Finite Group Databasebeta

← #6506563

Subgroups — #6506563

Order 512.

1892 conjugacy classes of subgroups.

The subgroup lattice has 752 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35019C211yesmaximal, characteristic
2C2×#2439C211yesmaximal, characteristic, complemented
3#29336C212yesmaximal, complemented
4#29330C212yesmaximal, complemented
5#27977C212yesmaximal, complemented
6#27705C214yesmaximal, complemented
7#27703C214yesmaximal, complemented
8#27285C214yesmaximal, complemented
9#27281C214yesmaximal, complemented
10#25908C212yesmaximal, complemented
11#21765C211yesmaximal, characteristic, complemented
12#21764C211yesmaximal, characteristic
13#20912C211yesmaximal, characteristic
14#20911C211yesmaximal, characteristic, complemented
15#20524C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C2×#489C2214yescomplemented
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#2439—22no—
34#2439—22no—
35#2439C2214yes—
36#2439C2214yescomplemented
37#2436C2212yes—
38#2435C2212yes—
39C2×#396—21no—
40C2×#393—21no—
41#2308—21no—
42#2308—21no—
43C2×#391—21no—
44C2×#389C2211yescharacteristic, complemented
45C2×#389C2211yescharacteristic, complemented
46C2×#387—21no—
47C2×#382C2211yescharacteristic, complemented
48C2×#380C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2214yescomplemented
51#2101C2214yescomplemented
52#2074C2214yescomplemented
53#2073C2214yescomplemented
54#2042C2214yescomplemented
55#2042C2214yescomplemented
56#2024C2214yescomplemented
57#2023C2214yescomplemented
58#1993C2214yescomplemented
59#1958C2214yescomplemented
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#1835C2214yescomplemented
77#1824C2212yes—
78#1823C2212yescomplemented
79#1787C2212yescomplemented
80#1786C2212yes—
81#1753C2214yes—
82#1753C2214yescomplemented
83#1749C2212yescomplemented
84C4×C22⋊C8C2212yescomplemented
85#1736C2212yes—
86#1736C2212yescomplemented
87C23×Q8—21no—
88C23×Q8C2311yescharacteristic
89C23×D4—21no—
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×M16C2312yes—
93C22×M16C2312yes—
94C2×C4⋊Q8—22no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8C2314yes—
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×C22⋊Q8—21no—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8C2312yes—
104C2×C22⋊Q8C2312yes—
105C2×#121—21no—
106C2×#121—21no—
107C2×#121—24no—
108C2×#121—24no—
109C2×#121C2312yes—
110C2×#121C2312yes—
111C2×#117—21no—
112C2×#117—21no—
113C2×#117—44no—
114C2×#117C2312yes—
115C2×#117C2312yes—
116C22×C4⋊C4—21no—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2311yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C22⋊C4—21no—
127C22×C22⋊C4—21no—
128C22×C22⋊C4—21no—
129C22×C22⋊C4—21no—
130C22×C22⋊C4D411yescharacteristic
131C22×C22⋊C4D411yescharacteristic
132C22×C42—21no—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#489—48no—
136#419—44no—
137#418—44no—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#103C2314yes—
141C2×#103C2314yes—
142C2×#102C2314yes—
143C2×#102C2314yes—
144C2×#102C2314yes—
145C2×#102C2314yes—
146#407—24no—
147#407—24no—
148C2×C22⋊C8C2312yes—
149C2×C22⋊C8C2312yes—
150C2×C22⋊C8C2312yes—
151C2×C22⋊C8C2312yes—
152C2×#97C2314yes—
153C2×C4×C8C2314yes—
154#400—22no—
155#400—22no—
156#397—22no—
157#397—22no—
158#396—24no—
159#396—24no—
160#394—22no—
161#394—22no—
162#393—24no—
163#393—24no—
164#393—22no—
165#393—22no—
166#391—24no—
167#391—24no—
168#391C2312yes—
169#391C2312yes—
170#390C2312yes—
171#390C2312yes—
172#389—22no—
173#389—22no—
174#389C2314yes—
175#389C2314yes—
176#388—22no—
177#388—22no—
178#388—22no—
179#388—22no—
180#387—24no—
181#387—24no—
182#386—22no—
183#386—22no—
184#386—22no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#382—22no—
191#382C2314yes—
192#382C2312yes—
193#382C2312yes—
194#381—22no—
195#381—22no—
196#380—21no—
197#380—21no—
198#380C2312yes—
199#380C2312yes—
200#379—22no—
201#379—22no—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4D412yes—
204C4×C22⋊C4C2312yes—
205C4×C22⋊C4C2312yes—
206C2×#95—21no—
207C2×#95—21no—
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—24no—
226C22×Q8—21no—
227C22×Q8—22no—
228C22×Q8—22no—
229C22×Q8—44no—
230C22×Q8—44no—
231C22×Q8—42no—
232C22×Q8—42no—
233C22×Q8—82no—
234C22×Q8C2412yes—
235C22×D4—22no—
236C22×D4—22no—
237C22×D4—21no—
238C22×D4—22no—
239C22×D4—22no—
240C22×D4—42no—
241C22×D4—42no—
242C22×D4—44no—
243C22×D4—44no—
244C22×D4—44no—
245C22×D4C2412yes—
246C23×C4—21no—
247C23×C4—21no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—42no—
254C23×C4—42no—
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4#4114yes—
263C23×C4#4114yes—
264C23×C4C2411yescharacteristic
265C23×C4C2411yescharacteristic
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—42no—
270C2×M16—42no—
271C2×M16—42no—
272C2×M16—42no—
273C22×C8C2414yes—
274C22×C8C2414yes—
275C4⋊Q8—28no—
276C4⋊Q8—44no—
277C4⋊Q8—48no—
278#124—44no—
279#124—44no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C22⋊Q8—24no—
284C22⋊Q8—44no—
285C22⋊Q8—44no—
286#121—24no—
287#121—24no—
288#121—24no—
289#121—24no—
290#121—44no—
291#121—44no—
292#117—24no—
293#117—24no—
294#117—42no—
295#117—42no—
296#117—44no—
297#117—44no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—42no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—42no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4C2412yes—
322C2×C4⋊C4C2412yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2412yes—
326C2×C4⋊C4C2412yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
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—42no—
344C2×C22⋊C4—42no—
345C2×C22⋊C4—44no—
346C2×C22⋊C4—44no—
347C2×C22⋊C4—44no—
348C2×C22⋊C4—82no—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C42—22no—
354C2×C42—22no—
355C2×C42—42no—
356C2×C42—42no—
357C2×C42—44no—
358C2×C42C2×D412yes—
359C2×C42C2×D412yes—
360C2×C42C2412yes—
361C2×C42C2412yes—
362#103—28no—
363#103—44no—
364#103—44no—
365#103—44no—
366#102—28no—
367#102—44no—
368#102—44no—
369#102—44no—
370C22⋊C8—24no—
371C22⋊C8—24no—
372C22⋊C8—24no—
373C22⋊C8—24no—
374#97—44no—
375C4×C8—44no—
376#95—22no—
377#95—22no—
378#95—24no—
379#95—24no—
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—48no—
418C2×Q8—48no—
419C2×Q8—42no—
420C2×Q8—81no—
421C2×Q8—81no—
422C2×Q8—84no—
423C2×Q8—84no—
424C2×Q8—816no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—48no—
430C2×D4—44no—
431C2×D4—44no—
432C2×D4—44no—
433C2×D4—44no—
434C2×D4—44no—
435C2×D4—44no—
436C2×D4—42no—
437C2×D4—84no—
438C2×D4—81no—
439C2×D4—81no—
440C2×D4—816no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—48no—
460C22×C4—48no—
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—41no—
472C22×C4—41no—
473C22×C4—41no—
474C22×C4—41no—
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—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—42no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4#12414yes—
496C22×C4#12414yes—
497C22×C4#12414yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C2×#4112yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2511yescharacteristic
512M16—84no—
513M16—84no—
514C2×C8—24no—
515C2×C8—24no—
516C2×C8—24no—
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—24no—
528C4⋊C4—22no—
529C4⋊C4—24no—
530C4⋊C4—48no—
531C4⋊C4—42no—
532C4⋊C4—48no—
533C4⋊C4—42no—
534C4⋊C4—42no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—44no—
538C4⋊C4—44no—
539C4⋊C4—84no—
540C4⋊C4H(5,𝔽2)14yes—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
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—
560C22⋊C4—82no—
561C42—24no—
562C42—24no—
563C42—42no—
564C42—42no—
565C42—44no—
566C42—44no—
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—21no—
583C23—28no—
584C23—44no—
585C23—44no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—82no—
594C23—82no—
595C23—82no—
596C23—82no—
597C23—84no—
598C23—84no—
599C23—84no—
600C23—162no—
601C23—162no—
602C23—162no—
603C23—162no—
604C23C2×#12412yes—
605C23C2×#12412yes—
606C23C2×#12412yes—
607C23#57512yes—
608C23#57512yes—
609C23C23×D411yescharacteristic
610Q8—44no—
611Q8—44no—
612Q8—84no—
613Q8—816no—
614D4—44no—
615D4—44no—
616D4—84no—
617D4—88no—
618D4—88no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—22no—
643C2×C4—22no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—48no—
649C2×C4—44no—
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—42no—
669C2×C4—42no—
670C2×C4—82no—
671C2×C4—82no—
672C2×C4—84no—
673C2×C4—84no—
674C2×C4—84no—
675C2×C4—84no—
676C2×C4—84no—
677C2×C4—84no—
678C2×C4C2×#12412yes—
679C2×C4C2×#12412yes—
680C2×C4C2×#12412yes—
681C2×C4C22×#4111yescharacteristic
682C2×C4C22×#4111yescharacteristic
683C2×C4C2×H(5,𝔽2)12yes—
684C2×C4#58512yes—
685C8—44no—
686C8—44no—
687C22—22no—
688C22—22no—
689C22—22no—
690C22—22no—
691C22—22no—
692C22—22no—
693C22—22no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—41no—
700C22—41no—
701C22—41no—
702C22—41no—
703C22—41no—
704C22—41no—
705C22—41no—
706C22—41no—
707C22—88no—
708C22—82no—
709C22—82no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—168no—
716C22#314014yes—
717C22C22×#12411yescharacteristic
718C22#348411yescharacteristic
719C22#348511yescharacteristic
720C22C2×#57511yescharacteristic
721C22C2×#57511yescharacteristic
722C22#358511yescharacteristic
723C22#358611yescharacteristic
724C4—22no—
725C4—22no—
726C4—22no—
727C4—42no—
728C4—42no—
729C4—42no—
730C4—44no—
731C4—42no—
732C4—82no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—82no—
737C4—82no—
738C4#350512yes—
739C4#350512yes—
740C2—24no—
741C2—42no—
742C2—42no—
743C2—162no—
744C2#3537711yescharacteristic
745C2C2×#314011yescharacteristic
746C2#6062111yescharacteristic
747C2#6158111yescharacteristic
748C2#6158311yescharacteristic
749C2#6171011yescharacteristic
750C2#6171211yescharacteristic
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×#2439C2semidirect
3. #29336C2semidirect
4. #29330C2semidirect
5. #27977C2semidirect
6. #27705C2semidirect
7. #27703C2semidirect
8. #27285C2semidirect
9. #27281C2semidirect
10. #25908C2semidirect
11. #21765C2semidirect
14. #20911C2semidirect
15. #20524C2semidirect
16. C22×C4⋊Q8C22semidirect
20. C2×#489C22semidirect
21. C2×#419C22semidirect
36. #2439C22semidirect
44. C2×#389C22semidirect
45. C2×#389C22semidirect
47. C2×#382C22semidirect
50. #2101C22semidirect
51. #2101C22semidirect
52. #2074C22semidirect
53. #2073C22semidirect
54. #2042C22semidirect
55. #2042C22semidirect
56. #2024C22semidirect
57. #2023C22semidirect
58. #1993C22semidirect
59. #1958C22semidirect
60. #1936C22semidirect
61. #1935C22semidirect
62. #1934C22semidirect
63. #1933C22semidirect
72. #1872C22semidirect
73. #1872C22semidirect
74. #1872C22semidirect
75. #1872C22semidirect
76. #1835C22semidirect
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×#24392
#293362
#293302
#279772
#277052
#277032
#272852
#272812
#259082
#217652
#217642
#209122
#209112
#205242