Finite Group Databasebeta

← #6495913

Subgroups — #6495913

Order 512.

1580 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33174C211yesmaximal, characteristic
4#33172C211yesmaximal, characteristic, complemented
5#33049C211yesmaximal, characteristic, complemented
6#33042C211yesmaximal, characteristic
7#30304C211yesmaximal, characteristic, complemented
8#30261C211yesmaximal, characteristic, complemented
9#30219C211yesmaximal, characteristic, complemented
10#30205C211yesmaximal, characteristic, complemented
11#29567C214yesmaximal, complemented
12#29511C214yesmaximal, complemented
13#28590C214yesmaximal, complemented
14#25514C214yesmaximal, complemented
15#22270C211yesmaximal, characteristic, complemented
16#22268C211yesmaximal, characteristic, complemented
17#22221C211yesmaximal, characteristic, complemented
18#22220C211yesmaximal, characteristic, complemented
19#20366C211yesmaximal, characteristic, complemented
20C2×#542C2212yes—
21C2×#494C2211yescharacteristic
22C2×#492C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#486C2211yescharacteristic
25C2×#468C2214yes—
26C2×#467C2214yes—
27C2×C4×M16C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2858C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2645C2212yes—
38#2640C2212yes—
39#2625C2212yes—
40#2620C2212yes—
41#2514—21no—
42#2512—21no—
43#2512—21no—
44#2509—21no—
45#2508—21no—
46#2495—21no—
47#2494—21no—
48#2491—21no—
49#2491—21no—
50#2489—21no—
51#2453—21no—
52#2452—21no—
53#2452—21no—
54#2422—21no—
55#2422—21no—
56#2421—21no—
57#2326C2212yes—
58#2323C2212yes—
59#2320C2212yes—
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×#127C2211yescharacteristic
64C4×#117C2211yescharacteristic
65#2083C2214yes—
66#2072C2214yes—
67#2057C2214yes—
68#2042C2214yes—
69#2042C2214yes—
70#2024C2214yes—
71#2015C2214yes—
72#2015C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1973C2214yes—
76#1969C2214yes—
77#1957C2211yescharacteristic
78#1957C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1935C2214yes—
84#1934C2214yes—
85#1928C2214yes—
86#1925C2214yes—
87#1894C2214yes—
88#1893C2214yes—
89#1880C2214yes—
90#1878C2214yes—
91#1878C2214yes—
92#1877C2214yes—
93#1863C2214yes—
94#1862C2214yes—
95#1845C2211yescharacteristic
96#1845C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1746C2212yes—
100#1746C2212yes—
101#1744C2211yescharacteristic
102#1744C2211yescharacteristic
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105#1729C2212yes—
106#1729C2212yes—
107C22×Q16C2314yes—
108C22×SD16C2314yes—
109C22×M16C2312yes—
110C23×C8C2312yes—
111#542—48no—
112C2×C4⋊Q8—21no—
113C2×C4⋊Q8—21no—
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2311yescharacteristic
116C2×C4⋊Q8C2311yescharacteristic
117C2×C4⋊Q8C2312yes—
118C2×#127C2311yescharacteristic
119C2×#126—22no—
120C2×#125—21no—
121C2×#124—22no—
122C2×#124C2312yes—
123C2×#124C2312yes—
124C2×#123—22no—
125C2×C22⋊Q8—21no—
126C2×C22⋊Q8—22no—
127C2×C22⋊Q8C2312yes—
128C2×C22⋊Q8C2312yes—
129C2×#121—21no—
130C2×#121—21no—
131C2×#121C2312yes—
132C2×C4×Q8—21no—
133C2×C4×Q8C2312yes—
134C2×C4×Q8C2312yes—
135C2×C4×D4—21no—
136C2×C4×D4—21no—
137C2×C4×D4C2312yes—
138C2×#117C2311yescharacteristic
139C2×#117C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#494—24no—
142#492—24no—
143#487—24no—
144#486—24no—
145#468—48no—
146#467—48no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C4×M16—24no—
158C2×#97C2312yes—
159C2×C4×C8—24no—
160C2×C4×C8C2312yes—
161C2×C4×C8C2311yescharacteristic
162C2×C4×C8C2311yescharacteristic
163C2×C4×C8C2311yescharacteristic
164C2×C4×C8C2311yescharacteristic
165#401—22no—
166#400—22no—
167#400—22no—
168#400—22no—
169#400—22no—
170#398—22no—
171#397—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#396—22no—
176#396—22no—
177#394—22no—
178#394—22no—
179#393—22no—
180#393—22no—
181#391—21no—
182#391—21no—
183#391—21no—
184#391—21no—
185#391C2312yes—
186#391C2312yes—
187#391C2312yes—
188#391C2312yes—
189#391C2312yes—
190#391C2312yes—
191#390—21no—
192#390—21no—
193#390—21no—
194#390—21no—
195#390—21no—
196#390—21no—
197#390—21no—
198#390—21no—
199#390C2312yes—
200#390C2312yes—
201#389—22no—
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#389—21no—
213#389—21no—
214#389C2312yes—
215#389C2312yes—
216#389C2312yes—
217#389C2312yes—
218#388—22no—
219#388—22no—
220#388—22no—
221#388—22no—
222#387—22no—
223#387—22no—
224#387—22no—
225#385—22no—
226#385—22no—
227#384—21no—
228#384—21no—
229#384C2311yescharacteristic
230#384C2311yescharacteristic
231#384C2311yescharacteristic
232#384C2311yescharacteristic
233#384C2312yes—
234#383—22no—
235#382—21no—
236#382—21no—
237#382—21no—
238#382—22no—
239#380—21no—
240#380—21no—
241#380—21no—
242#380—21no—
243#380—21no—
244#380C2312yes—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259#376C2311yescharacteristic
260#376C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8—22no—
277C22×Q8C2412yes—
278C22×Q8C2412yes—
279C22×D4—22no—
280C22×D4—21no—
281C22×D4—21no—
282C22×D4C2412yes—
283C23×C4—22no—
284C23×C4C2×D411yescharacteristic
285C23×C4C2×D411yescharacteristic
286C23×C4C2411yescharacteristic
287C2×Q16—28no—
288C2×Q16—44no—
289C2×Q16—44no—
290C2×SD16—28no—
291C2×SD16—44no—
292C2×SD16—44no—
293C2×M16—24no—
294C2×M16—24no—
295C2×M16—42no—
296C22×C8—24no—
297C22×C8—24no—
298C22×C8—42no—
299C22×C8C2412yes—
300C22×C8C2412yes—
301C22×C8C2412yes—
302C22×C8C2412yes—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—22no—
307C4⋊Q8—22no—
308C4⋊Q8—22no—
309C4⋊Q8—44no—
310C4⋊Q8—82no—
311C4⋊Q8—82no—
312#127—22no—
313#127—22no—
314#126—48no—
315#125—82no—
316#124—44no—
317#124—44no—
318#124—48no—
319#123—48no—
320C22⋊Q8—28no—
321C22⋊Q8—28no—
322C22⋊Q8—48no—
323C22⋊Q8—82no—
324#121—28no—
325#121—82no—
326#121—82no—
327C4×Q8—44no—
328C4×Q8—44no—
329C4×Q8—82no—
330C4×D4—44no—
331C4×D4—82no—
332C4×D4—82no—
333#117—24no—
334#117—42no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
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—21no—
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⋊C4C2×D411yescharacteristic
360C2×C4⋊C4C2×D411yescharacteristic
361C2×C4⋊C4C2×D411yescharacteristic
362C2×C4⋊C4C2×D411yescharacteristic
363C2×C4⋊C4C2412yes—
364C2×C4⋊C4C2412yes—
365C2×C4⋊C4C2412yes—
366C2×C4⋊C4C2412yes—
367C2×C4⋊C4C2412yes—
368C2×C22⋊C4—22no—
369C2×C22⋊C4—22no—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
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⋊C4C2×D411yescharacteristic
382C2×C22⋊C4C2×D411yescharacteristic
383C2×C22⋊C4C2×D411yescharacteristic
384C2×C22⋊C4C2×D411yescharacteristic
385C2×C42—22no—
386C2×C42—22no—
387C2×C42—22no—
388C2×C42—22no—
389C2×C42—22no—
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406#107—44no—
407#107—44no—
408#103—28no—
409#103—44no—
410#103—44no—
411#103—44no—
412#103—44no—
413#102—28no—
414#102—44no—
415#102—44no—
416#97—24no—
417C4×C8—24no—
418C4×C8—22no—
419C4×C8—22no—
420C4×C8#4114yes—
421C4×C8#4114yes—
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#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448C24—22no—
449C24C22×D411yescharacteristic
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—22no—
454C2×Q8—24no—
455C2×Q8—24no—
456C2×Q8—24no—
457C2×Q8—24no—
458C2×Q8—44no—
459C2×Q8—42no—
460C2×Q8—42no—
461C2×Q8—42no—
462C2×Q8—82no—
463C2×Q8—82no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—24no—
467C2×D4—24no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4—82no—
476C2×D4—82no—
477C22×C4—21no—
478C22×C4—21no—
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—22no—
493C22×C4—44no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4C22×D411yescharacteristic
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×C4H(5,𝔽2)14yes—
510C22×C4C2511yescharacteristic
511Q16—88no—
512SD16—88no—
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—48no—
525C2×C8C2×#4114yes—
526C2×C8C2×#4114yes—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—24no—
532C4⋊C4—24no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—41no—
536C4⋊C4—41no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C4⋊C4—44no—
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—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C4⋊C4—82no—
561C22⋊C4—22no—
562C22⋊C4—22no—
563C22⋊C4—41no—
564C22⋊C4—41no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C22⋊C4—82no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—21no—
589C42—21no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—22no—
594C42—44no—
595C42—44no—
596C42—82no—
597C42C2×#4112yes—
598C42C2×#4112yes—
599C23—21no—
600C23—21no—
601C23—21no—
602C23—21no—
603C23—21no—
604C23—21no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23#53512yes—
613C23C23×D411yescharacteristic
614Q8—44no—
615Q8—44no—
616Q8—44no—
617Q8—44no—
618Q8—84no—
619Q8—164no—
620D4—44no—
621D4—44no—
622D4—84no—
623D4—84no—
624D4—164no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—24no—
628C2×C4—24no—
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—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—44no—
660C2×C4—44no—
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—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—164no—
683C2×C4#53212yes—
684C2×C4#53212yes—
685C2×C4#53512yes—
686C2×C4#57612yes—
687C2×C4#57712yes—
688C2×C4#57812yes—
689C2×C4#57812yes—
690C2×C4C22×#4111yescharacteristic
691C2×C4C22×#4111yescharacteristic
692C2×C4C2×H(5,𝔽2)12yes—
693C2×C4#58512yes—
694C8—24no—
695C8—24no—
696C8—42no—
697C8—42no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22C2×#53211yescharacteristic
713C22C2×#53511yescharacteristic
714C22#344911yescharacteristic
715C22C2×#57611yescharacteristic
716C22C2×#57711yescharacteristic
717C22C2×#57811yescharacteristic
718C22C2×#57811yescharacteristic
719C4—22no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—41no—
729C4—41no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4#317514yes—
738C4#317814yes—
739C4#349412yes—
740C4#349412yes—
741C2—42no—
742C2—162no—
743C2#3530311yescharacteristic
744C2C2×#317511yescharacteristic
745C2C2×#317811yescharacteristic
746C2#6082111yescharacteristic
747C2#6082211yescharacteristic
748C2#6100811yescharacteristic
749C2#6101411yescharacteristic
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
4. #33172C2semidirect
5. #33049C2semidirect
7. #30304C2semidirect
8. #30261C2semidirect
9. #30219C2semidirect
10. #30205C2semidirect
11. #29567C2semidirect
12. #29511C2semidirect
13. #28590C2semidirect
14. #25514C2semidirect
15. #22270C2semidirect
16. #22268C2semidirect
17. #22221C2semidirect
18. #22220C2semidirect
19. #20366C2semidirect

Maximal subgroups

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

subgroupindex
#352912
#352772
#331742
#331722
#330492
#330422
#303042
#302612
#302192
#302052
#295672
#295112
#285902
#255142
#222702
#222682
#222212
#222202
#203662