Finite Group Databasebeta

← #6432676

Subgroups — #6432676

Order 512.

2020 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34275C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic, complemented
3#23828C214yesmaximal, complemented
4#23827C214yesmaximal, complemented
5#23822C212yesmaximal, complemented
6#23821C212yesmaximal, complemented
7#23817C212yesmaximal, complemented
8#23815C212yesmaximal, complemented
9#23749C212yesmaximal, complemented
10#23628C211yesmaximal, characteristic, complemented
11#23627C211yesmaximal, characteristic, complemented
12#23511C212yesmaximal
13#23452C211yesmaximal, characteristic, complemented
14C2×#2023C211yesmaximal, characteristic, complemented
15#20416C212yesmaximal, complemented
16#20415C212yesmaximal, complemented
17#20368C211yesmaximal, characteristic, complemented
18C2×#535C2211yescharacteristic
19C22×C4⋊Q8C2211yescharacteristic
20C2×#520C2211yescharacteristic
21C2×#518C2211yescharacteristic
22C2×#417C2212yes—
23C2×#418C2212yes—
24C2×#416C2212yes—
25C22×#102C2212yes—
26C2×#405C2212yes—
27#2369—21no—
28#2369—21no—
29#2365—21no—
30#2365—21no—
31#2363—21no—
32#2363—21no—
33#2361—21no—
34#2361—21no—
35#2358—22no—
36#2358—22no—
37#2345—22no—
38Q8×C22⋊C4—22no—
39#2331C2214yescomplemented
40#2328C2212yes—
41#2326C2212yes—
42#2317C2211yescharacteristic
43#2316C2212yes—
44C4×C4⋊Q8C2212yescomplemented
45C4×C4⋊Q8C2212yescomplemented
46C4×#127C2212yes—
47C4×#124C2212yes—
48#2297C2211yescharacteristic
49C2×#390C2211yescharacteristic
50#2285C2211yescharacteristic
51#2274C2212yes—
52C4×#117C2211yescharacteristic
53#2101C2214yescomplemented
54#2101C2214yescomplemented
55#2100C2214yescomplemented
56#2099C2214yescomplemented
57#2074C2212yescomplemented
58#2073C2214yes—
59#2073C2212yescomplemented
60#2072C2211yescharacteristic, complemented
61#2072C2211yescharacteristic, complemented
62#2071C2212yes—
63#2071C2212yes—
64#2071C2211yescharacteristic, complemented
65#2071C2211yescharacteristic, complemented
66#2042C2214yes—
67#2042C2214yes—
68#2041C2214yes—
69#2041C2214yes—
70#2024C2212yescomplemented
71#2024C2212yescomplemented
72#2023C2212yes—
73#2023C2212yes—
74#2023C2214yes—
75#2023C2212yescomplemented
76#2023C2212yescomplemented
77#1763C2214yescomplemented
78#1762C2214yes—
79#1762C2214yes—
80#1762C2214yescomplemented
81C4×#103C2214yescomplemented
82C4×#102C2214yes—
83C4×#102C2214yes—
84C4×#102C2214yescomplemented
85#1746C2214yescomplemented
86#1745C2212yescomplemented
87#1744C2212yescomplemented
88#1736C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92#1729C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C23×Q8—21no—
97C23×D4C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100#535—24no—
101#535—24no—
102C2×C4⋊Q8—28no—
103C2×C4⋊Q8—21no—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2312yes—
107C2×#127C2312yes—
108C2×#124C2312yes—
109C2×#121—21no—
110C2×#121—21no—
111C2×#120—22no—
112#520—28no—
113#518—28no—
114C2×C4×Q8—22no—
115C2×C4×Q8—22no—
116C2×C4×Q8C2312yes—
117C2×C4×D4—21no—
118C2×C4×D4—21no—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—22no—
122C2×C4×D4C2312yes—
123C2×C4×D4C2312yes—
124C2×C4×D4C2312yes—
125C2×#117—21no—
126C2×#117C2312yes—
127C2×#117C2311yescharacteristic
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C22×C4⋊C4—21no—
131C22×C4⋊C4—21no—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—21no—
136C22×C22⋊C4—22no—
137C22×C42C2311yescharacteristic
138#418—28no—
139#417—28no—
140#416—28no—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#102—28no—
145C2×#102C2314yes—
146C2×#102C2314yes—
147C2×#102C2312yes—
148C2×#102C2312yes—
149C2×#102C2314yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152#405—28no—
153C2×#97C2314yes—
154C2×C4×C8C2314yes—
155#391—22no—
156#391—22no—
157#391—24no—
158#391—24no—
159#391C2311yescharacteristic
160#391C2311yescharacteristic
161#390—22no—
162#390—22no—
163#390—21no—
164#390—21no—
165#390C2311yescharacteristic
166#390C2312yes—
167#390C2312yes—
168#390C2311yescharacteristic
169#389—22no—
170#389—22no—
171#389—21no—
172#389—21no—
173#389—22no—
174#389—22no—
175#389C2314yes—
176#389C2314yes—
177#388—22no—
178#388—22no—
179#388—22no—
180#388—22no—
181#387—24no—
182#387—24no—
183#386—21no—
184#386—21no—
185#386—22no—
186#386—22no—
187#386—22no—
188#386—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382C2312yes—
196#382C2312yes—
197#380—21no—
198#380—21no—
199#380—24no—
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⋊C4D411yescharacteristic
207C4×C4⋊C4D411yescharacteristic
208C4×C4⋊C4D411yescharacteristic
209C4×C4⋊C4D411yescharacteristic
210C4×C4⋊C4C2314yes—
211C4×C4⋊C4C2314yes—
212C4×C22⋊C4—24no—
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4D412yes—
218C4×C22⋊C4D412yes—
219#376C2312yes—
220C43C2312yes—
221#340C2314yes—
222#340C2314yes—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#336C2312yes—
232#336C2311yescharacteristic
233#336C2311yescharacteristic
234#336C2312yes—
235#336C2312yes—
236#336C2314yes—
237#336C2314yes—
238C25—21no—
239C2×#41—22no—
240C2×#41—24no—
241C2×#41—24no—
242C2×#41#4114yes—
243C22×Q8—28no—
244C22×Q8—22no—
245C22×Q8—22no—
246C22×Q8—82no—
247C22×Q8—82no—
248C22×Q8C2411yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—22no—
252C22×D4—22no—
253C22×D4—24no—
254C22×D4—24no—
255C22×D4—22no—
256C22×D4#4114yes—
257C22×D4C2412yes—
258C22×D4C2411yescharacteristic
259C23×C4—22no—
260C23×C4—21no—
261C23×C4—21no—
262C23×C4—22no—
263C23×C4—22no—
264C23×C4—22no—
265C23×C4C2×D411yescharacteristic
266C23×C4C2×D411yescharacteristic
267C23×C4C2411yescharacteristic
268C2×M16—24no—
269C2×M16—24no—
270C2×M16—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8C2412yes—
275C22×C8C2412yes—
276C22×C8C2414yes—
277C4⋊Q8—28no—
278C4⋊Q8—28no—
279C4⋊Q8—28no—
280C4⋊Q8—28no—
281#127—24no—
282#127—24no—
283#124—24no—
284#124—24no—
285#121—28no—
286#121—28no—
287#120—216no—
288C4×Q8—28no—
289C4×Q8—84no—
290C4×Q8—84no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—28no—
294C4×D4—44no—
295C4×D4—44no—
296C4×D4—48no—
297C4×D4—84no—
298C4×D4—84no—
299#117—22no—
300#117—22no—
301#117—24no—
302#117—24no—
303#117—24no—
304#117—44no—
305#117#4118yes—
306C2×C4⋊C4—28no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—24no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—44no—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4C2×D411yescharacteristic
326C2×C4⋊C4C2×D411yescharacteristic
327C2×C4⋊C4#4118yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2411yescharacteristic
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2412yes—
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2414yes—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
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—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—48no—
350C2×C22⋊C4—82no—
351C2×C22⋊C4—82no—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—24no—
355C2×C42—22no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D412yes—
365C2×C42C2×D412yes—
366C2×C42C2412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2411yescharacteristic
369C2×C42C2412yes—
370#103—24no—
371#103—24no—
372#103—28no—
373#102—24no—
374#102—24no—
375#102—28no—
376#102—28no—
377#102—28no—
378#102—28no—
379#102—28no—
380#97—28no—
381C4×C8—28no—
382#95—21no—
383#95—21no—
384#95—24no—
385#95—24no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95C2×D412yes—
395#95C2×D412yes—
396#95C2×D412yes—
397#95C2×D412yes—
398C24—22no—
399C24—21no—
400C24—24no—
401C24—42no—
402C24—42no—
403C24—81no—
404C24—81no—
405C24—81no—
406C24—81no—
407C24C22×D411yescharacteristic
408#41—216no—
409#41—48no—
410C2×Q8—28no—
411C2×Q8—28no—
412C2×Q8—21no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—44no—
416C2×Q8—44no—
417C2×Q8—82no—
418C2×Q8—82no—
419C2×Q8—816no—
420C2×Q8C2×#4112yes—
421C2×D4—24no—
422C2×D4—28no—
423C2×D4—28no—
424C2×D4—24no—
425C2×D4—216no—
426C2×D4—24no—
427C2×D4—22no—
428C2×D4—24no—
429C2×D4—22no—
430C2×D4—22no—
431C2×D4—21no—
432C2×D4—42no—
433C2×D4—42no—
434C2×D4—42no—
435C2×D4—42no—
436C2×D4—48no—
437C2×D4—82no—
438C2×D4—82no—
439C2×D4—82no—
440C2×D4—82no—
441C2×D4C2×#4114yes—
442C2×D4C2×#4112yes—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—82no—
465C22×C4—82no—
466C22×C4—82no—
467C22×C4—82no—
468C22×C4—82no—
469C22×C4—82no—
470C22×C4—82no—
471C22×C4—82no—
472C22×C4—81no—
473C22×C4—81no—
474C22×C4C22×D411yescharacteristic
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4114yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4114yes—
487C22×C4C2511yescharacteristic
488M16—48no—
489C2×C8—22no—
490C2×C8—24no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—24no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—48no—
499C4⋊C4—28no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—84no—
521C4⋊C4—84no—
522C4⋊C4#12414yes—
523C4⋊C4#12414yes—
524C4⋊C4C2×#4118yes—
525C22⋊C4—28no—
526C22⋊C4—28no—
527C22⋊C4—24no—
528C22⋊C4—44no—
529C22⋊C4—42no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—42no—
535C22⋊C4—44no—
536C22⋊C4—42no—
537C22⋊C4—81no—
538C22⋊C4—81no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4#12418yes—
542C42—22no—
543C42—22no—
544C42—24no—
545C42—24no—
546C42—22no—
547C42—22no—
548C42—22no—
549C42—22no—
550C42—42no—
551C42—42no—
552C42—44no—
553C42—42no—
554C42—44no—
555C42—84no—
556C42—84no—
557C42C2×#4118yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—24no—
563C23—22no—
564C23—41no—
565C23—42no—
566C23—41no—
567C23—42no—
568C23—41no—
569C23—42no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—42no—
575C23—41no—
576C23—48no—
577C23—84no—
578C23—84no—
579C23—162no—
580C23—162no—
581C23—162no—
582C23—162no—
583C23#52512yes—
584C23#52512yes—
585C23C2×#12412yes—
586C23C23×D411yescharacteristic
587Q8—24no—
588Q8—416no—
589Q8—42no—
590Q8—88no—
591D4—28no—
592D4—24no—
593D4—48no—
594D4—48no—
595D4—44no—
596D4—42no—
597D4—88no—
598D4—88no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—21no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—48no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—88no—
644C2×C4—164no—
645C2×C4—164no—
646C2×C4—164no—
647C2×C4—162no—
648C2×C4#52512yes—
649C2×C4#52512yes—
650C2×C4C2×#12414yes—
651C2×C4C2×#12414yes—
652C2×C4C2×#12412yes—
653C2×C4#52918yes—
654C2×C4#52918yes—
655C2×C4#57612yes—
656C2×C4#57712yes—
657C2×C4C22×#4111yescharacteristic
658C2×C4C22×#4111yescharacteristic
659C2×C4C22×#4112yes—
660C2×C4C22×#4112yes—
661C8—42no—
662C8—42no—
663C8—44no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—42no—
671C22—44no—
672C22—42no—
673C22—44no—
674C22—42no—
675C22—44no—
676C22—81no—
677C22—82no—
678C22—81no—
679C22—82no—
680C22—81no—
681C22—82no—
682C22—82no—
683C22—81no—
684C22—168no—
685C22#239814yes—
686C22C2×#52511yescharacteristic
687C22C2×#52511yescharacteristic
688C22C22×#12411yescharacteristic
689C22C2×#57611yescharacteristic
690C22C2×#57711yescharacteristic
691C22#358511yescharacteristic
692C22#358611yescharacteristic
693C4—22no—
694C4—22no—
695C4—24no—
696C4—24no—
697C4—41no—
698C4—48no—
699C4—41no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—82no—
704C4—82no—
705C4—81no—
706C4—82no—
707C4—81no—
708C4#239814yes—
709C4C2×#52912yes—
710C4C2×#52912yes—
711C2—24no—
712C2—82no—
713C2—84no—
714C2C2×#239811yescharacteristic
715C2#6057811yescharacteristic
716C2#6058211yescharacteristic
717C2#6059611yescharacteristic
718C2#6059711yescharacteristic
719C2#6062211yescharacteristic
720C2#6062511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34216C2semidirect
3. #23828C2semidirect
4. #23827C2semidirect
5. #23822C2semidirect
6. #23821C2semidirect
7. #23817C2semidirect
8. #23815C2semidirect
9. #23749C2semidirect
10. #23628C2semidirect
11. #23627C2semidirect
13. #23452C2semidirect
14. C2×#2023C2semidirect
15. #20416C2semidirect
16. #20415C2semidirect
17. #20368C2semidirect
39. #2331C22semidirect
44. C4×C4⋊Q8C22semidirect
45. C4×C4⋊Q8C22semidirect
53. #2101C22semidirect
54. #2101C22semidirect
55. #2100C22semidirect
56. #2099C22semidirect
57. #2074C22semidirect
59. #2073C22semidirect
60. #2072C22semidirect
61. #2072C22semidirect
64. #2071C22semidirect
65. #2071C22semidirect
70. #2024C22semidirect
71. #2024C22semidirect
75. #2023C22semidirect
76. #2023C22semidirect
77. #1763C22semidirect
80. #1762C22semidirect
81. C4×#103C22semidirect
84. C4×#102C22semidirect
85. #1746C22semidirect
86. #1745C22semidirect
87. #1744C22semidirect

Maximal subgroups

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

subgroupindex
#342752
#342162
#238282
#238272
#238222
#238212
#238172
#238152
#237492
#236282
#236272
#235112
#234522
C2×#20232
#204162
#204152
#203682