Finite Group Databasebeta

← #6506549

Subgroups — #6506549

Order 512.

1860 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35019C211yesmaximal, characteristic
2C2×#2441C211yesmaximal, characteristic, complemented
3#29336C212yesmaximal, complemented
4#29330C212yesmaximal, complemented
5#27978C212yesmaximal, complemented
6#27725C214yesmaximal, complemented
7#27723C214yesmaximal, complemented
8#27275C214yesmaximal, complemented
9#27271C214yesmaximal, complemented
10#25907C212yesmaximal, complemented
11#21765C211yesmaximal, characteristic, complemented
12#21764C211yesmaximal, characteristic
13#20912C211yesmaximal, characteristic
14#20911C211yesmaximal, characteristic, complemented
15#20524C211yesmaximal, characteristic, complemented
16C22×#125C2211yescharacteristic, complemented
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#488C2214yescomplemented
21C2×#419C2212yescomplemented
22C2×#418C2212yes—
23C2×#407C2211yescharacteristic
24C2×#407C2211yescharacteristic
25#2850C2212yes—
26#2850C2212yes—
27#2639—21no—
28#2639—21no—
29#2591—22no—
30#2561—22no—
31#2504C2212yes—
32#2503C2212yes—
33#2441—22no—
34#2441—22no—
35#2441C2214yes—
36#2441C2214yescomplemented
37#2436C2212yes—
38#2435C2212yes—
39C2×#399—21no—
40C2×#396—21no—
41#2308—21no—
42#2308—21no—
43C2×#389—21no—
44C2×#389C2211yescharacteristic, complemented
45C2×#389C2211yescharacteristic, complemented
46C2×#387—21no—
47C2×#384C2211yescharacteristic, complemented
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2214yescomplemented
51#2101C2214yescomplemented
52#2072C2214yescomplemented
53#2071C2214yescomplemented
54#2042C2214yescomplemented
55#2042C2214yescomplemented
56#2026C2214yescomplemented
57#2025C2214yescomplemented
58#1991C2214yescomplemented
59#1959C2214yescomplemented
60#1936C2214yescomplemented
61#1935C2214yescomplemented
62#1934C2214yescomplemented
63#1933C2214yescomplemented
64#1895C2214yes—
65#1894C2214yes—
66#1875C2214yes—
67#1874C2212yes—
68#1874C2212yes—
69#1874C2212yes—
70#1874C2212yes—
71#1873C2214yes—
72#1872C2212yescomplemented
73#1872C2212yescomplemented
74#1872C2212yescomplemented
75#1872C2212yescomplemented
76#1836C2214yescomplemented
77#1824C2212yes—
78#1823C2212yescomplemented
79#1787C2212yescomplemented
80#1786C2212yes—
81#1753C2214yes—
82#1753C2214yescomplemented
83#1749C2212yescomplemented
84C4×C22⋊C8C2212yescomplemented
85#1736C2212yes—
86#1736C2212yescomplemented
87C23×Q8C2311yescharacteristic
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C22×M16C2312yes—
92C22×M16C2312yes—
93C2×#125—22no—
94C2×#125—44no—
95C2×#125C2314yes—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×C22⋊Q8—21no—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8C2312yes—
103C2×C22⋊Q8C2312yes—
104C2×#121—21no—
105C2×#121—21no—
106C2×#121—24no—
107C2×#121—24no—
108C2×#121C2312yes—
109C2×#121C2312yes—
110C2×#117—21no—
111C2×#117—21no—
112C2×#117—44no—
113C2×#117C2312yes—
114C2×#117C2312yes—
115C22×C4⋊C4—21no—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2311yescharacteristic
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4—21no—
131C22×C22⋊C4—21no—
132C22×C22⋊C4D411yescharacteristic
133C22×C22⋊C4D411yescharacteristic
134C22×C42—21no—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137#488—48no—
138#419—44no—
139#418—44no—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#103C2314yes—
143C2×#103C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2314yes—
148#407—24no—
149#407—24no—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×C22⋊C8C2312yes—
153C2×C22⋊C8C2312yes—
154C2×#97C2314yes—
155C2×C4×C8C2314yes—
156#400—22no—
157#400—22no—
158#399—24no—
159#399—24no—
160#397—22no—
161#397—22no—
162#396—24no—
163#396—24no—
164#394—22no—
165#394—22no—
166#393—22no—
167#393—22no—
168#391C2312yes—
169#391C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389—22no—
174#389—24no—
175#389—24no—
176#389C2314yes—
177#389C2314yes—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#387—24no—
183#387—24no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#384—22no—
193#384C2314yes—
194#382C2312yes—
195#382C2312yes—
196#381—22no—
197#381—22no—
198#380—21no—
199#380—21no—
200#380C2312yes—
201#380C2312yes—
202#379—22no—
203#379—22no—
204C4×C22⋊C4D412yes—
205C4×C22⋊C4D412yes—
206C4×C22⋊C4C2312yes—
207C4×C22⋊C4C2312yes—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95—21no—
211C2×#95D411yescharacteristic
212C2×#95D411yescharacteristic
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#340C2314yes—
220#340C2314yes—
221#336C2314yes—
222#336C2314yes—
223C25—21no—
224C25C2×D411yescharacteristic
225C22×Q8—21no—
226C22×Q8—22no—
227C22×Q8—22no—
228C22×Q8—42no—
229C22×Q8—42no—
230C22×Q8C2412yes—
231C22×D4—22no—
232C22×D4—22no—
233C22×D4—21no—
234C22×D4—22no—
235C22×D4—22no—
236C22×D4—42no—
237C22×D4—42no—
238C22×D4—44no—
239C22×D4—44no—
240C22×D4—44no—
241C22×D4C2412yes—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—21no—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—42no—
250C23×C4—42no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4#4114yes—
259C23×C4#4114yes—
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×M16—24no—
264C2×M16—24no—
265C2×M16—42no—
266C2×M16—42no—
267C2×M16—42no—
268C2×M16—42no—
269C22×C8C2414yes—
270C22×C8C2414yes—
271#125—28no—
272#125—44no—
273#125—48no—
274#124—44no—
275#124—44no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—44no—
281C22⋊Q8—44no—
282#121—24no—
283#121—24no—
284#121—24no—
285#121—24no—
286#121—44no—
287#121—44no—
288#117—24no—
289#117—24no—
290#117—42no—
291#117—42no—
292#117—44no—
293#117—44no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—44no—
321C2×C4⋊C4—82no—
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—22no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
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—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—42no—
346C2×C22⋊C4—42no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—44no—
350C2×C22⋊C4—82no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—22no—
356C2×C42—22no—
357C2×C42—42no—
358C2×C42—42no—
359C2×C42—44no—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2412yes—
363C2×C42C2412yes—
364#103—28no—
365#103—44no—
366#103—44no—
367#103—44no—
368#102—28no—
369#102—44no—
370#102—44no—
371#102—44no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374C22⋊C8—24no—
375C22⋊C8—24no—
376#97—44no—
377C4×C8—44no—
378#95—22no—
379#95—22no—
380#95—24no—
381#95—24no—
382#95—22no—
383#95—22no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95C2×D412yes—
391#95C2×D412yes—
392#95C2×D412yes—
393#95C2×D412yes—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—41no—
398C24—41no—
399C24—41no—
400C24—41no—
401C24—42no—
402C24—42no—
403C24—42no—
404C24—81no—
405C24—81no—
406C24—81no—
407C24—81no—
408C24#12414yes—
409C24C22×D411yescharacteristic
410C24C22×D411yescharacteristic
411C24C22×D411yescharacteristic
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—24no—
415C2×Q8—24no—
416C2×Q8—48no—
417C2×Q8—42no—
418C2×Q8—81no—
419C2×Q8—81no—
420C2×Q8—84no—
421C2×D4—22no—
422C2×D4—22no—
423C2×D4—24no—
424C2×D4—24no—
425C2×D4—48no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—44no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—42no—
433C2×D4—84no—
434C2×D4—81no—
435C2×D4—81no—
436C2×D4—816no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—41no—
460C22×C4—41no—
461C22×C4—41no—
462C22×C4—41no—
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4—41no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—42no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—81no—
488C22×C4—81no—
489C22×C4—81no—
490C22×C4—81no—
491C22×C4#12414yes—
492C22×C4#12414yes—
493C22×C4#12414yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C2×#4112yes—
502C22×C4C2×#4112yes—
503C22×C4C2×#4112yes—
504C22×C4C2×#4112yes—
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4C2511yescharacteristic
508M16—84no—
509M16—84no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—24no—
524C4⋊C4—22no—
525C4⋊C4—24no—
526C4⋊C4—48no—
527C4⋊C4—48no—
528C4⋊C4—42no—
529C4⋊C4—48no—
530C4⋊C4—48no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—44no—
536C4⋊C4—44no—
537C4⋊C4—84no—
538C4⋊C4—84no—
539C4⋊C4H(5,𝔽2)14yes—
540C22⋊C4—42no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—82no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—82no—
559C22⋊C4—82no—
560C42—24no—
561C42—24no—
562C42—42no—
563C42—42no—
564C42—44no—
565C42—44no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—28no—
583C23—44no—
584C23—44no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—82no—
593C23—82no—
594C23—82no—
595C23—82no—
596C23—84no—
597C23—84no—
598C23—84no—
599C23—162no—
600C23—162no—
601C23—162no—
602C23—162no—
603C23C2×#12412yes—
604C23C2×#12412yes—
605C23C2×#12412yes—
606C23#57512yes—
607C23#57512yes—
608C23C23×D411yescharacteristic
609Q8—44no—
610Q8—44no—
611Q8—84no—
612D4—44no—
613D4—44no—
614D4—84no—
615D4—88no—
616D4—88no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—22no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—24no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—48no—
648C2×C4—42no—
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—82no—
669C2×C4—82no—
670C2×C4—84no—
671C2×C4—84no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4C2×#12412yes—
677C2×C4C2×#12412yes—
678C2×C4C2×#12412yes—
679C2×C4C22×#4111yescharacteristic
680C2×C4C22×#4111yescharacteristic
681C2×C4C2×H(5,𝔽2)12yes—
682C2×C4#58512yes—
683C8—44no—
684C8—44no—
685C22—22no—
686C22—22no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—41no—
698C22—41no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—88no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—168no—
714C22#314014yes—
715C22C22×#12411yescharacteristic
716C22#348411yescharacteristic
717C22#348511yescharacteristic
718C22C2×#57511yescharacteristic
719C22C2×#57511yescharacteristic
720C22#358511yescharacteristic
721C22#358611yescharacteristic
722C4—22no—
723C4—22no—
724C4—22no—
725C4—42no—
726C4—42no—
727C4—42no—
728C4—44no—
729C4—42no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4#350512yes—
737C4#350512yes—
738C2—24no—
739C2—42no—
740C2—42no—
741C2—162no—
742C2#3537711yescharacteristic
743C2C2×#314011yescharacteristic
744C2#6062111yescharacteristic
745C2#6157511yescharacteristic
746C2#6157711yescharacteristic
747C2#6171611yescharacteristic
748C2#6171811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
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×#2441C2semidirect
3. #29336C2semidirect
4. #29330C2semidirect
5. #27978C2semidirect
6. #27725C2semidirect
7. #27723C2semidirect
8. #27275C2semidirect
9. #27271C2semidirect
10. #25907C2semidirect
11. #21765C2semidirect
14. #20911C2semidirect
15. #20524C2semidirect
16. C22×#125C22semidirect
20. C2×#488C22semidirect
21. C2×#419C22semidirect
36. #2441C22semidirect
44. C2×#389C22semidirect
45. C2×#389C22semidirect
47. C2×#384C22semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2072C22semidirect
53. #2071C22semidirect
54. #2042C22semidirect
55. #2042C22semidirect
56. #2026C22semidirect
57. #2025C22semidirect
58. #1991C22semidirect
59. #1959C22semidirect
60. #1936C22semidirect
61. #1935C22semidirect
62. #1934C22semidirect
63. #1933C22semidirect
72. #1872C22semidirect
73. #1872C22semidirect
74. #1872C22semidirect
75. #1872C22semidirect
76. #1836C22semidirect
78. #1823C22semidirect
79. #1787C22semidirect
82. #1753C22semidirect
83. #1749C22semidirect
84. C4×C22⋊C8C22semidirect
86. #1736C22semidirect

Maximal subgroups

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

subgroupindex
#350192
C2×#24412
#293362
#293302
#279782
#277252
#277232
#272752
#272712
#259072
#217652
#217642
#209122
#209112
#205242