Finite Group Databasebeta

← #6499012

Subgroups — #6499012

Order 512.

1509 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33297C211yesmaximal, characteristic, complemented
4#33252C211yesmaximal, characteristic, complemented
5#33049C211yesmaximal, characteristic, complemented
6#33048C211yesmaximal, characteristic
7#30205C211yesmaximal, characteristic, complemented
8#30204C211yesmaximal, characteristic, complemented
9#29878C211yesmaximal, characteristic, complemented
10#29807C211yesmaximal, characteristic
11#29094C214yesmaximal, complemented
12#29091C214yesmaximal, complemented
13#28998C214yesmaximal, complemented
14#28995C214yesmaximal, complemented
15#22268C211yesmaximal, characteristic, complemented
16#22247C211yesmaximal, characteristic, complemented
17C4×#487C211yesmaximal, characteristic, complemented
18C4×#486C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic, complemented
20C2×#542—21no—
21C2×#500C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×C4×M16C2211yescharacteristic
27#2868C2211yescharacteristic
28#2862C2211yescharacteristic
29#2859C2211yescharacteristic
30#2858C2211yescharacteristic
31#2857C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2856C2211yescharacteristic
35#2645—21no—
36#2640—21no—
37#2625—21no—
38#2620—21no—
39#2514C2212yes—
40#2512—21no—
41#2512C2212yes—
42#2509—21no—
43#2508—21no—
44#2495—21no—
45#2494—21no—
46#2491—21no—
47#2491C2212yes—
48#2489C2212yes—
49#2453C2212yes—
50#2452—21no—
51#2452C2212yes—
52#2422—21no—
53#2422C2212yes—
54#2421C2212yes—
55#2326—21no—
56#2323—21no—
57#2320—21no—
58C4×C4⋊Q8C2211yescharacteristic
59C4×C4⋊Q8C2211yescharacteristic
60C4×C4⋊Q8C2211yescharacteristic
61C4×#127C2211yescharacteristic
62C4×#117C2211yescharacteristic
63#2087C2214yes—
64#2076C2214yes—
65#2073C2214yes—
66#2072C2214yes—
67#2055C2214yes—
68#2025C2214yes—
69#2024C2214yes—
70#2000C2214yes—
71#1993C2214yes—
72#1992C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1973C2214yes—
78#1973C2214yes—
79#1957C2211yescharacteristic
80#1957C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1936C2214yes—
84#1935C2214yes—
85#1934C2214yes—
86#1933C2214yes—
87#1918C2214yes—
88#1918C2214yes—
89#1895C2214yes—
90#1894C2214yes—
91#1880C2214yes—
92#1877C2214yes—
93#1834C2211yescharacteristic
94#1834C2211yescharacteristic
95#1834C2211yescharacteristic
96#1834C2211yescharacteristic
97#1771C2212yes—
98#1771C2212yes—
99C4×#103C2214yes—
100C4×#102C2214yes—
101#1753C2212yes—
102#1753C2212yes—
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105C4×#97C2211yescharacteristic
106C42×C8C2211yescharacteristic
107C22×SD16C2314yes—
108C22×M16C2312yes—
109#542—48no—
110C2×C4⋊Q8—21no—
111C2×C4⋊Q8—21no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×#127C2311yescharacteristic
117C2×#126—22no—
118C2×#125C2312yes—
119C2×#124—22no—
120C2×#124—21no—
121C2×#124—21no—
122C2×#123—22no—
123C2×C22⋊Q8—21no—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8—22no—
126C2×C22⋊Q8C2312yes—
127C2×#121—21no—
128C2×#121—21no—
129C2×#121C2312yes—
130C2×C4×Q8—21no—
131C2×C4×Q8—21no—
132C2×C4×Q8C2312yes—
133C2×C4×D4—21no—
134C2×C4×D4—21no—
135C2×C4×D4C2312yes—
136C2×#117D411yescharacteristic
137C2×#117D411yescharacteristic
138C22×C42C2311yescharacteristic
139#500—24no—
140#492—24no—
141#487—24no—
142#486—24no—
143#431—24no—
144C2×#107C2314yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×C22⋊C8C2312yes—
153C4×M16—24no—
154C2×#97C2312yes—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159C2×C4×C8C2312yes—
160C2×C4×C8C2312yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163#401—22no—
164#400—22no—
165#400—22no—
166#400—22no—
167#400—22no—
168#398—22no—
169#397—22no—
170#397—22no—
171#397—22no—
172#397—22no—
173#396—22no—
174#396—22no—
175#394—22no—
176#394—22no—
177#393—22no—
178#393—22no—
179#391—21no—
180#391—21no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#390—21no—
190#390—21no—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390C2312yes—
196#390C2312yes—
197#390C2312yes—
198#390C2312yes—
199#389—22no—
200#389—21no—
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—21no—
210#389—21no—
211#389—21no—
212#389C2312yes—
213#389C2312yes—
214#389C2312yes—
215#389C2312yes—
216#388—22no—
217#388—22no—
218#388—22no—
219#388—22no—
220#387—22no—
221#387—22no—
222#387—22no—
223#385—22no—
224#385—22no—
225#384—21no—
226#384—21no—
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2312yes—
232#383—22no—
233#382—21no—
234#382—21no—
235#382—22no—
236#382C2312yes—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—21no—
241#380C2312yes—
242#380C2312yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4D411yescharacteristic
248C4×C4⋊C4D411yescharacteristic
249C4×C4⋊C4C2312yes—
250C4×C4⋊C4C2312yes—
251C4×C4⋊C4C2311yescharacteristic
252C4×C4⋊C4C2311yescharacteristic
253C4×C22⋊C4D411yescharacteristic
254C4×C22⋊C4D411yescharacteristic
255C4×C22⋊C4C2311yescharacteristic
256C4×C22⋊C4C2311yescharacteristic
257#376D411yescharacteristic
258#376D411yescharacteristic
259C43C2311yescharacteristic
260C43C2311yescharacteristic
261#340C2314yes—
262#340C2314yes—
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#340C2314yes—
268#340C2314yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273C22×Q8—21no—
274C22×Q8—21no—
275C22×Q8—22no—
276C22×Q8C2412yes—
277C22×D4—22no—
278C22×D4—21no—
279C22×D4—21no—
280C22×D4C2412yes—
281C23×C4—22no—
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C2×SD16—28no—
286C2×SD16—44no—
287C2×SD16—44no—
288C2×M16—24no—
289C2×M16—42no—
290C2×M16—42no—
291C22×C8C2412yes—
292C22×C8C2412yes—
293C22×C8C2412yes—
294C22×C8C2412yes—
295C4⋊Q8—22no—
296C4⋊Q8—22no—
297C4⋊Q8—22no—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—44no—
302C4⋊Q8—44no—
303C4⋊Q8—82no—
304#127—22no—
305#127—22no—
306#126—48no—
307#125—44no—
308#124—44no—
309#124—44no—
310#124—48no—
311#123—48no—
312C22⋊Q8—48no—
313C22⋊Q8—44no—
314C22⋊Q8—44no—
315C22⋊Q8—44no—
316#121—44no—
317#121—44no—
318#121—82no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8—44no—
322C4×D4—44no—
323C4×D4—44no—
324C4×D4—82no—
325#117—24no—
326#117—42no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4C2×D411yescharacteristic
351C2×C4⋊C4C2×D411yescharacteristic
352C2×C4⋊C4C2×D411yescharacteristic
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2412yes—
355C2×C4⋊C4C2412yes—
356C2×C4⋊C4C2412yes—
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—22no—
373C2×C22⋊C4C2×D411yescharacteristic
374C2×C22⋊C4C2×D411yescharacteristic
375C2×C22⋊C4C2×D411yescharacteristic
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C42—22no—
378C2×C42—22no—
379C2×C42—22no—
380C2×C42—22no—
381C2×C42—22no—
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398#107—28no—
399C4⋊C8—24no—
400#103—28no—
401#103—44no—
402#103—44no—
403#102—28no—
404#102—44no—
405#102—44no—
406C22⋊C8—24no—
407#97—24no—
408#97—42no—
409#97—42no—
410C4×C8—22no—
411C4×C8—22no—
412C4×C8—42no—
413C4×C8—42no—
414C4×C8#4114yes—
415C4×C8#4114yes—
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#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442C24—22no—
443C24C22×D411yescharacteristic
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—24no—
447C2×Q8—24no—
448C2×Q8—44no—
449C2×Q8—42no—
450C2×Q8—42no—
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—82no—
456C2×Q8—82no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—24no—
460C2×D4—24no—
461C2×D4—44no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—42no—
468C2×D4—82no—
469C2×D4—82no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—21no—
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—44no—
488C22×C4—82no—
489C22×C4—82no—
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C2511yescharacteristic
504SD16—88no—
505M16—48no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8C2×#4114yes—
517C2×C8C2×#4114yes—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—44no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—44no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—82no—
548C4⋊C4—82no—
549C4⋊C4—82no—
550C4⋊C4—82no—
551C4⋊C4—82no—
552C4⋊C4—82no—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—41no—
556C22⋊C4—41no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—82no—
566C22⋊C4—82no—
567C22⋊C4—82no—
568C22⋊C4—82no—
569C22⋊C4—82no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—21no—
580C42—21no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—44no—
586C42—44no—
587C42—82no—
588C42AGL(1,ℤ/8)14yes—
589C42C2×#4112yes—
590C42C2×#4112yes—
591C23—21no—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—21no—
597C23—21no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23C23×D411yescharacteristic
606Q8—44no—
607Q8—44no—
608Q8—84no—
609Q8—84no—
610Q8—164no—
611D4—44no—
612D4—44no—
613D4—84no—
614D4—84no—
615D4—164no—
616C2×C4—21no—
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—21no—
632C2×C4—21no—
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—44no—
648C2×C4—44no—
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—42no—
672C2×C4—164no—
673C2×C4#53212yes—
674C2×C4#53212yes—
675C2×C4C2×AGL(1,ℤ/8)12yes—
676C2×C4C2×AGL(1,ℤ/8)12yes—
677C2×C4C2×AGL(1,ℤ/8)12yes—
678C2×C4#57512yes—
679C2×C4#57712yes—
680C2×C4#57812yes—
681C2×C4C22×#4111yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4#58512yes—
684C2×C4#58512yes—
685C8—24no—
686C8—24no—
687C8—42no—
688C8—42no—
689C22—41no—
690C22—41no—
691C22—41no—
692C22—41no—
693C22—41no—
694C22—41no—
695C22—41no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22C2×#53211yescharacteristic
704C22#350011yescharacteristic
705C22#350011yescharacteristic
706C22C22×AGL(1,ℤ/8)11yescharacteristic
707C22C2×#57511yescharacteristic
708C22C2×#57711yescharacteristic
709C22C2×#57811yescharacteristic
710C4—22no—
711C4—22no—
712C4—41no—
713C4—41no—
714C4—41no—
715C4—41no—
716C4—41no—
717C4—41no—
718C4—41no—
719C4—41no—
720C4—41no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4#317114yes—
729C4#317814yes—
730C4#345512yes—
731C4#351012yes—
732C2—81no—
733C2—162no—
734C2#3533111yescharacteristic
735C2C2×#317111yescharacteristic
736C2C2×#317811yescharacteristic
737C2#6253811yescharacteristic
738C2#6253911yescharacteristic
739C2#6254311yescharacteristic
740C2#6254511yescharacteristic
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
1. #35291C2semidirect
3. #33297C2semidirect
4. #33252C2semidirect
5. #33049C2semidirect
7. #30205C2semidirect
8. #30204C2semidirect
9. #29878C2semidirect
11. #29094C2semidirect
12. #29091C2semidirect
13. #28998C2semidirect
14. #28995C2semidirect
15. #22268C2semidirect
16. #22247C2semidirect
17. C4×#487C2semidirect
18. C4×#486C2semidirect
19. #20472C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352772
#332972
#332522
#330492
#330482
#302052
#302042
#298782
#298072
#290942
#290912
#289982
#289952
#222682
#222472
C4×#4872
C4×#4862
#204722