Finite Group Databasebeta

← #6494677

Subgroups — #6494677

Order 512.

1593 conjugacy classes of subgroups.

The subgroup lattice has 749 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35291C211yesmaximal, characteristic, complemented
2#35277C211yesmaximal, characteristic
3#33316C211yesmaximal, characteristic, complemented
4#33252C211yesmaximal, characteristic, complemented
5#33212C211yesmaximal, characteristic, complemented
6#33174C211yesmaximal, characteristic
7#30205C211yesmaximal, characteristic, complemented
8#30204C211yesmaximal, characteristic, complemented
9#29844C211yesmaximal, characteristic, complemented
10#29843C211yesmaximal, characteristic
11#28019C214yesmaximal, complemented
12#27796C214yesmaximal, complemented
13#26486C214yesmaximal, complemented
14#25036C214yesmaximal, complemented
15#22220C211yesmaximal, characteristic, complemented
16#22218C211yesmaximal, characteristic, complemented
17C4×#500C211yesmaximal, characteristic, complemented
18C4×#492C211yesmaximal, characteristic, complemented
19#20472C211yesmaximal, characteristic, complemented
20C2×#542—21no—
21C2×#501C2214yes—
22C2×#500C2211yescharacteristic
23C2×#492C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#486C2211yescharacteristic
26C2×#468C2214yes—
27C2×C4×SD16C2214yes—
28C2×#431C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30#2868C2211yescharacteristic
31#2862C2211yescharacteristic
32#2859C2211yescharacteristic
33#2858C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2857C2211yescharacteristic
37#2856C2211yescharacteristic
38#2645—21no—
39#2640—21no—
40#2625—21no—
41#2620—21no—
42#2514C2212yes—
43#2512—21no—
44#2512C2212yes—
45#2509—21no—
46#2508—21no—
47#2495—21no—
48#2494—21no—
49#2491—21no—
50#2491C2212yes—
51#2489C2212yes—
52#2453C2212yes—
53#2452—21no—
54#2452C2212yes—
55#2422—21no—
56#2422C2212yes—
57#2421C2212yes—
58#2326—21no—
59#2323—21no—
60#2320—21no—
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×C4⋊Q8C2211yescharacteristic
64C4×#127C2211yescharacteristic
65C4×#117C2211yescharacteristic
66#2073C2214yes—
67#2072C2214yes—
68#2068C2214yes—
69#2065C2214yes—
70#2036C2214yes—
71#2034C2214yes—
72#2025C2214yes—
73#2024C2214yes—
74#1990C2211yescharacteristic
75#1990C2211yescharacteristic
76#1990C2211yescharacteristic
77#1990C2211yescharacteristic
78#1973C2214yes—
79#1973C2214yes—
80#1973C2214yes—
81#1968C2214yes—
82#1957C2211yescharacteristic
83#1957C2211yescharacteristic
84#1956C2211yescharacteristic
85#1956C2211yescharacteristic
86#1944C2214yes—
87#1942C2214yes—
88#1927C2214yes—
89#1925C2214yes—
90#1865C2214yes—
91#1862C2214yes—
92#1847C2214yes—
93#1845C2211yescharacteristic
94#1845C2211yescharacteristic
95#1844C2211yescharacteristic
96#1844C2211yescharacteristic
97#1809C2214yes—
98C4×#107C2214yes—
99#1771C2212yes—
100#1771C2212yes—
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#501—48no—
140#500—24no—
141#492—24no—
142#487—24no—
143#486—24no—
144#468—48no—
145C4×SD16—48no—
146#431—24no—
147C2×#107C2314yes—
148C2×#107C2314yes—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×#106C2314yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#102C2314yes—
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#401—22no—
171#400—22no—
172#400—22no—
173#400—22no—
174#400—22no—
175#398—22no—
176#397—22no—
177#397—22no—
178#397—22no—
179#397—22no—
180#396—22no—
181#396—22no—
182#394—22no—
183#394—22no—
184#393—22no—
185#393—22no—
186#391—21no—
187#391—21no—
188#391—21no—
189#391—21no—
190#391—21no—
191#391—21no—
192#391C2312yes—
193#391C2312yes—
194#391C2312yes—
195#391C2312yes—
196#390—21no—
197#390—21no—
198#390—21no—
199#390—21no—
200#390—21no—
201#390—21no—
202#390C2312yes—
203#390C2312yes—
204#390C2312yes—
205#390C2312yes—
206#389—22no—
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#389C2312yes—
220#389C2312yes—
221#389C2312yes—
222#389C2312yes—
223#388—22no—
224#388—22no—
225#388—22no—
226#388—22no—
227#387—22no—
228#387—22no—
229#387—22no—
230#385—22no—
231#385—22no—
232#384—21no—
233#384—21no—
234#384C2311yescharacteristic
235#384C2311yescharacteristic
236#384C2311yescharacteristic
237#384C2311yescharacteristic
238#384C2312yes—
239#383—22no—
240#382—21no—
241#382—21no—
242#382—22no—
243#382C2312yes—
244#380—21no—
245#380—21no—
246#380—21no—
247#380—21no—
248#380C2312yes—
249#380C2312yes—
250C4×C4⋊C4—21no—
251C4×C4⋊C4—21no—
252C4×C4⋊C4—21no—
253C4×C4⋊C4—21no—
254C4×C4⋊C4D411yescharacteristic
255C4×C4⋊C4D411yescharacteristic
256C4×C4⋊C4C2312yes—
257C4×C4⋊C4C2312yes—
258C4×C4⋊C4C2311yescharacteristic
259C4×C4⋊C4C2311yescharacteristic
260C4×C22⋊C4D411yescharacteristic
261C4×C22⋊C4D411yescharacteristic
262C4×C22⋊C4C2311yescharacteristic
263C4×C22⋊C4C2311yescharacteristic
264#376D411yescharacteristic
265#376D411yescharacteristic
266C43C2311yescharacteristic
267C43C2311yescharacteristic
268#340C2314yes—
269#340C2314yes—
270#340C2314yes—
271#340C2314yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275#336C2312yes—
276C22×Q8—21no—
277C22×Q8—21no—
278C22×Q8—22no—
279C22×Q8C2412yes—
280C22×D4—22no—
281C22×D4—21no—
282C22×D4—21no—
283C22×D4C2412yes—
284C23×C4—22no—
285C23×C4C2×D411yescharacteristic
286C23×C4C2×D411yescharacteristic
287C23×C4C2411yescharacteristic
288C2×SD16—28no—
289C2×SD16—28no—
290C2×SD16—28no—
291C2×M16—24no—
292C2×M16—24no—
293C2×M16—24no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—44no—
305C4⋊Q8—44no—
306C4⋊Q8—82no—
307#127—22no—
308#127—22no—
309#126—48no—
310#125—44no—
311#124—44no—
312#124—44no—
313#124—48no—
314#123—48no—
315C22⋊Q8—48no—
316C22⋊Q8—44no—
317C22⋊Q8—44no—
318C22⋊Q8—44no—
319#121—44no—
320#121—44no—
321#121—82no—
322C4×Q8—44no—
323C4×Q8—44no—
324C4×Q8—44no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—82no—
328#117—24no—
329#117—42no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—22no—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—22no—
350C2×C4⋊C4—22no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4C2×D411yescharacteristic
354C2×C4⋊C4C2×D411yescharacteristic
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2412yes—
358C2×C4⋊C4C2412yes—
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
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⋊C4—22no—
374C2×C22⋊C4—22no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4C2×D411yescharacteristic
377C2×C22⋊C4C2×D411yescharacteristic
378C2×C22⋊C4C2×D411yescharacteristic
379C2×C22⋊C4C2×D411yescharacteristic
380C2×C42—22no—
381C2×C42—22no—
382C2×C42—22no—
383C2×C42—22no—
384C2×C42—22no—
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2411yescharacteristic
396C2×C42C2411yescharacteristic
397C2×C42C2411yescharacteristic
398C2×C42C2411yescharacteristic
399C2×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401#107—28no—
402#107—28no—
403#107—44no—
404#106—28no—
405#106—44no—
406C4⋊C8—24no—
407#103—28no—
408#103—44no—
409#103—44no—
410#102—28no—
411#102—44no—
412#102—44no—
413C22⋊C8—24no—
414#97—24no—
415#97—24no—
416#97—24no—
417C4×C8—24no—
418C4×C8—24no—
419C4×C8—22no—
420C4×C8—22no—
421C4×C8#4114yes—
422C4×C8#4114yes—
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#95—22no—
441#95C2×D411yescharacteristic
442#95C2×D411yescharacteristic
443#95C2×D411yescharacteristic
444#95C2×D411yescharacteristic
445#95C2×D411yescharacteristic
446#95C2×D411yescharacteristic
447#95C2×D411yescharacteristic
448#95C2×D411yescharacteristic
449C24—22no—
450C24C22×D411yescharacteristic
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—44no—
456C2×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
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—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—21no—
484C22×C4—21no—
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—22no—
494C22×C4—44no—
495C22×C4—82no—
496C22×C4—82no—
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×C4C22×D411yescharacteristic
510C22×C4C2511yescharacteristic
511SD16—416no—
512M16—48no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8C2×#4114yes—
522C2×C8C2×#4114yes—
523C2×C8H(5,𝔽2)14yes—
524C2×C82−1+414yes—
525C4⋊C4—22no—
526C4⋊C4—22no—
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—42no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—44no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4—42no—
554C4⋊C4—82no—
555C4⋊C4—82no—
556C4⋊C4—82no—
557C4⋊C4—82no—
558C4⋊C4—82no—
559C4⋊C4—82no—
560C22⋊C4—22no—
561C22⋊C4—22no—
562C22⋊C4—41no—
563C22⋊C4—41no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—82no—
573C22⋊C4—82no—
574C22⋊C4—82no—
575C22⋊C4—82no—
576C22⋊C4—82no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—21no—
587C42—21no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—44no—
593C42—44no—
594C42—82no—
595C42AGL(1,ℤ/8)14yes—
596C42C2×#4112yes—
597C42C2×#4112yes—
598C23—21no—
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—
612C23C23×D411yescharacteristic
613Q8—44no—
614Q8—44no—
615Q8—84no—
616Q8—84no—
617Q8—164no—
618D4—44no—
619D4—44no—
620D4—84no—
621D4—84no—
622D4—164no—
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—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—21no—
654C2×C4—44no—
655C2×C4—44no—
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—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—164no—
680C2×C4#53212yes—
681C2×C4#53212yes—
682C2×C4C2×AGL(1,ℤ/8)12yes—
683C2×C4C2×AGL(1,ℤ/8)12yes—
684C2×C4C2×AGL(1,ℤ/8)12yes—
685C2×C4#57512yes—
686C2×C4#57712yes—
687C2×C4#57812yes—
688C2×C4C22×#4111yescharacteristic
689C2×C4C22×#4111yescharacteristic
690C2×C4C2×H(5,𝔽2)12yes—
691C2×C4C2×2−1+412yes—
692C8—24no—
693C8—42no—
694C8—42no—
695C8#54218yescomplemented
696C22—41no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22C2×#53211yescharacteristic
711C22#349811yescharacteristic
712C22#349911yescharacteristic
713C22C22×AGL(1,ℤ/8)11yescharacteristic
714C22C2×#57511yescharacteristic
715C22C2×#57711yescharacteristic
716C22C2×#57811yescharacteristic
717C4—22no—
718C4—22no—
719C4—41no—
720C4—41no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—41no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4#317114yes—
736C4#317814yes—
737C4C2×#54212yes—
738C4#351112yes—
739C2—81no—
740C2—162no—
741C2#3527711yescharacteristic
742C2C2×#317111yescharacteristic
743C2C2×#317811yescharacteristic
744C2#6242111yescharacteristic
745C2#6242511yescharacteristic
746C2#6247911yescharacteristic
747C2#6248311yescharacteristic
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. #33316C2semidirect
4. #33252C2semidirect
5. #33212C2semidirect
7. #30205C2semidirect
8. #30204C2semidirect
9. #29844C2semidirect
11. #28019C2semidirect
12. #27796C2semidirect
13. #26486C2semidirect
14. #25036C2semidirect
15. #22220C2semidirect
16. #22218C2semidirect
17. C4×#500C2semidirect
18. C4×#492C2semidirect
19. #20472C2semidirect
695. C8#542semidirect

Maximal subgroups

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

subgroupindex
#352912
#352772
#333162
#332522
#332122
#331742
#302052
#302042
#298442
#298432
#280192
#277962
#264862
#250362
#222202
#222182
C4×#5002
C4×#4922
#204722