Finite Group Databasebeta

← #6500727

Subgroups — #6500727

Order 512.

2560 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2407C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#27586C211yesmaximal, characteristic, complemented
4#27585C211yesmaximal, characteristic, complemented
5#27583C211yesmaximal, characteristic, complemented
6#27582C211yesmaximal, characteristic, complemented
7#24876C214yesmaximal, complemented
8#24874C214yesmaximal, complemented
9#24872C214yesmaximal, complemented
10#24870C214yesmaximal, complemented
11#24010C212yesmaximal, complemented
12#24009C212yesmaximal, complemented
13#21941C212yesmaximal, complemented
14#21738C211yesmaximal, characteristic, complemented
15#20597C212yesmaximal, complemented
16C23×#41C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C2×#485C2214yescomplemented
21C2×#482C2214yescomplemented
22C2×#416C2214yescomplemented
23C2×#416C2214yescomplemented
24C22×C22⋊C8C2212yes—
25#2407—22no—
26#2407—22no—
27#2407C2214yes—
28#2407C2214yes—
29C2×#394—21no—
30C2×#392—21no—
31#2357—22no—
32#2357—22no—
33#2354C2212yescomplemented
34#2354C2212yescomplemented
35#2354C2212yescomplemented
36#2354C2212yescomplemented
37#2312—21no—
38#2312—21no—
39#2309—21no—
40C2×#391C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#386—21no—
43C2×#385—21no—
44#2285C2211yescharacteristic
45#2285C2211yescharacteristic
46#2280C2212yes—
47#2280C2212yes—
48C2×#380C2211yescharacteristic
49#2274—21no—
50#2096C2212yescomplemented
51#2095C2212yescomplemented
52#2094C2212yescomplemented
53#2093C2212yescomplemented
54#2072C2212yescomplemented
55#2072C2212yescomplemented
56#2071C2212yescomplemented
57#2071C2212yescomplemented
58#2036C2214yescomplemented
59#2035C2214yescomplemented
60#2034C2214yescomplemented
61#2033C2214yescomplemented
62#1928C2214yescomplemented
63#1927C2214yescomplemented
64#1926C2214yescomplemented
65#1925C2214yescomplemented
66#1875C2214yescomplemented
67#1873C2214yescomplemented
68#1869C2214yescomplemented
69#1869C2214yescomplemented
70#1854C2214yescomplemented
71#1854C2214yescomplemented
72#1853C2212yes—
73#1853C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1817C2211yescharacteristic
77#1817C2211yescharacteristic
78#1808C2214yescomplemented
79#1807C2214yescomplemented
80#1788C2214yescomplemented
81#1788C2214yescomplemented
82#1787C2212yes—
83#1787C2212yes—
84#1786C2212yes—
85#1786C2212yes—
86#1779C2212yes—
87C22×#41—22no—
88C22×#41—28no—
89C22×#41—44no—
90C22×#41—44no—
91C22×#41C2314yes—
92C23×Q8C2311yescharacteristic
93C23×D4—21no—
94C23×D4—21no—
95C23×D4D411yescharacteristic
96C23×D4C2311yescharacteristic
97C24×C4—21no—
98C24×C4C2311yescharacteristic
99C24×C4C2311yescharacteristic
100C23×C8C2314yes—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—44no—
103C2×C22⋊Q8C2314yes—
104C2×#121—22no—
105C2×#121—44no—
106C2×#121C2314yes—
107C2×C4×D4—28no—
108C2×C4×D4D412yes—
109C2×C4×D4D412yes—
110C2×C4×D4C2312yes—
111C2×C4×D4C2312yes—
112C2×#117—21no—
113C2×#117—21no—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C2×#117C2312yes—
117C2×#117C2312yes—
118C22×C4⋊C4—21no—
119C22×C4⋊C4D411yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C22⋊C4—21no—
123C22×C22⋊C4—21no—
124C22×C22⋊C4—21no—
125C22×C22⋊C4—21no—
126C22×C22⋊C4—21no—
127C22×C22⋊C4D411yescharacteristic
128C22×C22⋊C4D411yescharacteristic
129C22×C42C2311yescharacteristic
130#485—48no—
131#482—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×#106C2314yes—
136#416—48no—
137#416—48no—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146C2×C22⋊C8—28no—
147C2×C22⋊C8C2312yes—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151C2×C22⋊C8C2314yes—
152#394—24no—
153#394—24no—
154#392—24no—
155#392—24no—
156#391—21no—
157#391—21no—
158#391C2312yes—
159#391C2312yes—
160#390—22no—
161#390—22no—
162#390—22no—
163#390—22no—
164#390—21no—
165#390—21no—
166#390C2312yes—
167#390C2312yes—
168#388—22no—
169#388—22no—
170#388—22no—
171#388—22no—
172#388—22no—
173#388—22no—
174#388—22no—
175#388—22no—
176#386—24no—
177#386—24no—
178#385—24no—
179#385—24no—
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#382—21no—
185#382—21no—
186#381—22no—
187#381—22no—
188#380—21no—
189#380—21no—
190#380—22no—
191#380C2312yes—
192#380C2314yes—
193#380C2312yes—
194#380C2312yes—
195#380C2312yes—
196#380C2312yes—
197#380C2312yes—
198#379—22no—
199#379—22no—
200C4×C4⋊C4—22no—
201C4×C4⋊C4—22no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—21no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4C2312yes—
207C4×C4⋊C4C2312yes—
208C4×C22⋊C4—22no—
209C4×C22⋊C4—22no—
210C4×C22⋊C4—22no—
211C4×C22⋊C4—22no—
212C2×#95—21no—
213C2×#95—21no—
214C2×#95—21no—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#340C2314yes—
219#336C2312yes—
220#336C2312yes—
221#336C2312yes—
222#336C2312yes—
223C25—21no—
224C25C2×D411yescharacteristic
225C25C2×D411yescharacteristic
226C2×#41—28no—
227C2×#41—28no—
228C2×#41—28no—
229C2×#41—28no—
230C2×#41—44no—
231C2×#41—44no—
232C2×#41—48no—
233C2×#41—48no—
234C2×#41—464no—
235C2×#41—48no—
236C22×Q8—24no—
237C22×Q8—21no—
238C22×Q8—24no—
239C22×Q8—42no—
240C22×Q8C2412yes—
241C22×D4—24no—
242C22×D4—24no—
243C22×D4—24no—
244C22×D4—24no—
245C22×D4—24no—
246C22×D4—21no—
247C22×D4—24no—
248C22×D4—42no—
249C22×D4—42no—
250C22×D4—44no—
251C22×D4—44no—
252C22×D4—44no—
253C22×D4—44no—
254C22×D4—44no—
255C22×D4—82no—
256C22×D4C2×D412yes—
257C22×D4C2×D412yes—
258C22×D4C2412yes—
259C23×C4—21no—
260C23×C4—24no—
261C23×C4—21no—
262C23×C4—21no—
263C23×C4—24no—
264C23×C4—42no—
265C23×C4—42no—
266C23×C4—42no—
267C23×C4—42no—
268C23×C4—44no—
269C23×C4—44no—
270C23×C4—44no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D412yes—
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D412yes—
277C23×C4#4118yes—
278C23×C4C2412yes—
279C23×C4C2411yescharacteristic
280C23×C4C2412yes—
281C22×C8—28no—
282C22×C8—44no—
283C22×C8—44no—
284C22×C8C2414yes—
285C22×C8C2414yes—
286C22⋊Q8—28no—
287C22⋊Q8—44no—
288C22⋊Q8—48no—
289#121—28no—
290#121—44no—
291#121—48no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—24no—
297C4×D4—24no—
298#117—44no—
299#117—44no—
300#117—44no—
301#117—44no—
302#117—44no—
303#117—44no—
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—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4C2×D412yes—
320C2×C4⋊C4C2×D412yes—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—22no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—24no—
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—22no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—44no—
349C2×C22⋊C4—82no—
350C2×C22⋊C4—82no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C22⋊C4C2×D412yes—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2411yescharacteristic
365C2×C42C2411yescharacteristic
366#107—44no—
367#107—44no—
368#106—44no—
369#106—44no—
370#103—28no—
371#103—28no—
372#103—44no—
373#103—44no—
374#102—28no—
375#102—28no—
376#102—44no—
377#102—44no—
378C22⋊C8—24no—
379C22⋊C8—24no—
380C22⋊C8—28no—
381C22⋊C8—44no—
382C22⋊C8—44no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—22no—
389#95—22no—
390#95—22no—
391#95—22no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—22no—
396C24—22no—
397C24—22no—
398C24—24no—
399C24—22no—
400C24—22no—
401C24—22no—
402C24—22no—
403C24—41no—
404C24—41no—
405C24—41no—
406C24—41no—
407C24—41no—
408C24—41no—
409C24—41no—
410C24—41no—
411C24—42no—
412C24—42no—
413C24—42no—
414C24—42no—
415C24—42no—
416C24—42no—
417C24#12314yes—
418C24C22×D412yes—
419C24C22×D412yes—
420C24C22×D411yescharacteristic
421#41—232no—
422#41—416no—
423#41—416no—
424#41—416no—
425#41—464no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—24no—
429C2×Q8—24no—
430C2×Q8—416no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8H(5,𝔽2)14yes—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—416no—
443C2×D4—42no—
444C2×D4—42no—
445C2×D4—416no—
446C2×D4—48no—
447C2×D4—42no—
448C2×D4—48no—
449C2×D4—432no—
450C2×D4—48no—
451C2×D4—48no—
452C2×D4—48no—
453C2×D4—42no—
454C2×D4—84no—
455C2×D4—816no—
456C2×D4H(5,𝔽2)14yes—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—416no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—48no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—41no—
490C22×C4—48no—
491C22×C4—42no—
492C22×C4—432no—
493C22×C4—42no—
494C22×C4—48no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—42no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—42no—
501C22×C4—88no—
502C22×C4#12314yes—
503C22×C4#12318yes—
504C22×C4#13514yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C2×#4114yes—
513C22×C4C2×#4114yes—
514C22×C4C2×#4114yes—
515C22×C4C2511yescharacteristic
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—88no—
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—48no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—81no—
541C4⋊C4—81no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—48no—
547C22⋊C4—48no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—82no—
553C22⋊C4—82no—
554C22⋊C4—82no—
555C22⋊C4—82no—
556C22⋊C4—82no—
557C22⋊C4—82no—
558C22⋊C4—84no—
559C22⋊C4—82no—
560C22⋊C4—82no—
561C22⋊C4—82no—
562C22⋊C4—82no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—44no—
568C42—44no—
569C42—82no—
570C42—82no—
571C42—82no—
572C42—82no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—28no—
584C23—22no—
585C23—22no—
586C23—22no—
587C23—22no—
588C23—24no—
589C23—22no—
590C23—22no—
591C23—22no—
592C23—44no—
593C23—44no—
594C23—48no—
595C23—44no—
596C23—48no—
597C23—48no—
598C23—44no—
599C23—42no—
600C23—42no—
601C23—44no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—82no—
607C23—82no—
608C23—82no—
609C23—82no—
610C23—82no—
611C23—82no—
612C23—82no—
613C23—82no—
614C23—84no—
615C23—84no—
616C23—84no—
617C23—84no—
618C23C2×#12314yes—
619C23C2×#12312yes—
620C23C2×#13512yes—
621C23#57512yes—
622C23C23×D411yescharacteristic
623Q8—28no—
624Q8—48no—
625Q8—44no—
626D4—28no—
627D4—48no—
628D4—48no—
629D4—48no—
630D4—432no—
631D4—44no—
632D4—816no—
633C2×C4—28no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—28no—
643C2×C4—28no—
644C2×C4—28no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—24no—
655C2×C4—44no—
656C2×C4—48no—
657C2×C4—432no—
658C2×C4—48no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
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—84no—
682C2×C4—82no—
683C2×C4—82no—
684C2×C4—84no—
685C2×C4—84no—
686C2×C4C2×#12312yes—
687C2×C4C2×#12314yes—
688C2×C4C2×#13512yes—
689C2×C4C2×#13512yes—
690C2×C4C22×#4112yes—
691C2×C4C2×H(5,𝔽2)12yes—
692C2×C4C2×H(5,𝔽2)12yes—
693C8—44no—
694C8—44no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—22no—
701C22—24no—
702C22—24no—
703C22—24no—
704C22—42no—
705C22—42no—
706C22—42no—
707C22—42no—
708C22—44no—
709C22—42no—
710C22—42no—
711C22—42no—
712C22—42no—
713C22—44no—
714C22—42no—
715C22—42no—
716C22—44no—
717C22—42no—
718C22—88no—
719C22—88no—
720C22—816no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—82no—
725C22#309114yes—
726C22C22×#12311yescharacteristic
727C22#348411yescharacteristic
728C22#348411yescharacteristic
729C22C22×#13511yescharacteristic
730C22C2×#57511yescharacteristic
731C22#358511yescharacteristic
732C22#358611yescharacteristic
733C4—24no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—42no—
738C4—44no—
739C4—41no—
740C4—41no—
741C4—82no—
742C4—82no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4#349014yes—
748C2—24no—
749C2—44no—
750C2—82no—
751C2—84no—
752C2#3535011yescharacteristic
753C2C2×#309111yescharacteristic
754C2#6052011yescharacteristic
755C2#6156511yescharacteristic
756C2#6156611yescharacteristic
757C2#6157611yescharacteristic
758C2#6157811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2407C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34186C2semidirect
3. #27586C2semidirect
4. #27585C2semidirect
5. #27583C2semidirect
6. #27582C2semidirect
7. #24876C2semidirect
8. #24874C2semidirect
9. #24872C2semidirect
10. #24870C2semidirect
11. #24010C2semidirect
12. #24009C2semidirect
13. #21941C2semidirect
14. #21738C2semidirect
15. #20597C2semidirect
20. C2×#485C22semidirect
21. C2×#482C22semidirect
22. C2×#416C22semidirect
23. C2×#416C22semidirect
33. #2354C22semidirect
34. #2354C22semidirect
35. #2354C22semidirect
36. #2354C22semidirect
50. #2096C22semidirect
51. #2095C22semidirect
52. #2094C22semidirect
53. #2093C22semidirect
54. #2072C22semidirect
55. #2072C22semidirect
56. #2071C22semidirect
57. #2071C22semidirect
58. #2036C22semidirect
59. #2035C22semidirect
60. #2034C22semidirect
61. #2033C22semidirect
62. #1928C22semidirect
63. #1927C22semidirect
64. #1926C22semidirect
65. #1925C22semidirect
66. #1875C22semidirect
67. #1873C22semidirect
68. #1869C22semidirect
69. #1869C22semidirect
70. #1854C22semidirect
71. #1854C22semidirect
78. #1808C22semidirect
79. #1807C22semidirect
80. #1788C22semidirect
81. #1788C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24072
#341862
#275862
#275852
#275832
#275822
#248762
#248742
#248722
#248702
#240102
#240092
#219412
#217382
#205972