Finite Group Databasebeta

← #6506544

Subgroups — #6506544

Order 512.

2012 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35032C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic, complemented
3#29336C212yesmaximal, complemented
4#29329C211yesmaximal, characteristic, complemented
5#29328C211yesmaximal, characteristic, complemented
6#27978C212yesmaximal, complemented
7#27722C214yesmaximal, complemented
8#27720C214yesmaximal, complemented
9#27274C214yesmaximal, complemented
10#27270C214yesmaximal, complemented
11#25903C211yesmaximal, characteristic, complemented
12#25902C211yesmaximal, characteristic, complemented
13#21769C212yesmaximal, complemented
14#20909C212yesmaximal, complemented
15#20524C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C2×#525C2212yes—
18C22×C22⋊Q8—21no—
19C23×C4⋊C4C2211yescharacteristic
20C2×#487C2212yescomplemented
21C2×#486C2212yescomplemented
22C2×#417C2214yes—
23C2×#407C2212yes—
24#2857C2211yescharacteristic
25#2856C2211yescharacteristic
26#2850C2212yes—
27#2617—21no—
28#2617—21no—
29#2602—22no—
30#2602—22no—
31#2500C2212yes—
32#2499C2212yes—
33#2440—22no—
34#2440—22no—
35#2440C2214yes—
36#2440C2214yes—
37#2432C2212yes—
38#2431C2212yes—
39C2×#398—21no—
40C2×#395—21no—
41#2325—21no—
42#2325—21no—
43C2×#389C2211yescharacteristic
44C2×#389C2211yescharacteristic
45C2×#385—21no—
46C2×#384C2211yescharacteristic
47C2×#380C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49#2101C2214yescomplemented
50#2100C2212yescomplemented
51#2099C2212yescomplemented
52#2072C2214yescomplemented
53#2071C2214yescomplemented
54#2043C2212yescomplemented
55#2042C2214yescomplemented
56#2041C2212yescomplemented
57#2024C2214yescomplemented
58#2023C2214yescomplemented
59#1990C2212yescomplemented
60#1990C2212yescomplemented
61#1959C2214yescomplemented
62#1936C2214yescomplemented
63#1935C2214yescomplemented
64#1934C2214yescomplemented
65#1933C2214yescomplemented
66#1895C2214yes—
67#1894C2214yes—
68#1874C2212yes—
69#1874C2212yes—
70#1874C2212yes—
71#1874C2214yes—
72#1874C2212yes—
73#1872C2212yes—
74#1872C2212yes—
75#1872C2212yes—
76#1872C2214yes—
77#1872C2212yes—
78#1836C2214yescomplemented
79#1825C2214yes—
80#1788C2214yes—
81#1753C2214yes—
82#1753C2214yes—
83#1749C2212yes—
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2211yescharacteristic
86#1736C2214yes—
87C22×#41C2312yes—
88C23×Q8—21no—
89C24×C4—21no—
90C24×C4C2311yescharacteristic
91C22×M16C2314yes—
92C2×C4⋊Q8—21no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—44no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2312yes—
97C2×C4⋊Q8C2312yes—
98C2×#127C2311yescharacteristic
99C2×#124C2312yes—
100#525—28no—
101#525—28no—
102C2×#123—22no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8—24no—
105C2×C22⋊Q8—48no—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×#121—21no—
109C2×#121—21no—
110C2×#121C2312yes—
111C2×#121C2312yes—
112C2×#120—22no—
113C2×C4×D4—22no—
114C2×C4×D4—22no—
115C2×#117C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—22no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—44no—
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C4⋊C4C2311yescharacteristic
129C22×C22⋊C4—21no—
130C22×C22⋊C4D411yescharacteristic
131C22×C22⋊C4D411yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#487—44no—
135#486—44no—
136#417—48no—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#102C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145#407—28no—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×#97C2314yes—
150C2×C4×C8C2312yes—
151C2×C4×C8C2312yes—
152#398—24no—
153#398—24no—
154#397—22no—
155#397—22no—
156#395—24no—
157#395—24no—
158#394—22no—
159#394—22no—
160#394—22no—
161#394—22no—
162#392—22no—
163#392—22no—
164#391C2312yes—
165#391C2312yes—
166#390—21no—
167#390—21no—
168#390C2312yes—
169#390C2312yes—
170#389—21no—
171#389—21no—
172#389—22no—
173#389—21no—
174#389—21no—
175#389C2312yes—
176#389C2312yes—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—22no—
183#387—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#385—24no—
189#385—24no—
190#385—22no—
191#385—22no—
192#384—22no—
193#384C2314yes—
194#384C2311yescharacteristic
195#384C2311yescharacteristic
196#382C2312yes—
197#380—21no—
198#380—21no—
199#380—22no—
200#380C2314yes—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4C2312yes—
204C4×C22⋊C4C2312yes—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95D411yescharacteristic
209C2×#95D411yescharacteristic
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2312yes—
220#336C2312yes—
221C25C2×D411yescharacteristic
222C2×#41—24no—
223C2×#41—24no—
224C2×#41—44no—
225C2×#41—44no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—44no—
229C22×Q8—44no—
230C22×Q8—82no—
231C22×Q8C2412yes—
232C22×D4—22no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—21no—
237C22×D4C2412yes—
238C23×C4—21no—
239C23×C4—21no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—24no—
243C23×C4—22no—
244C23×C4—42no—
245C23×C4—42no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—82no—
249C23×C4C2×D411yescharacteristic
250C23×C4C2×D411yescharacteristic
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D412yes—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4#4118yes—
256C23×C4C2411yescharacteristic
257C23×C4C2412yes—
258C2×M16—28no—
259C2×M16—44no—
260C2×M16—44no—
261C22×C8C2414yes—
262C22×C8C2414yes—
263C4⋊Q8—24no—
264C4⋊Q8—24no—
265C4⋊Q8—42no—
266C4⋊Q8—42no—
267C4⋊Q8—42no—
268C4⋊Q8—44no—
269C4⋊Q8—44no—
270#127—42no—
271#124—44no—
272#123—48no—
273C22⋊Q8—24no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276C22⋊Q8—24no—
277C22⋊Q8—416no—
278C22⋊Q8—416no—
279#121—24no—
280#121—24no—
281#121—24no—
282#121—24no—
283#121—44no—
284#121—44no—
285#120—48no—
286C4×D4—48no—
287C4×D4—48no—
288#117—24no—
289#117—24no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—28no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—44no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—48no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—44no—
322C2×C4⋊C4—82no—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2412yes—
328C2×C4⋊C4C2412yes—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—22no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—22no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4C2×D412yes—
345C2×C22⋊C4C2×D412yes—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C42—22no—
349C2×C42—42no—
350C2×C42—42no—
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D412yes—
354C2×C42C2412yes—
355C2×C42C2411yescharacteristic
356C2×C42C2411yescharacteristic
357#103—28no—
358#103—44no—
359#103—44no—
360#103—44no—
361#102—28no—
362#102—44no—
363#102—44no—
364#102—44no—
365C22⋊C8—28no—
366C22⋊C8—24no—
367C22⋊C8—24no—
368#97—44no—
369C4×C8—42no—
370C4×C8—42no—
371#95—24no—
372#95—24no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95C2×D412yes—
382#95C2×D412yes—
383#95C2×D412yes—
384#95C2×D412yes—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—41no—
389C24—41no—
390C24—41no—
391C24—41no—
392C24#12414yes—
393C24C22×D412yes—
394C24C22×D411yescharacteristic
395#41—416no—
396#41—88no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—24no—
400C2×Q8—24no—
401C2×Q8—44no—
402C2×Q8—44no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—82no—
406C2×Q8—82no—
407C2×Q8—816no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—44no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4—81no—
421C2×D4—81no—
422C2×D4—81no—
423C2×D4—82no—
424C2×D4—82no—
425C2×D4—82no—
426C2×D4—82no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—416no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—42no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—48no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—48no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—44no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—84no—
471C22×C4—816no—
472C22×C4#12414yes—
473C22×C4#12418yes—
474C22×C4AGL(1,ℤ/8)14yes—
475C22×C4#13714yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4114yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4114yes—
487C22×C4C2511yescharacteristic
488M16—88no—
489C2×C8—24no—
490C2×C8—24no—
491C2×C8—24no—
492C2×C8—24no—
493C2×C8—24no—
494C2×C8—24no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—81no—
521C4⋊C4—81no—
522C4⋊C4—84no—
523C4⋊C4—81no—
524C4⋊C4—81no—
525C4⋊C4H(5,𝔽2)14yes—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—48no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C42—22no—
543C42—22no—
544C42—24no—
545C42—42no—
546C42—41no—
547C42—41no—
548C42—44no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—22no—
554C23—22no—
555C23—28no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23C2×#12414yes—
574C23C2×#12412yes—
575C23C2×AGL(1,ℤ/8)12yes—
576C23C2×#13712yes—
577C23C23×D411yescharacteristic
578Q8—44no—
579Q8—44no—
580Q8—88no—
581Q8—88no—
582D4—44no—
583D4—44no—
584D4—88no—
585D4—88no—
586D4—88no—
587D4—162no—
588D4—162no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—22no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—44no—
612C2×C4—48no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—48no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—88no—
639C2×C4—816no—
640C2×C4—84no—
641C2×C4—84no—
642C2×C4—84no—
643C2×C4C2×#12414yes—
644C2×C4C2×#12412yes—
645C2×C4C2×AGL(1,ℤ/8)12yes—
646C2×C4C2×AGL(1,ℤ/8)12yes—
647C2×C4C2×#13712yes—
648C2×C4C2×#13712yes—
649C2×C4C22×#4112yes—
650C2×C4C2×H(5,𝔽2)12yes—
651C2×C4#58512yes—
652C8—44no—
653C8—44no—
654C22—22no—
655C22—22no—
656C22—22no—
657C22—22no—
658C22—22no—
659C22—22no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—416no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—42no—
668C22—82no—
669C22—82no—
670C22—82no—
671C22—82no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22#314114yes—
676C22C22×#12411yescharacteristic
677C22#348411yescharacteristic
678C22#348511yescharacteristic
679C22C22×AGL(1,ℤ/8)11yescharacteristic
680C22C22×#13711yescharacteristic
681C22#358511yescharacteristic
682C22#358611yescharacteristic
683C4—24no—
684C4—22no—
685C4—22no—
686C4—22no—
687C4—22no—
688C4—22no—
689C4—42no—
690C4—42no—
691C4—44no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4#350514yes—
699C2—24no—
700C2—44no—
701C2—162no—
702C2#3537711yescharacteristic
703C2C2×#314111yescharacteristic
704C2#6062111yescharacteristic
705C2#6157111yescharacteristic
706C2#6157411yescharacteristic
707C2#6171711yescharacteristic
708C2#6171911yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35032C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2440C2semidirect
3. #29336C2semidirect
4. #29329C2semidirect
5. #29328C2semidirect
6. #27978C2semidirect
7. #27722C2semidirect
8. #27720C2semidirect
9. #27274C2semidirect
10. #27270C2semidirect
11. #25903C2semidirect
12. #25902C2semidirect
13. #21769C2semidirect
14. #20909C2semidirect
15. #20524C2semidirect
20. C2×#487C22semidirect
21. C2×#486C22semidirect
49. #2101C22semidirect
50. #2100C22semidirect
51. #2099C22semidirect
52. #2072C22semidirect
53. #2071C22semidirect
54. #2043C22semidirect
55. #2042C22semidirect
56. #2041C22semidirect
57. #2024C22semidirect
58. #2023C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1959C22semidirect
62. #1936C22semidirect
63. #1935C22semidirect
64. #1934C22semidirect
65. #1933C22semidirect
78. #1836C22semidirect

Maximal subgroups

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

subgroupindex
#350322
C2×#24402
#293362
#293292
#293282
#279782
#277222
#277202
#272742
#272702
#259032
#259022
#217692
#209092
#205242