Finite Group Databasebeta

← #6019435

Subgroups — #6019435

Order 512.

1620 conjugacy classes of subgroups.

The subgroup lattice has 757 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17304C211yesmaximal, characteristic, complemented
2#17288C212yesmaximal, complemented
3#17251C212yesmaximal, complemented
4#17246C212yesmaximal, complemented
5#17235C211yesmaximal, characteristic, complemented
6#17219C211yesmaximal, characteristic, complemented
7#16995C212yesmaximal, complemented
8#16937C212yesmaximal, complemented
9#16718C212yesmaximal
10#16617C212yesmaximal, complemented
11#16396C212yesmaximal
12#16262C212yesmaximal
13#16088C211yesmaximal, characteristic, complemented
14#15953C212yesmaximal
15#15915C212yesmaximal
16#15486C212yesmaximal
17#15019C211yesmaximal, characteristic
18#14936C211yesmaximal, characteristic
19#14914C211yesmaximal, characteristic
20C22×C22⋊Q8C2211yescharacteristic
21C2×#397C2212yes—
22C2×#397C2212yes—
23C2×#395C2212yes—
24C2×#388C2212yes—
25C2×#387C2212yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#381C2212yes—
29C2×#381C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C22×#95C2211yescharacteristic
33#1446C2211yescharacteristic
34#1446C2211yescharacteristic
35#1446C2212yes—
36#1445C2212yes—
37#1445C2212yes—
38#1445C2212yes—
39#1445C2212yes—
40#1445C2212yes—
41#1445C2211yescharacteristic
42#1445C2211yescharacteristic
43#1445C2212yes—
44#1445C2211yescharacteristic
45#1445C2211yescharacteristic
46#1444C2212yes—
47#1444C2211yescharacteristic
48#1444C2211yescharacteristic
49#1444C2211yescharacteristic
50#1444C2212yes—
51#1444C2211yescharacteristic
52#1444C2212yes—
53#1443C2212yes—
54#1443C2212yes—
55#1443C2212yes—
56#1443C2212yes—
57#1443C2212yes—
58#1443C2212yes—
59#1443C2212yes—
60#1443C2212yes—
61#1443C2212yes—
62#1443C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1442C2211yescharacteristic
66#1442C2212yes—
67#1442C2212yes—
68#1442C2212yes—
69#1442C2212yes—
70#1442C2212yes—
71#1442C2212yes—
72#1442C2211yescharacteristic
73#1442C2212yes—
74#1442C2212yes—
75#1442C2212yes—
76#1442C2212yes—
77#1442C2212yes—
78#1442C2212yes—
79#1441C2212yes—
80#1441C2212yes—
81#1441C2212yes—
82#1441C2212yes—
83#1441C2212yes—
84#1441C2212yes—
85#1441C2212yes—
86#1441C2212yes—
87#1441C2212yes—
88#1441C2212yes—
89#1441C2212yes—
90#1441C2212yes—
91#1441C2212yes—
92#1440C2212yes—
93#1440C2212yes—
94#1440C2212yes—
95#1439C2212yes—
96#1439C2212yes—
97#1436C2212yes—
98#1436C2212yes—
99#1436C2212yes—
100#1436C2212yes—
101C4⋊#95C2212yes—
102C22⋊#95C2211yescharacteristic
103C22⋊#95C2212yes—
104C22⋊#95C2211yescharacteristic
105C22⋊#95C2211yescharacteristic
106C22⋊#95C2212yes—
107C23×Q8C2311yescharacteristic
108C24×C4C2311yescharacteristic
109C24×C4C2311yescharacteristic
110C24×C4C2311yescharacteristic
111C2×C22⋊Q8—44no—
112C2×C22⋊Q8—42no—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2311yescharacteristic
127C22×C4⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C42C2312yes—
135#397—44no—
136#397—44no—
137#395—44no—
138#388—44no—
139#387—44no—
140#385—44no—
141#385—44no—
142#381—44no—
143#381—44no—
144#380—42no—
145#380—42no—
146C2×#95—42no—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2311yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2311yescharacteristic
198C2×#95C2311yescharacteristic
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2311yescharacteristic
207C2×#95C2311yescharacteristic
208C25C2411yescharacteristic
209C22×Q8—22no—
210C22×Q8—41no—
211C22×Q8—42no—
212C22×Q8—41no—
213C22×Q8—42no—
214C23×C4—22no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—42no—
223C23×C4—41no—
224C23×C4—41no—
225C23×C4—41no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2412yes—
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C22⋊Q8—416no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—41no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—41no—
278C2×C4⋊C4—41no—
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—42no—
287C2×C4⋊C4—42no—
288C2×C4⋊C4—42no—
289C2×C4⋊C4—42no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—42no—
311#95—24no—
312#95—24no—
313#95—22no—
314#95—24no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—41no—
320#95—41no—
321#95—41no—
322#95—41no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—41no—
329#95—42no—
330#95—41no—
331#95—41no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—42no—
336#95—42no—
337#95—42no—
338#95—42no—
339#95—42no—
340#95—42no—
341#95—42no—
342#95—42no—
343#95—42no—
344#95—42no—
345#95—42no—
346#95—42no—
347#95—42no—
348#95—42no—
349#95—42no—
350#95—42no—
351#95—42no—
352#95—42no—
353#95—42no—
354#95—42no—
355#95—42no—
356#95—42no—
357#95—42no—
358#95—42no—
359#95—42no—
360#95—42no—
361#95—42no—
362#95—42no—
363#95—42no—
364#95—42no—
365#95—42no—
366#95—42no—
367#95—42no—
368#95—42no—
369#95—42no—
370#95—42no—
371#95—42no—
372C24—21no—
373C24—22no—
374C24—21no—
375C24—21no—
376C24—21no—
377C24—21no—
378C24—21no—
379C24—22no—
380C24—22no—
381C24H(5,𝔽2)14yes—
382C24H(5,𝔽2)12yes—
383C24C2511yescharacteristic
384C2×Q8—44no—
385C2×Q8—84no—
386C2×Q8—82no—
387C2×Q8—82no—
388C2×Q8—84no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
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—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—48no—
500C22×C4—44no—
501C22×C4—44no—
502C22×C4—48no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—84no—
508C22×C4—82no—
509C22×C4—84no—
510C22×C4—82no—
511C22×C4—82no—
512C22×C4—84no—
513C22×C4—84no—
514C22×C4H(5,𝔽2)12yes—
515C22×C42−1+412yes—
516C22×C42−1+414yes—
517C22×C42−1+412yes—
518C4⋊C4—48no—
519C4⋊C4—44no—
520C4⋊C4—48no—
521C4⋊C4—84no—
522C4⋊C4—84no—
523C4⋊C4—84no—
524C4⋊C4—82no—
525C4⋊C4—82no—
526C4⋊C4—82no—
527C4⋊C4—82no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C4⋊C4—84no—
531C4⋊C4—84no—
532C4⋊C4—84no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C42—84no—
540C23—24no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—22no—
546C23—24no—
547C23—22no—
548C23—22no—
549C23—24no—
550C23—22no—
551C23—24no—
552C23—42no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23#56014yes—
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×H(5,𝔽2)12yes—
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×2−1+411yescharacteristic
567C23C2×2−1+412yes—
568C23C2×2−1+411yescharacteristic
569C23#58512yes—
570C23#58511yescharacteristic
571C23#58512yes—
572C23#58511yescharacteristic
573C23#58511yescharacteristic
574Q8—88no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—41no—
655C2×C4—41no—
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—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—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—88no—
690C2×C4—88no—
691C2×C4—88no—
692C22—24no—
693C22—24no—
694C22—24no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22C2×#56011yescharacteristic
704C22#353311yescharacteristic
705C22#353312yes—
706C22#353411yescharacteristic
707C22#353412yes—
708C22#353511yescharacteristic
709C22#353512yes—
710C22#353511yescharacteristic
711C22#353512yes—
712C22#354112yes—
713C22#354411yescharacteristic
714C22#354412yes—
715C22#354511yescharacteristic
716C22#354711yescharacteristic
717C22#355411yescharacteristic
718C22#355511yescharacteristic
719C22#355712yes—
720C22#355712yes—
721C22#356012yes—
722C22#356312yes—
723C22#356412yes—
724C22#356612yes—
725C22#356811yescharacteristic
726C4—42no—
727C4—42no—
728C4—42no—
729C4—44no—
730C4—82no—
731C4—81no—
732C4—81no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4—81no—
737C4—82no—
738C4—82no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—82no—
744C2—44no—
745C2#5258611yescharacteristic
746C2#5274111yescharacteristic
747C2#5291311yescharacteristic
748C2#5329911yescharacteristic
749C2#5523212yes—
750C2#5561011yescharacteristic
751C2#5596112yes—
752C2#5647112yes—
753C2#5660011yescharacteristic
754C2#5754312yes—
755C2#5838911yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #17304C2semidirect
2. #17288C2semidirect
3. #17251C2semidirect
4. #17246C2semidirect
5. #17235C2semidirect
6. #17219C2semidirect
7. #16995C2semidirect
8. #16937C2semidirect
10. #16617C2semidirect
13. #16088C2semidirect

Maximal subgroups

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

subgroupindex
#173042
#172882
#172512
#172462
#172352
#172192
#169952
#169372
#167182
#166172
#163962
#162622
#160882
#159532
#159152
#154862
#150192
#149362
#149142