1845C2211yescharacteristic
90#1844C2211yescharacteristic
91#1834C2211yescharacteristic
92#1834C2211yescharacteristic
93#1825C2214yes—
94#1776C2214yes—
95C4×#107C2214yes—
96#1768C2212yes—
97C4×C4⋊C8C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99#1763C2214yes—
100C4×#103C2214yes—
101#1750C2212yes—
102C4×C22⋊C8C2211yescharacteristic
103C4×C22⋊C8C2211yescharacteristic
104#1746C2212yes—
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C23×Q8—21no—
108C2×C4⋊Q8—24no—
109C2×C4⋊Q8—24no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2312yes—
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×#125—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8C2312yes—
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132#500—24no—
133#500—24no—
134#487—24no—
135#487—24no—
136#485—216no—
137#433—28no—
138C2×#107C2314yes—
139C2×#107C2314yes—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2312yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×#97C2312yes—
151C2×#97C2312yes—
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154C2×C4×C8C2311yescharacteristic
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2312yes—
157C2×C4×C8C2312yes—
158#401—22no—
159#400—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#398—22no—
164#397—22no—
165#397—22no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391—22no—
175#391—22no—
176#391—22no—
177#391—22no—
178#391C2312yes—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#389—22no—
183#389—22no—
184#389—22no—
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#389—21no—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#389C2312yes—
201#389C2312yes—
202#389C2312yes—
203#389C2312yes—
204#389C2312yes—
205#388—22no—
206#388—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#384C2311yescharacteristic
215#384C2311yescharacteristic
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#383—22no—
219#382—21no—
220#382—21no—
221#382—21no—
222#382—22no—
223#382C2312yes—
224#380—21no—
225#380—21no—
226#380—21no—
227#380—21no—
228#380—21no—
229#380—22no—
230#380—22no—
231#380C2312yes—
232C4×C4⋊C4—22no—
233C4×C4⋊C4—22no—
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—21no—
242C4×C4⋊C4—21no—
243C4×C4⋊C4D411yescharacteristic
244C4×C4⋊C4D411yescharacteristic
245C4×C4⋊C4D411yescharacteristic
246C4×C4⋊C4D411yescharacteristic
247C4×C4⋊C4C2312yes—
248C4×C4⋊C4C2312yes—
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2312yes—
252C4×C22⋊C4—22no—
253C4×C22⋊C4—22no—
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258#376C2311yescharacteristic
259#376C2311yescharacteristic
260C43C2311yescharacteristic
261C43C2311yescharacteristic
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#340C2314yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274C22×Q8—28no—
275C22×Q8—21no—
276C22×Q8—21no—
277C22×Q8—22no—
278C22×Q8—21no—
279C22×Q8—82no—
280C22×Q8—82no—
281C22×Q8C2412yes—
282C23×C4—22no—
283C23×C4C2×D411yescharacteristic
284C23×C4C2×D411yescharacteristic
285C23×C4C2411yescharacteristic
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2412yes—
289C22×C8C2412yes—
290C4⋊Q8—28no—
291C4⋊Q8—28no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—22no—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8#4114yes—
301C4⋊Q8#4114yes—
302#125—82no—
303C22⋊Q8—28no—
304C22⋊Q8—82no—
305C4×Q8—28no—
306C4×Q8—82no—
307C4×Q8—82no—
308C4×Q8—82no—
309C4×Q8—82no—
310C4×Q8—82no—
311#117—24no—
312#117—24no—
313#117—28no—
314C2×C4⋊C4—28no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—24no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
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—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C4⋊C4C2412yes—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4C2×D411yescharacteristic
357C2×C22⋊C4C2×D411yescharacteristic
358C2×C22⋊C4C2×D411yescharacteristic
359C2×C22⋊C4C2×D411yescharacteristic
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—22no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
378C2×C42C2×D411yescharacteristic
379C2×C42C2411yescharacteristic
380C2×C42C2411yescharacteristic
381C2×C42C2411yescharacteristic
382C2×C42C2411yescharacteristic
383C2×C42C2411yescharacteristic
384C2×C42C2411yescharacteristic
385#107—28no—
386#107—28no—
387#107—28no—
388#106—28no—
389C4⋊C8—24no—
390C4⋊C8—24no—
391#103—28no—
392#103—28no—
393#103—28no—
394#103—28no—
395C22⋊C8—24no—
396C22⋊C8#4118yes—
397#97—24no—
398#97—24no—
399C4×C8—24no—
400C4×C8—22no—
401C4×C8—22no—
402C4×C8—22no—
403C4×C8—22no—
404C4×C8—24no—
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#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95C2×D411yescharacteristic
430#95C2×D411yescharacteristic
431#95C2×D411yescharacteristic
432#95C2×D411yescharacteristic
433#95C2×D411yescharacteristic
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437C24C22×D411yescharacteristic
438C2×Q8—28no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—82no—
453C2×Q8—82no—
454C2×Q8—816no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—28no—
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×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4114yes—
489C22×C4C2511yescharacteristic
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8C2×#4114yes—
501C2×C8C2×#4114yes—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—24no—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4C2×#4114yes—
540C4⋊C4C2×#4114yes—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—82no—
548C42—22no—
549C42—22no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—21no—
561C42—21no—
562C42—22no—
563C42—44no—
564C42—44no—
565C42—82no—
566C42—82no—
567C42—82no—
568C42—82no—
569C42—82no—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C422−1+414yes—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23#53212yes—
580C23C23×D411yescharacteristic
581Q8—48no—
582Q8—48no—
583Q8—44no—
584Q8—44no—
585Q8—88no—
586Q8—84no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—24no—
598C2×C4—24no—
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—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
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—88no—
646C2×C4#52512yes—
647C2×C4#52512yes—
648C2×C4#53212yes—
649C2×C4#53612yes—
650C2×C4#53612yes—
651C2×C4#57612yes—
652C2×C4#57712yes—
653C2×C4#57812yes—
654C2×C4#57812yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C22×#4112yes—
658C2×C4C2×2−1+412yes—
659C8—24no—
660C8—24no—
661C8—42no—
662C8—42no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22C2×#52511yescharacteristic
671C22C2×#53211yescharacteristic
672C22C2×#53611yescharacteristic
673C22C2×#57611yescharacteristic
674C22C2×#57711yescharacteristic
675C22C2×#57811yescharacteristic
676C22C2×#57811yescharacteristic
677C4—22no—
678C4—22no—
679C4—22no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—44no—
686C4—44no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#311814yes—
697C4#312714yes—
698C4#348212yes—
699C4#348212yes—
700C2—42no—
701C2#3497211yescharacteristic
702C2C2×#311811yescharacteristic
703C2C2×#312711yescharacteristic
704C2#6071811yescharacteristic
705C2#6072311yescharacteristic
706C2#6087811yescharacteristic
707C2#6088011yescharacteristic
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. #29526C2semidirect
4. #29178C2semidirect
5. #29176C2semidirect
6. #28656C2semidirect
7. #28656C2semidirect
13. #23788C2semidirect
14. #21726C2semidirect
15. C4×#500C2semidirect
16. #21647C2semidirect
17. C4×#487C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#342162
#295262
#291782
#291762
#286562
#279622
#279082
#259022
#258132
#246132
#237882
#217262
C4×#5002
#216472
C4×#4872
#215892
C4×#4332