Finite Group Databasebeta

← #6345465

Subgroups — #6345465

Order 512.

1785 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60949C211yesmaximal, characteristic
2C2×#2969C211yesmaximal, characteristic, complemented
3#60144C211yesmaximal, characteristic, complemented
4#18076C218yesmaximal, complemented
5#18006C218yesmaximal, complemented
6#17935C212yesmaximal, complemented
7#17929C212yesmaximal, complemented
8#17908C212yesmaximal, complemented
9#17900C212yesmaximal, complemented
10#17809C211yesmaximal, characteristic, complemented
11#17492C211yesmaximal, characteristic, complemented
12#17463C211yesmaximal, characteristic, complemented
13#17334C211yesmaximal, characteristic, complemented
14C2×#568—21no—
15C2×#532C2211yescharacteristic
16C2×#529C2211yescharacteristic
17#3251C414yes—
18#3239C412yes—
19#3234C412yes—
20#3217C412yes—
21#3213C412yes—
22#3193C412yes—
23#3192C412yes—
24#3171C2214yes—
25#3105C2214yes—
26#3103C2212yes—
27#3100C2212yes—
28#2969—22no—
29#2969—22no—
30#2969C2214yes—
31#2969C2214yescomplemented
32C2×#446C2211yescharacteristic, complemented
33C2×C8×Q8C2211yescharacteristic, complemented
34#2935—22no—
35#2935C2214yescomplemented
36D4×M16—22no—
37D4×M16C2214yescomplemented
38#2932—21no—
39#2932—21no—
40#2931C2212yes—
41#2931C2212yes—
42C2×#436—21no—
43C2×#436C2211yescharacteristic, complemented
44C2×#435—21no—
45C2×C8×D4C2211yescharacteristic, complemented
46#2920C2211yescharacteristic
47#2920C2211yescharacteristic
48C2×#433—21no—
49C2×#431C2211yescharacteristic
50C2×#423C2211yescharacteristic
51#2879—21no—
52C2×#405—21no—
53C2×C4×M16C2211yescharacteristic
54#1594C2218yescomplemented
55#1590C2218yescomplemented
56#1588C2218yescomplemented
57#1586C2218yescomplemented
58#1582C2218yescomplemented
59#1581C2218yescomplemented
60SD16⋊C8C2218yescomplemented
61C8×D8C2218yescomplemented
62#1574C2214yescomplemented
63#1571C2212yescomplemented
64#1570C2212yescomplemented
65#1565C2214yescomplemented
66#1495C2214yes—
67#1487C2212yescomplemented
68#1486C2212yescomplemented
69#1484C2214yescomplemented
70#1483C2214yescomplemented
71#1480C2212yes—
72#1480C2212yes—
73#1478C2214yescomplemented
74#1455C2214yescomplemented
75#1453C2214yescomplemented
76#1452C2212yescomplemented
77#1450C2212yescomplemented
78C22×#41—21no—
79#568—48no—
80C2×#131—24no—
81C2×#131—22no—
82C2×#131—22no—
83C22×M16—21no—
84C22×M16—24no—
85C22×M16—21no—
86C22×M16D411yescharacteristic
87C22×M16D411yescharacteristic
88C23×C8—21no—
89#532—21no—
90#532—21no—
91#532—21no—
92#532—21no—
93#532C2×C412yes—
94#532C2×C412yes—
95#532C2312yes—
96#532C2312yes—
97C2×C4⋊Q8C2×C411yescharacteristic
98C2×#127C2×C411yescharacteristic
99#529—21no—
100#529—21no—
101#529—21no—
102#529—21no—
103#529C2×C412yes—
104#529C2×C412yes—
105#529C2312yes—
106#529C2312yes—
107C2×#126—21no—
108C2×#125C2×C411yescharacteristic
109C2×#124—21no—
110C2×#124C2×C411yescharacteristic
111C2×#123—21no—
112C2×C22⋊Q8C2×C411yescharacteristic
113C2×#121—21no—
114C2×#121C2×C411yescharacteristic
115C2×#121C2×C412yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4C2311yescharacteristic
120C2×C4×D4C2312yes—
121C2×#117—21no—
122C22×C42C2311yescharacteristic
123#501C2×C414yes—
124#496C2×C414yes—
125#490C2×C414yes—
126#489C2×C414yes—
127#484C2×C414yes—
128#481C2×C414yes—
129#481C2×C414yes—
130#480C2×C414yes—
131#478C2×C418yes—
132#475C2×C414yes—
133#474C2×C414yes—
134#469C2×C418yes—
135#468C2×C418yes—
136#463C2×C418yes—
137#460C2×C414yes—
138#459C2×C414yes—
139#450C2×C414yes—
140#448C2×C414yes—
141#447C2×C414yes—
142#447C2×C414yes—
143#446—22no—
144#446C2314yes—
145C8×Q8—22no—
146C8×Q8C2314yes—
147C4×SD16C2318yes—
148C4×D8C2318yes—
149#436—24no—
150#436—24no—
151#436—22no—
152#436—22no—
153#436—22no—
154#436C2314yes—
155#436C2314yes—
156#436C2314yes—
157#435—24no—
158#435—24no—
159#435—24no—
160#435—24no—
161#435—24no—
162#435—24no—
163C8×D4—22no—
164C8×D4—22no—
165C8×D4—22no—
166C8×D4C2314yes—
167C8×D4C2314yes—
168C8×D4C2314yes—
169#433—24no—
170#433—24no—
171#432—21no—
172#432—21no—
173#432C2×C412yes—
174#432C2×C412yes—
175#431—21no—
176#431—21no—
177#431C2312yes—
178#431C2312yes—
179#431C2312yes—
180#431C2312yes—
181#431C2312yes—
182#431C2312yes—
183#423—21no—
184#423—21no—
185#423—21no—
186#423C2×C412yes—
187#423C2312yes—
188#423C2312yes—
189C2×C4⋊C8—21no—
190C2×C4⋊C8—21no—
191C2×C4⋊C8—21no—
192C2×C4⋊C8C2312yes—
193C2×C4⋊C8C2311yescharacteristic
194C2×C4⋊C8C2311yescharacteristic
195#408—24no—
196#408—22no—
197#408—22no—
198#407—21no—
199#407—24no—
200#407C2×C412yes—
201C2×C22⋊C8—21no—
202C2×C22⋊C8—21no—
203C2×C22⋊C8—21no—
204#405—24no—
205#405—24no—
206C4×M16D412yes—
207C4×M16D412yes—
208C4×M16C2312yes—
209C4×M16C2312yes—
210C4×M16C2312yes—
211C4×M16C2312yes—
212C2×#97—21no—
213C2×#97D411yescharacteristic
214C2×#97D411yescharacteristic
215C2×C4×C8—21no—
216C2×C4×C8—21no—
217C2×C4×C8C2311yescharacteristic
218C2×C4×C8C2311yescharacteristic
219#335C2318yes—
220#334C2318yes—
221Q8⋊C8C2318yes—
222D4⋊C8C2318yes—
223D4⋊C8C2318yes—
224D4⋊C8C2318yes—
225#322C2314yes—
226#322C2314yes—
227#322C2314yes—
228C82C2314yes—
229C2×#41—22no—
230C2×#41—22no—
231C2×#41—22no—
232C2×#41—22no—
233C2×#41—44no—
234C2×#41—44no—
235C22×Q8—21no—
236C22×Q8C22×C411yescharacteristic
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4C22×C411yescharacteristic
242C22×D4C22×C412yes—
243C23×C4—21no—
244C23×C4—22no—
245C23×C4C22×C411yescharacteristic
246C23×C4C2×D411yescharacteristic
247C23×C4#4111yescharacteristic
248C2×SD16C22×C418yes—
249C2×D8C22×C418yes—
250#131—48no—
251#131—416no—
252#131—48no—
253C2×M16—22no—
254C2×M16—22no—
255C2×M16—22no—
256C2×M16—24no—
257C2×M16—24no—
258C2×M16—24no—
259C2×M16—24no—
260C2×M16—42no—
261C2×M16—44no—
262C2×M16—416no—
263C2×M16—44no—
264C2×M16C22×C414yes—
265C2×M16C2×D412yes—
266C2×M16C2×D412yes—
267C2×M16C2×D412yes—
268C2×M16C2×D412yes—
269C22×C8—21no—
270C22×C8—22no—
271C22×C8—21no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8—44no—
276C22×C8C2×D411yescharacteristic
277C22×C8C2×D411yescharacteristic
278C22×C8C2×D411yescharacteristic
279C22×C8C2×D411yescharacteristic
280C4⋊Q8—21no—
281C4⋊Q8—21no—
282C4⋊Q8C22×C412yes—
283C4⋊Q8C22×C412yes—
284#127—21no—
285#127—21no—
286#127C22×C412yes—
287#127C22×C412yes—
288#126—22no—
289#126—22no—
290#126—22no—
291#126—22no—
292#125—21no—
293#125—21no—
294#125C22×C412yes—
295#125C22×C412yes—
296#124—22no—
297#124—22no—
298#124—21no—
299#124—21no—
300#124—22no—
301#124—22no—
302#124C22×C412yes—
303#124C22×C412yes—
304#123—22no—
305#123—22no—
306#123—22no—
307#123—22no—
308C22⋊Q8—21no—
309C22⋊Q8—21no—
310C22⋊Q8C22×C412yes—
311C22⋊Q8C22×C412yes—
312#121—22no—
313#121—22no—
314#121—21no—
315#121—21no—
316#121—22no—
317#121—22no—
318#121—22no—
319#121—22no—
320#121C22×C412yes—
321#121C22×C412yes—
322#121C22×C414yes—
323#121C22×C414yes—
324C4×Q8—21no—
325C4×Q8—21no—
326C4×Q8C22×C412yes—
327C4×Q8C2412yes—
328C4×D4—22no—
329C4×D4—22no—
330C4×D4—22no—
331C4×D4—22no—
332C4×D4—22no—
333C4×D4—22no—
334C4×D4—22no—
335C4×D4—22no—
336C4×D4—21no—
337C4×D4—22no—
338C4×D4—21no—
339C4×D4—22no—
340C4×D4C22×C412yes—
341C4×D4C22×C414yes—
342C4×D4C2412yes—
343C4×D4C2414yes—
344#117—22no—
345#117—22no—
346#117—22no—
347#117—22no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4C22×C412yes—
354C2×C4⋊C4C22×C411yescharacteristic
355C2×C4⋊C4C22×C411yescharacteristic
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—21no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—21no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—21no—
363C2×C42—21no—
364C2×C42C22×C411yescharacteristic
365C2×C42C22×C411yescharacteristic
366C2×C42C22×C411yescharacteristic
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42#4111yescharacteristic
371C2×C42#4111yescharacteristic
372C2×C42#4111yescharacteristic
373C2×C42C2411yescharacteristic
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107C22×C418yes—
377#106C22×C418yes—
378C4⋊C8—24no—
379C4⋊C8—24no—
380C4⋊C8—22no—
381C4⋊C8—22no—
382C4⋊C8—22no—
383C4⋊C8—21no—
384C4⋊C8—21no—
385C4⋊C8—22no—
386C4⋊C8—22no—
387C4⋊C8C22×C414yes—
388C4⋊C8C22×C414yes—
389C4⋊C8C2412yes—
390C4⋊C8C2412yes—
391C4⋊C8C2414yes—
392#103C22×C418yes—
393#102C22×C418yes—
394#102C22×C418yes—
395#102C22×C418yes—
396C22⋊C8—22no—
397C22⋊C8—22no—
398C22⋊C8—22no—
399C22⋊C8—22no—
400C22⋊C8—24no—
401C22⋊C8—24no—
402C22⋊C8—24no—
403C22⋊C8—24no—
404C22⋊C8—24no—
405C22⋊C8—24no—
406C22⋊C8C22×C414yes—
407C22⋊C8C22×C414yes—
408#97—24no—
409#97—22no—
410#97C22×C414yes—
411#97C2×D412yes—
412#97C2×D412yes—
413#97C2×D412yes—
414#97C2×D412yes—
415C4×C8—24no—
416C4×C8—22no—
417C4×C8—22no—
418C4×C8C2×D412yes—
419C4×C8C2×D412yes—
420C4×C8C2414yes—
421C4×C8C2414yes—
422C4×C8C2412yes—
423C4×C8C2412yes—
424C24—21no—
425C24—22no—
426C24C4×D411yescharacteristic
427#41—44no—
428#41—48no—
429#41—48no—
430#41—44no—
431#41—44no—
432#41—44no—
433C2×Q8—21no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—42no—
437C2×Q8—44no—
438C2×Q8C23×C412yes—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—22no—
443C2×D4—21no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—42no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—48no—
453C2×D4—82no—
454C2×D4—82no—
455C2×D4C23×C412yes—
456C2×D4C23×C414yes—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—81no—
473C22×C4C4×D411yescharacteristic
474C22×C4C4×D411yescharacteristic
475C22×C4C4×D411yescharacteristic
476C22×C4C4×D411yescharacteristic
477C22×C4C4×D411yescharacteristic
478C22×C4C4×D411yescharacteristic
479C22×C4C4×D411yescharacteristic
480C22×C4C4×D411yescharacteristic
481C22×C4C4×D411yescharacteristic
482C22×C4C23×C411yescharacteristic
483C22×C4C23×C411yescharacteristic
484C22×C4C23×C411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4111yescharacteristic
489C22×C4C2×#4111yescharacteristic
490C22×C4C2×#4111yescharacteristic
491SD16—48no—
492D8—48no—
493M16—24no—
494M16—24no—
495M16—42no—
496M16—42no—
497M16—44no—
498M16—48no—
499M16—48no—
500M16—44no—
501M16—44no—
502M16—48no—
503M16—82no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—48no—
512C2×C8—44no—
513C2×C8—42no—
514C2×C8—42no—
515C2×C8—48no—
516C2×C8—48no—
517C2×C8AGL(1,ℤ/8)14yes—
518C2×C8C23×C418yes—
519C2×C8C22×D412yes—
520C2×C8C22×D412yes—
521C2×C8C22×D412yes—
522C2×C8C22×D412yes—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—22no—
536C4⋊C4C23×C412yes—
537C4⋊C4C23×C412yes—
538C4⋊C4C23×C414yes—
539C22⋊C4—22no—
540C22⋊C4—22no—
541C22⋊C4—22no—
542C22⋊C4—22no—
543C22⋊C4—22no—
544C22⋊C4—22no—
545C22⋊C4—24no—
546C22⋊C4—24no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—24no—
550C22⋊C4—22no—
551C22⋊C4—24no—
552C22⋊C4—22no—
553C42—22no—
554C42—22no—
555C42C4×D411yescharacteristic
556C42C4×D411yescharacteristic
557C42C4×D411yescharacteristic
558C42C4×D411yescharacteristic
559C42C4×D411yescharacteristic
560C42C4×D411yescharacteristic
561C42C23×C411yescharacteristic
562C42C23×C411yescharacteristic
563C42C23×C411yescharacteristic
564C42C22×D411yescharacteristic
565C42C22×D411yescharacteristic
566C42C22×D411yescharacteristic
567C42C2×#4111yescharacteristic
568C42C2×#4111yescharacteristic
569C42C2×#4111yescharacteristic
570C42C2511yescharacteristic
571C23—22no—
572C23—22no—
573C23—24no—
574C23—41no—
575C23—41no—
576C23—42no—
577C23—42no—
578C23—44no—
579C23—42no—
580C23—82no—
581C23—82no—
582C23C2×C4×D411yescharacteristic
583C23C2×C4×D411yescharacteristic
584C23C2×C4×D411yescharacteristic
585Q8—24no—
586Q8—42no—
587Q8—44no—
588Q8—44no—
589D4—24no—
590D4—28no—
591D4—42no—
592D4—44no—
593D4—44no—
594D4—48no—
595D4—48no—
596D4—44no—
597D4—82no—
598D4—82no—
599D4—82no—
600D4—82no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4C2×C4×D411yescharacteristic
635C2×C4C2×C4×D411yescharacteristic
636C2×C4C2×C4×D411yescharacteristic
637C2×C4C2×C4×D411yescharacteristic
638C2×C4C2×C4×D411yescharacteristic
639C2×C4C2×C4×D411yescharacteristic
640C2×C4C2×C4×D411yescharacteristic
641C2×C4C2×C4×D411yescharacteristic
642C2×C4C2×C4×D411yescharacteristic
643C2×C4C2×AGL(1,ℤ/8)12yes—
644C2×C4#57512yes—
645C2×C4C24×C411yescharacteristic
646C2×C4C23×D411yescharacteristic
647C2×C4C22×#4111yescharacteristic
648C8—28no—
649C8—22no—
650C8—22no—
651C8—22no—
652C8—44no—
653C8—42no—
654C8—44no—
655C8—42no—
656C8C2×AGL(1,ℤ/8)14yes—
657C22—21no—
658C22—21no—
659C22—21no—
660C22—21no—
661C22—21no—
662C22—21no—
663C22—21no—
664C22—21no—
665C22—44no—
666C22—42no—
667C22—42no—
668C22—44no—
669C22—44no—
670C22—42no—
671C22—82no—
672C22—84no—
673C22—164no—
674C22C4×AGL(1,ℤ/8)12yes—
675C22C22×C4×D411yescharacteristic
676C4—21no—
677C4—21no—
678C4—21no—
679C4—21no—
680C4—21no—
681C4—21no—
682C4—21no—
683C4—21no—
684C4—21no—
685C4—42no—
686C4—42no—
687C4—42no—
688C4—44no—
689C4C4×AGL(1,ℤ/8)12yes—
690C4#357312yes—
691C4C22×AGL(1,ℤ/8)11yescharacteristic
692C4C2×#57511yescharacteristic
693C2—22no—
694C2—41no—
695C2—41no—
696C2—82no—
697C2—84no—
698C2#6014211yescharacteristic
699C2C2×C4×AGL(1,ℤ/8)11yescharacteristic
700C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC42C25
socleC22C22×C4×D4
prime radical#60949C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2969C2semidirect
3. #60144C2semidirect
4. #18076C2semidirect
5. #18006C2semidirect
6. #17935C2semidirect
7. #17929C2semidirect
8. #17908C2semidirect
9. #17900C2semidirect
10. #17809C2semidirect
11. #17492C2semidirect
12. #17463C2semidirect
13. #17334C2semidirect
31. #2969C22semidirect
32. C2×#446C22semidirect
33. C2×C8×Q8C22semidirect
35. #2935C22semidirect
37. D4×M16C22semidirect
43. C2×#436C22semidirect
45. C2×C8×D4C22semidirect
54. #1594C22semidirect
55. #1590C22semidirect
56. #1588C22semidirect
57. #1586C22semidirect
58. #1582C22semidirect
59. #1581C22semidirect
60. SD16⋊C8C22semidirect
61. C8×D8C22semidirect
62. #1574C22semidirect
63. #1571C22semidirect
64. #1570C22semidirect
65. #1565C22semidirect
67. #1487C22semidirect
68. #1486C22semidirect
69. #1484C22semidirect
70. #1483C22semidirect
73. #1478C22semidirect
74. #1455C22semidirect
75. #1453C22semidirect
76. #1452C22semidirect
77. #1450C22semidirect

Maximal subgroups

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

subgroupindex
#609492
C2×#29692
#601442
#180762
#180062
#179352
#179292
#179082
#179002
#178092
#174922
#174632
#173342