Finite Group Databasebeta

← #6434104

Subgroups — #6434104

Order 512.

1834 conjugacy classes of subgroups.

The subgroup lattice has 729 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35311C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#29919C214yesmaximal, complemented
4#29918C214yesmaximal, complemented
5#29913C212yesmaximal, complemented
6#29913C212yesmaximal, complemented
7#29909C212yesmaximal, complemented
8#29908C212yesmaximal, complemented
9#23748C212yesmaximal, complemented
10#23736C212yesmaximal, complemented
11#23662C212yesmaximal, complemented
12#23633C212yesmaximal, complemented
13#20911C211yesmaximal, characteristic
14C2×#1872C211yesmaximal, characteristic
15#20844C211yesmaximal, characteristic
16#20843C211yesmaximal, characteristic
17#20685C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C2×#428C2212yes—
23C2×#427C2212yes—
24C2×#419C2212yes—
25C22×#103C2212yes—
26C2×#407C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28#2855C2212yes—
29#2855C2212yes—
30#2625C2212yes—
31#2625C2212yes—
32#2503—21no—
33#2502—21no—
34#2499—21no—
35#2499—21no—
36#2484—22no—
37#2483—22no—
38#2317—22no—
39#2317—21no—
40#2317—21no—
41#2317C2211yescharacteristic
42#2317C2211yescharacteristic
43#2317C2212yes—
44#2317C2212yes—
45#2317C2214yes—
46#2310C2211yescharacteristic
47#2310C2211yescharacteristic
48C2×#389—21no—
49C2×#389—21no—
50C2×#385—22no—
51C2×#384C2212yes—
52C2×#382C2212yes—
53#2280C2211yescharacteristic
54#2101C2214yescomplemented
55#2101C2214yescomplemented
56#2100C2214yescomplemented
57#2099C2214yescomplemented
58#2097C2214yescomplemented
59#2097C2214yescomplemented
60#2096C2214yescomplemented
61#2095C2214yescomplemented
62#2085C2214yescomplemented
63#2085C2214yescomplemented
64#2084C2214yescomplemented
65#2083C2214yescomplemented
66#2074C2214yescomplemented
67#2074C2214yescomplemented
68#2072C2214yescomplemented
69#2072C2214yescomplemented
70#1873C2214yes—
71#1872C2212yes—
72#1872C2212yes—
73#1872C2214yes—
74#1872C2212yes—
75#1872C2212yes—
76#1854C2212yes—
77#1854C2212yes—
78#1853C2211yescharacteristic
79#1853C2211yescharacteristic
80#1853C2212yes—
81#1853C2212yes—
82#1852C2211yescharacteristic
83#1852C2211yescharacteristic
84#1852C2212yes—
85#1852C2212yes—
86#1818C2214yes—
87#1817C2212yes—
88#1817C2212yes—
89#1736C2212yes—
90#1736C2212yes—
91#1735C2212yes—
92C2×#340C2212yes—
93#1729C2212yes—
94#1729C2212yes—
95C23×Q8C2311yescharacteristic
96C24×C4—21no—
97C24×C4C2311yescharacteristic
98C22×M16C2312yes—
99C23×C8C2312yes—
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C2×#117—24no—
107C2×#117—24no—
108C2×#117—21no—
109C2×#117—21no—
110C2×#117—44no—
111C2×#117D412yes—
112C2×#117D411yescharacteristic
113C2×#117D411yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C2×#117C2312yes—
117C2×#117C2312yes—
118C2×#117C2311yescharacteristic
119C2×#117C2311yescharacteristic
120C2×#117C2312yes—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4—22no—
126C22×C4⋊C4—22no—
127C22×C4⋊C4—44no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C4⋊C4C2312yes—
133C22×C4⋊C4C2312yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2311yescharacteristic
136C22×C22⋊C4—21no—
137C22×C22⋊C4—21no—
138C22×C22⋊C4D412yes—
139C22×C42—21no—
140C22×C42C2312yes—
141#428—44no—
142#427—44no—
143C2×#107C2312yes—
144C2×#107C2314yes—
145C2×#107C2312yes—
146C2×#106C2312yes—
147C2×#106C2314yes—
148C2×#106C2312yes—
149#419—44no—
150C2×#103—44no—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155#407—24no—
156C2×C22⋊C8—24no—
157C2×C22⋊C8C2311yescharacteristic
158C2×C22⋊C8C2312yes—
159C2×C22⋊C8C2311yescharacteristic
160C2×C22⋊C8C2312yes—
161C2×C22⋊C8C2312yes—
162#400—22no—
163#400—22no—
164#398—22no—
165#398—22no—
166#397—22no—
167#397—22no—
168#397—22no—
169#397—22no—
170#391—21no—
171#391—21no—
172#391—21no—
173#391—21no—
174#391C2312yes—
175#391C2312yes—
176#391C2312yes—
177#391C2312yes—
178#389—22no—
179#389—22no—
180#389—22no—
181#389—22no—
182#389—24no—
183#389—24no—
184#389C2312yes—
185#389C2312yes—
186#389C2312yes—
187#389C2312yes—
188#387—22no—
189#387—22no—
190#387—22no—
191#387—22no—
192#385—22no—
193#385—22no—
194#385—24no—
195#385—24no—
196#385—24no—
197#385—24no—
198#385—22no—
199#385—22no—
200#384—21no—
201#384—21no—
202#384—22no—
203#384—22no—
204#384C2314yes—
205#384C2314yes—
206#382—21no—
207#382—21no—
208#382—22no—
209#382—22no—
210#382C2314yes—
211#382C2314yes—
212#381—22no—
213#381—22no—
214#381—21no—
215#381—22no—
216#381—22no—
217#380C2311yescharacteristic
218#380C2311yescharacteristic
219#380C2311yescharacteristic
220#380C2311yescharacteristic
221#379—21no—
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C2×#95—22no—
225C2×#95—22no—
226#340C2312yes—
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2314yes—
240C25C2×D411yescharacteristic
241C22×Q8—21no—
242C22×Q8—22no—
243C22×Q8—22no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8C2412yes—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—21no—
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—22no—
253C23×C4—21no—
254C23×C4—22no—
255C23×C4—22no—
256C23×C4—44no—
257C23×C4—82no—
258C23×C4—82no—
259C23×C4C2×D412yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4#4114yes—
265C23×C4#4114yes—
266C23×C4C2411yescharacteristic
267C23×C4C2411yescharacteristic
268C23×C4C2411yescharacteristic
269C2×M16—24no—
270C2×M16—24no—
271C2×M16—24no—
272C22×C8—24no—
273C22×C8—24no—
274C22×C8—24no—
275C22×C8C2412yes—
276C22×C8C2412yes—
277C22×C8C2414yes—
278C22⋊Q8—24no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—44no—
283C22⋊Q8—44no—
284#117—28no—
285#117—24no—
286#117—24no—
287#117—24no—
288#117—24no—
289#117—22no—
290#117—22no—
291#117—22no—
292#117—22no—
293#117—24no—
294#117—24no—
295#117—24no—
296#117—24no—
297#117—42no—
298#117—42no—
299#117—44no—
300#117—44no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—24no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—24no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—44no—
324C2×C4⋊C4—44no—
325C2×C4⋊C4—42no—
326C2×C4⋊C4—42no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—44no—
330C2×C4⋊C4—44no—
331C2×C4⋊C4—84no—
332C2×C4⋊C4—84no—
333C2×C4⋊C4C2×D411yescharacteristic
334C2×C4⋊C4C2×D411yescharacteristic
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D412yes—
338C2×C4⋊C4C2×D412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2411yescharacteristic
344C2×C4⋊C4C2411yescharacteristic
345C2×C4⋊C4C2411yescharacteristic
346C2×C4⋊C4C2411yescharacteristic
347C2×C22⋊C4—28no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—44no—
359C2×C22⋊C4—44no—
360C2×C22⋊C4C2×D412yes—
361C2×C22⋊C4C2×D412yes—
362C2×C22⋊C4C2×D412yes—
363C2×C22⋊C4C2×D412yes—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42—21no—
369C2×C42—21no—
370C2×C42—44no—
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2412yes—
374C2×C42C2412yes—
375#107—42no—
376#107—42no—
377#107—44no—
378#106—42no—
379#106—42no—
380#106—44no—
381#103—44no—
382#103—44no—
383#103—44no—
384#103—44no—
385C22⋊C8—22no—
386C22⋊C8—22no—
387C22⋊C8—24no—
388C22⋊C8—24no—
389C22⋊C8—24no—
390#95—24no—
391#95—24no—
392#95—24no—
393#95—24no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—41no—
415C24—41no—
416C24#12314yes—
417C24C22×D411yescharacteristic
418C24C22×D411yescharacteristic
419C24C22×D411yescharacteristic
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—24no—
423C2×Q8—24no—
424C2×Q8—42no—
425C2×Q8—42no—
426C2×Q8—42no—
427C2×Q8—48no—
428C2×Q8—48no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—42no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—42no—
461C22×C4—44no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—44no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—42no—
471C22×C4—81no—
472C22×C4—81no—
473C22×C4—82no—
474C22×C4—82no—
475C22×C4—82no—
476C22×C4—82no—
477C22×C4—816no—
478C22×C4#12314yes—
479C22×C4#12314yes—
480C22×C4#12314yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)14yes—
495C22×C42−1+414yes—
496C22×C4C2511yescharacteristic
497M16—48no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—24no—
507C2×C8—48no—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4—82no—
537C4⋊C4—82no—
538C4⋊C4—84no—
539C4⋊C4—84no—
540C4⋊C4—84no—
541C4⋊C4—84no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—42no—
551C22⋊C4—42no—
552C22⋊C4—42no—
553C22⋊C4—42no—
554C22⋊C4—81no—
555C22⋊C4—81no—
556C42—24no—
557C42—24no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—44no—
563C42—44no—
564C42—42no—
565C42—42no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—28no—
577C23—28no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23C2×#12312yes—
588C23C2×#12312yes—
589C23C2×#12312yes—
590C23#53512yes—
591C23#53612yes—
592C23#57712yes—
593C23#57812yes—
594C23C23×D411yescharacteristic
595Q8—44no—
596Q8—44no—
597Q8—84no—
598C2×C4—28no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—84no—
651C2×C4—84no—
652C2×C4—88no—
653C2×C4—84no—
654C2×C4—88no—
655C2×C4—88no—
656C2×C4—84no—
657C2×C4C2×#12312yes—
658C2×C4C2×#12312yes—
659C2×C4C2×#12312yes—
660C2×C4#53512yes—
661C2×C4#53612yes—
662C2×C4#53918yes—
663C2×C4#53918yes—
664C2×C4C22×#4111yescharacteristic
665C2×C4C22×#4111yescharacteristic
666C2×C4C2×H(5,𝔽2)12yes—
667C2×C4C2×2−1+412yes—
668C8—42no—
669C8—42no—
670C8—44no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—22no—
686C22—416no—
687C22—41no—
688C22—41no—
689C22—41no—
690C22—41no—
691C22#262314yes—
692C22#309714yes—
693C22C22×#12311yescharacteristic
694C22C2×#53511yescharacteristic
695C22C2×#53611yescharacteristic
696C22C2×#57711yescharacteristic
697C22C2×#57811yescharacteristic
698C22#358511yescharacteristic
699C22#358611yescharacteristic
700C4—22no—
701C4—22no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—42no—
706C4—82no—
707C4—82no—
708C4—81no—
709C4—82no—
710C4—84no—
711C4—81no—
712C4—82no—
713C4—81no—
714C4—81no—
715C4#262314yes—
716C4C2×#53912yes—
717C4C2×#53912yes—
718C2—24no—
719C2—24no—
720C2—42no—
721C2C2×#262311yescharacteristic
722C2C2×#309711yescharacteristic
723C2#6051711yescharacteristic
724C2#6081211yescharacteristic
725C2#6081411yescharacteristic
726C2#6087211yescharacteristic
727C2#6087411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29919C2semidirect
4. #29918C2semidirect
5. #29913C2semidirect
6. #29913C2semidirect
7. #29909C2semidirect
8. #29908C2semidirect
9. #23748C2semidirect
10. #23736C2semidirect
11. #23662C2semidirect
12. #23633C2semidirect
54. #2101C22semidirect
55. #2101C22semidirect
56. #2100C22semidirect
57. #2099C22semidirect
58. #2097C22semidirect
59. #2097C22semidirect
60. #2096C22semidirect
61. #2095C22semidirect
62. #2085C22semidirect
63. #2085C22semidirect
64. #2084C22semidirect
65. #2083C22semidirect
66. #2074C22semidirect
67. #2074C22semidirect
68. #2072C22semidirect
69. #2072C22semidirect

Maximal subgroups

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

subgroupindex
#353112
C2×#23172
#299192
#299182
#299132
#299092
#299082
#237482
#237362
#236622
#236332
#209112
C2×#18722
#208442
#208432
#206852