Finite Group Databasebeta

← #6494782

Subgroups — #6494782

Order 512.

1637 conjugacy classes of subgroups.

The subgroup lattice has 759 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35288C211yesmaximal, characteristic
2#35284C211yesmaximal, characteristic, complemented
3#33305C211yesmaximal, characteristic, complemented
4#33263C211yesmaximal, characteristic, complemented
5#33176C211yesmaximal, characteristic, complemented
6#33174C211yesmaximal, characteristic
7#30220C211yesmaximal, characteristic, complemented
8#30219C211yesmaximal, characteristic, complemented
9#29808C211yesmaximal, characteristic, complemented
10#29807C211yesmaximal, characteristic
11#26259C214yesmaximal, complemented
12#26256C214yesmaximal, complemented
13#25418C214yesmaximal, complemented
14#25417C214yesmaximal, complemented
15#22270C211yesmaximal, characteristic, complemented
16#22268C211yesmaximal, characteristic, complemented
17C4×#494C211yesmaximal, characteristic, complemented
18C4×#492C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic, complemented
20C2×#497C2214yes—
21C2×#497C2214yes—
22C2×#494C2211yescharacteristic
23C2×#494C2211yescharacteristic
24C2×#492C2211yescharacteristic
25C2×#492C2211yescharacteristic
26C2×#470C2214yes—
27C2×#468C2214yes—
28C2×C4×Q16C2214yes—
29C2×C4×SD16C2214yes—
30C2×#431C2211yescharacteristic
31C2×C4×M16C2211yescharacteristic
32#2868C2211yescharacteristic
33#2859C2211yescharacteristic
34#2859C2211yescharacteristic
35#2858C2211yescharacteristic
36#2857C2211yescharacteristic
37#2857C2211yescharacteristic
38#2857C2211yescharacteristic
39#2856C2211yescharacteristic
40#2647—21no—
41#2644—21no—
42#2629—21no—
43#2624—21no—
44#2513—21no—
45#2507—21no—
46#2507—21no—
47#2497—21no—
48#2497—21no—
49#2490—21no—
50#2454—21no—
51#2448C2212yes—
52#2447C2212yes—
53#2446C2212yes—
54#2446C2212yes—
55#2429C2212yes—
56#2429C2212yes—
57#2427C2212yes—
58#2426C2212yes—
59#2420—21no—
60#2330—21no—
61#2324—21no—
62#2319—21no—
63#2314—21no—
64C4×C4⋊Q8C2211yescharacteristic
65C4×C4⋊Q8C2211yescharacteristic
66C4×C4⋊Q8C2211yescharacteristic
67C4×#127C2211yescharacteristic
68C4×#117C2211yescharacteristic
69#2056C2214yes—
70#2055C2214yes—
71#2043C2214yes—
72#2043C2214yes—
73#2042C2214yes—
74#2042C2214yes—
75#2001C2214yes—
76#2000C2214yes—
77#1973C2214yes—
78#1971C2214yes—
79#1969C2214yes—
80#1968C2214yes—
81#1957C2211yescharacteristic
82#1957C2211yescharacteristic
83#1956C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1956C2211yescharacteristic
87#1956C2211yescharacteristic
88#1956C2211yescharacteristic
89#1944C2214yes—
90#1943C2214yes—
91#1880C2214yes—
92#1878C2214yes—
93#1878C2214yes—
94#1877C2214yes—
95#1845C2211yescharacteristic
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1771C2212yes—
100#1771C2212yes—
101#1753C2212yes—
102#1753C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×Q16C2314yes—
108C22×Q16C2314yes—
109C22×SD16C2314yes—
110C22×SD16C2314yes—
111C22×M16C2312yes—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2312yes—
117C2×#127C2311yescharacteristic
118C2×#125—21no—
119C2×#125—21no—
120C2×#124C2312yes—
121C2×#124C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×C22⋊Q8C2312yes—
124C2×C22⋊Q8C2312yes—
125C2×#121—21no—
126C2×#121—21no—
127C2×#121C2312yes—
128C2×C4×Q8C2312yes—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4C2312yes—
134C2×#117D411yescharacteristic
135C2×#117D411yescharacteristic
136C22×C42C2311yescharacteristic
137#497—48no—
138#497—48no—
139#494—24no—
140#494—24no—
141#492—24no—
142#492—24no—
143#470—48no—
144#468—48no—
145C4×Q16—48no—
146C4×SD16—48no—
147#431—24no—
148C2×#107C2314yes—
149C2×#106C2314yes—
150C2×C4⋊C8C2312yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×C22⋊C8C2312yes—
160C4×M16—24no—
161C2×#97C2312yes—
162C2×#97C2312yes—
163C2×#97C2312yes—
164C2×C4×C8C2311yescharacteristic
165C2×C4×C8C2311yescharacteristic
166C2×C4×C8C2312yes—
167C2×C4×C8C2312yes—
168C2×C4×C8C2311yescharacteristic
169C2×C4×C8C2311yescharacteristic
170#400—22no—
171#400—22no—
172#399—22no—
173#398—22no—
174#398—22no—
175#398—22no—
176#398—22no—
177#397—22no—
178#397—22no—
179#397—22no—
180#397—22no—
181#396—22no—
182#396—22no—
183#396—22no—
184#394—22no—
185#394—22no—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#391C2312yes—
192#390—22no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390C2312yes—
202#390C2312yes—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389—21no—
209#389—21no—
210#389—21no—
211#389—21no—
212#389—21no—
213#389—21no—
214#389—21no—
215#389—21no—
216#389—21no—
217#389—21no—
218#389—21no—
219#389—22no—
220#389—22no—
221#389C2312yes—
222#389C2312yes—
223#389C2312yes—
224#389C2312yes—
225#388—22no—
226#388—22no—
227#388—22no—
228#388—22no—
229#388—22no—
230#387—22no—
231#387—22no—
232#387—22no—
233#387—22no—
234#386—22no—
235#385—22no—
236#384—21no—
237#384—21no—
238#384—21no—
239#384—21no—
240#384C2311yescharacteristic
241#384C2311yescharacteristic
242#384C2311yescharacteristic
243#384C2311yescharacteristic
244#382—21no—
245#382—21no—
246#381—22no—
247#380—21no—
248#380—21no—
249#380—21no—
250#380—21no—
251#380—21no—
252#380—21no—
253#380—22no—
254C4×C4⋊C4—21no—
255C4×C4⋊C4—21no—
256C4×C4⋊C4—21no—
257C4×C4⋊C4—21no—
258C4×C4⋊C4—21no—
259C4×C4⋊C4—21no—
260C4×C4⋊C4D411yescharacteristic
261C4×C4⋊C4D411yescharacteristic
262C4×C4⋊C4C2311yescharacteristic
263C4×C4⋊C4C2311yescharacteristic
264C4×C22⋊C4D411yescharacteristic
265C4×C22⋊C4D411yescharacteristic
266C4×C22⋊C4C2311yescharacteristic
267C4×C22⋊C4C2311yescharacteristic
268#376D411yescharacteristic
269#376D411yescharacteristic
270C43C2311yescharacteristic
271C43C2311yescharacteristic
272#340C2314yes—
273#340C2314yes—
274#336C2312yes—
275#336C2312yes—
276#336C2312yes—
277#336C2312yes—
278C22×Q8C2412yes—
279C22×Q8C2412yes—
280C22×Q8C2412yes—
281C22×D4—22no—
282C22×D4—21no—
283C22×D4—21no—
284C22×D4C2412yes—
285C23×C4—22no—
286C23×C4C2×D411yescharacteristic
287C23×C4C2×D411yescharacteristic
288C23×C4C2411yescharacteristic
289C2×Q16—28no—
290C2×Q16—28no—
291C2×Q16—28no—
292C2×Q16—28no—
293C2×Q16—44no—
294C2×Q16—44no—
295C2×SD16—28no—
296C2×SD16—28no—
297C2×SD16—28no—
298C2×SD16—28no—
299C2×SD16—44no—
300C2×SD16—44no—
301C2×M16—24no—
302C2×M16—24no—
303C2×M16—24no—
304C22×C8C2412yes—
305C22×C8C2412yes—
306C22×C8C2412yes—
307C22×C8C2412yes—
308C4⋊Q8—22no—
309C4⋊Q8—22no—
310C4⋊Q8—22no—
311C4⋊Q8—22no—
312C4⋊Q8—22no—
313C4⋊Q8—22no—
314C4⋊Q8—44no—
315C4⋊Q8—44no—
316#127—22no—
317#127—22no—
318#125—44no—
319#125—82no—
320#124—44no—
321#124—44no—
322C22⋊Q8—44no—
323C22⋊Q8—44no—
324C22⋊Q8—44no—
325#121—44no—
326#121—44no—
327#121—82no—
328C4×Q8—44no—
329C4×Q8—44no—
330C4×Q8—44no—
331C4×D4—44no—
332C4×D4—44no—
333C4×D4—82no—
334#117—24no—
335#117—42no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
365C2×C4⋊C4C2×D411yescharacteristic
366C2×C4⋊C4C2×D411yescharacteristic
367C2×C4⋊C4C2×D411yescharacteristic
368C2×C4⋊C4C2412yes—
369C2×C4⋊C4C2412yes—
370C2×C4⋊C4C2412yes—
371C2×C4⋊C4C2412yes—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—22no—
378C2×C22⋊C4—22no—
379C2×C22⋊C4—22no—
380C2×C22⋊C4—22no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4—22no—
384C2×C22⋊C4—22no—
385C2×C22⋊C4C2×D411yescharacteristic
386C2×C22⋊C4C2×D411yescharacteristic
387C2×C22⋊C4C2×D411yescharacteristic
388C2×C22⋊C4C2×D411yescharacteristic
389C2×C42—22no—
390C2×C42—22no—
391C2×C42—22no—
392C2×C42—22no—
393C2×C42—22no—
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2×D411yescharacteristic
401C2×C42C2×D411yescharacteristic
402C2×C42C2×D411yescharacteristic
403C2×C42C2×D411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408C2×C42C2411yescharacteristic
409C2×C42C2411yescharacteristic
410#107—28no—
411#106—28no—
412C4⋊C8—24no—
413#103—44no—
414#103—44no—
415#103—44no—
416#103—44no—
417#103—44no—
418#103—44no—
419#102—44no—
420#102—44no—
421C22⋊C8—24no—
422#97—24no—
423#97—24no—
424#97—24no—
425C4×C8—24no—
426C4×C8—24no—
427C4×C8—22no—
428C4×C8—22no—
429C4×C8#4114yes—
430C4×C8#4114yes—
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#95—22no—
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449#95C2×D411yescharacteristic
450#95C2×D411yescharacteristic
451#95C2×D411yescharacteristic
452#95C2×D411yescharacteristic
453#95C2×D411yescharacteristic
454#95C2×D411yescharacteristic
455C24—22no—
456C24C22×D411yescharacteristic
457C2×Q8—22no—
458C2×Q8—22no—
459C2×Q8—22no—
460C2×Q8—22no—
461C2×Q8—22no—
462C2×Q8—22no—
463C2×Q8—24no—
464C2×Q8—24no—
465C2×Q8—24no—
466C2×Q8—24no—
467C2×Q8—24no—
468C2×Q8—24no—
469C2×D4—22no—
470C2×D4—22no—
471C2×D4—24no—
472C2×D4—24no—
473C2×D4—44no—
474C2×D4—42no—
475C2×D4—42no—
476C2×D4—42no—
477C2×D4—42no—
478C2×D4—42no—
479C2×D4—42no—
480C2×D4—82no—
481C2×D4—82no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—44no—
500C22×C4—82no—
501C22×C4—82no—
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×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C2511yescharacteristic
516Q16—416no—
517Q16—88no—
518SD16—416no—
519SD16—88no—
520M16—48no—
521C2×C8—22no—
522C2×C8—22no—
523C2×C8—22no—
524C2×C8—22no—
525C2×C8—22no—
526C2×C8—22no—
527C2×C8—22no—
528C2×C8—22no—
529C2×C8C2×#4114yes—
530C2×C8C2×#4114yes—
531C2×C8H(5,𝔽2)14yes—
532C2×C8H(5,𝔽2)14yes—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—41no—
540C4⋊C4—41no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—42no—
557C4⋊C4—82no—
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—
569C22⋊C4—22no—
570C22⋊C4—22no—
571C22⋊C4—41no—
572C22⋊C4—41no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C22⋊C4—82no—
579C22⋊C4—82no—
580C22⋊C4—82no—
581C22⋊C4—82no—
582C22⋊C4—82no—
583C22⋊C4—82no—
584C22⋊C4—82no—
585C22⋊C4—82no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—22no—
595C42—21no—
596C42—21no—
597C42—22no—
598C42—22no—
599C42—22no—
600C42—22no—
601C42—82no—
602C42—82no—
603C42—82no—
604C42#13714yes—
605C42C2×#4112yes—
606C42C2×#4112yes—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23C23×D411yescharacteristic
622Q8—44no—
623Q8—44no—
624Q8—44no—
625Q8—44no—
626Q8—44no—
627Q8—44no—
628D4—44no—
629D4—44no—
630D4—84no—
631D4—84no—
632D4—164no—
633C2×C4—21no—
634C2×C4—21no—
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—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—164no—
690C2×C4#53212yes—
691C2×C4#53212yes—
692C2×C4C2×#13712yes—
693C2×C4C2×#13712yes—
694C2×C4C2×#13712yes—
695C2×C4#57512yes—
696C2×C4#57712yes—
697C2×C4#57812yes—
698C2×C4C22×#4111yescharacteristic
699C2×C4C22×#4111yescharacteristic
700C2×C4C2×H(5,𝔽2)12yes—
701C2×C4C2×H(5,𝔽2)12yes—
702C8—24no—
703C8—42no—
704C8—42no—
705C8#54018yes—
706C22—41no—
707C22—41no—
708C22—41no—
709C22—41no—
710C22—41no—
711C22—41no—
712C22—41no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C2×#53211yescharacteristic
721C22#349811yescharacteristic
722C22#349811yescharacteristic
723C22C22×#13711yescharacteristic
724C22C2×#57511yescharacteristic
725C22C2×#57711yescharacteristic
726C22C2×#57811yescharacteristic
727C4—22no—
728C4—22no—
729C4—41no—
730C4—41no—
731C4—41no—
732C4—41no—
733C4—41no—
734C4—41no—
735C4—41no—
736C4—41no—
737C4—41no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4#317214yes—
746C4#317814yes—
747C4C2×#54012yes—
748C4#350812yes—
749C2—81no—
750C2—162no—
751C2#3527811yescharacteristic
752C2C2×#317211yescharacteristic
753C2C2×#317811yescharacteristic
754C2#6242211yescharacteristic
755C2#6242411yescharacteristic
756C2#6242511yescharacteristic
757C2#6242711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
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
2. #35284C2semidirect
3. #33305C2semidirect
4. #33263C2semidirect
5. #33176C2semidirect
7. #30220C2semidirect
8. #30219C2semidirect
9. #29808C2semidirect
11. #26259C2semidirect
12. #26256C2semidirect
13. #25418C2semidirect
14. #25417C2semidirect
15. #22270C2semidirect
16. #22268C2semidirect
17. C4×#494C2semidirect
18. C4×#492C2semidirect
19. #20472C2semidirect

Maximal subgroups

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

subgroupindex
#352882
#352842
#333052
#332632
#331762
#331742
#302202
#302192
#298082
#298072
#262592
#262562
#254182
#254172
#222702
#222682
C4×#4942
C4×#4922
#204722