Finite Group Databasebeta

← #5236739

Subgroups — #5236739

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 782 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17273C211yesmaximal, characteristic
2#16917C212yesmaximal, complemented
3#16860C212yesmaximal, complemented
4#16784C212yesmaximal, complemented
5#16772C212yesmaximal, complemented
6#16743C212yesmaximal, complemented
7#16416C212yesmaximal, complemented
8#16223C212yesmaximal, complemented
9#16143C212yesmaximal, complemented
10#16074C211yesmaximal, characteristic
11#15994C212yesmaximal, complemented
12#15987C212yesmaximal, complemented
13#15982C211yesmaximal, characteristic
14#15933C212yesmaximal, complemented
15#15492C212yesmaximal, complemented
16#15464C211yesmaximal, characteristic
17#14658C211yesmaximal, characteristic
18#14516C211yesmaximal, characteristic
19#14503C211yesmaximal, characteristic
20C22×#123C2211yescharacteristic
21C2×#401C2212yes—
22C2×#400C2211yescharacteristic
23C2×#398C2211yescharacteristic
24C2×#390C2211yescharacteristic
25C2×#386C2211yescharacteristic
26C2×#385C2211yescharacteristic
27C2×#383C2212yes—
28C2×#381C2211yescharacteristic
29C2×#380C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#1445C2212yes—
32#1445C2212yes—
33#1445C2212yes—
34#1445C2212yes—
35#1443C2212yes—
36#1443C2212yes—
37#1443C2212yes—
38#1443C2212yes—
39#1443C2212yes—
40#1443C2212yes—
41#1443C2212yes—
42#1443C2212yes—
43#1443C2212yes—
44#1442C2212yes—
45#1442C2212yes—
46#1442C2212yes—
47#1442C2212yes—
48#1442C2212yes—
49#1442C2212yes—
50#1442C2212yes—
51#1442C2212yes—
52#1442C2212yes—
53#1442C2212yes—
54#1442C2212yes—
55#1441C2212yes—
56#1441C2212yes—
57#1441C2212yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2212yes—
62#1440C2211yescharacteristic
63#1440C2212yes—
64#1440C2212yes—
65#1440C2212yes—
66#1440C2212yes—
67#1440C2211yescharacteristic
68#1440C2211yescharacteristic
69#1440C2212yes—
70#1440C2212yes—
71#1440C2212yes—
72#1440C2211yescharacteristic
73#1439C2212yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2212yes—
79#1439C2212yes—
80#1439C2212yes—
81#1439C2212yes—
82#1438C2211yescharacteristic
83#1438C2211yescharacteristic
84#1438C2211yescharacteristic
85#1438C2211yescharacteristic
86#1437C2212yes—
87#1437C2212yes—
88#1437C2212yes—
89#1437C2212yes—
90#1436C2211yescharacteristic
91#1436C2212yes—
92#1436C2212yes—
93#1436C2212yes—
94#1436C2212yes—
95#1436C2212yes—
96#1436C2212yes—
97#1436C2211yescharacteristic
98#1434C2212yes—
99#1434C2212yes—
100#1434C2212yes—
101#1434C2212yes—
102#1434C2212yes—
103C22⋊#95C2212yes—
104C22⋊#95C2212yes—
105C22⋊#95C2212yes—
106C22⋊#95C2212yes—
107C23×D4C2311yescharacteristic
108C24×C4C2311yescharacteristic
109C24×C4C2311yescharacteristic
110C2×#123—24no—
111C2×#123—42no—
112C2×#123—42no—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2311yescharacteristic
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142C22×C42C2312yes—
143#401—44no—
144#400—42no—
145#398—42no—
146#390—24no—
147#386—24no—
148#385—42no—
149#383—44no—
150#381—42no—
151#380—42no—
152#380—42no—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
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×#95C2312yes—
198C2×#95C2312yes—
199C2×#95C2312yes—
200C2×#95C2312yes—
201C2×#95C2312yes—
202C2×#95C2312yes—
203C2×#95C2312yes—
204C2×#95C2312yes—
205C2×#95C2312yes—
206C2×#95C2312yes—
207C25C2411yescharacteristic
208C25C2411yescharacteristic
209C22×D4—41no—
210C22×D4—41no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4#4114yes—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—42no—
219C23×C4—41no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—41no—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241#123—416no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
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—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—41no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—41no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
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×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4—41no—
313C2×C22⋊C4—41no—
314C2×C22⋊C4—41no—
315C2×C22⋊C4—41no—
316C2×C42—22no—
317C2×C42—42no—
318C2×C42—42no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—42no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—42no—
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—41no—
372#95—41no—
373#95—42no—
374#95—42no—
375#95—42no—
376#95—42no—
377#95—42no—
378C24—21no—
379C24—21no—
380C24—21no—
381C24—21no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24C2×#4112yes—
395C24C2×#4112yes—
396C24C2511yescharacteristic
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—84no—
401C2×D4—84no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—28no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
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—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
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—22no—
456C22×C4—22no—
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—22no—
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—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—22no—
509C22×C4—44no—
510C22×C4—84no—
511C22×C4—84no—
512C22×C4—82no—
513C22×C4—82no—
514C22×C4—82no—
515C22×C4—84no—
516C22×C4—84no—
517C22×C4—82no—
518C22×C4C2×#4112yes—
519C22×C4H(5,𝔽2)12yes—
520C22×C4H(5,𝔽2)12yes—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—82no—
524C4⋊C4—84no—
525C4⋊C4—84no—
526C4⋊C4—84no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—84no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—82no—
535C22⋊C4—84no—
536C22⋊C4—84no—
537C22⋊C4—84no—
538C22⋊C4—82no—
539C22⋊C4—82no—
540C22⋊C4—82no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C42—48no—
549C42—82no—
550C42—84no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—42no—
562C23—42no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23C22×#4111yescharacteristic
594C23C2×H(5,𝔽2)11yescharacteristic
595C23C2×H(5,𝔽2)11yescharacteristic
596C23#58512yes—
597C23#58512yes—
598C23#58512yes—
599C23#58512yes—
600C23#58512yes—
601C23#58512yes—
602D4—88no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
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—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—41no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—41no—
657C2×C4—41no—
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—42no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—42no—
701C2×C4—42no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—42no—
705C2×C4—42no—
706C2×C4—42no—
707C2×C4—42no—
708C2×C4—42no—
709C2×C4—42no—
710C2×C4—42no—
711C2×C4—42no—
712C2×C4—88no—
713C2×C4—88no—
714C22—42no—
715C22—42no—
716C22—42no—
717C22—42no—
718C22—42no—
719C22—42no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22#349011yescharacteristic
733C22#349111yescharacteristic
734C22#349111yescharacteristic
735C22#352112yes—
736C22#352112yes—
737C22#353912yes—
738C22#354011yescharacteristic
739C22#354011yescharacteristic
740C22#355111yescharacteristic
741C22#355111yescharacteristic
742C22#355112yes—
743C22#356012yes—
744C22#356012yes—
745C22#356612yes—
746C22#356612yes—
747C22#356612yes—
748C22#356612yes—
749C22#356612yes—
750C22#356612yes—
751C22#356712yes—
752C22#356812yes—
753C4—42no—
754C4—42no—
755C4—42no—
756C4—81no—
757C4—82no—
758C4—81no—
759C4—82no—
760C4—82no—
761C4—82no—
762C4—82no—
763C4—82no—
764C4—82no—
765C4—82no—
766C4—82no—
767C4—82no—
768C4—82no—
769C4—82no—
770C2—82no—
771C2—82no—
772C2#4638811yescharacteristic
773C2#4937312yes—
774C2#5021911yescharacteristic
775C2#5025511yescharacteristic
776C2#5057412yes—
777C2#5550212yes—
778C2#5598112yes—
779C2#5745212yes—
780C2#5836512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #16917C2semidirect
3. #16860C2semidirect
4. #16784C2semidirect
5. #16772C2semidirect
6. #16743C2semidirect
7. #16416C2semidirect
8. #16223C2semidirect
9. #16143C2semidirect
11. #15994C2semidirect
12. #15987C2semidirect
14. #15933C2semidirect
15. #15492C2semidirect

Maximal subgroups

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

subgroupindex
#172732
#169172
#168602
#167842
#167722
#167432
#164162
#162232
#161432
#160742
#159942
#159872
#159822
#159332
#154922
#154642
#146582
#145162
#145032