Finite Group Databasebeta

← #656995

Subgroups — #656995

Order 512.

4178 conjugacy classes of subgroups.

The subgroup lattice has 779 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C211yesmaximal, characteristic
2C2×#2342C211yesmaximal, characteristic
3C22×#385C211yesmaximal, characteristic
4C22×#381C211yesmaximal, characteristic
5#15717C214yesmaximal, complemented
6#15560C214yesmaximal, complemented
7#15073C212yesmaximal, complemented
8#14955C212yesmaximal, complemented
9#14935C212yesmaximal, complemented
10#14935C212yesmaximal, complemented
11#14681C211yesmaximal, characteristic
12#14149C212yesmaximal, complemented
13#13992C212yesmaximal, complemented
14#13972C212yesmaximal, complemented
15#13877C211yesmaximal, characteristic
16#13870C212yesmaximal, complemented
17#13869C211yesmaximal, characteristic
18C25×C4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2212yes—
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C22⋊C4C2211yescharacteristic
25#2483—28no—
26C2×#397C2212yes—
27C2×#397C2212yes—
28#2342—28no—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#385—28no—
32C2×#385C2212yes—
33C2×#385C2211yescharacteristic
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2211yescharacteristic
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#381—28no—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2212yes—
45C2×#381C2211yescharacteristic
46C2×#381C2211yescharacteristic
47C2×#381C2212yes—
48C2×#381C2212yes—
49C2×#381C2214yes—
50C2×#381C2211yescharacteristic
51C2×#381C2212yes—
52C2×#380C2212yes—
53C2×#380C2211yescharacteristic
54C2×#379C2211yescharacteristic
55C2×#379C2211yescharacteristic
56C2×C4×C22⋊C4C2212yes—
57C2×C4×C22⋊C4C2211yescharacteristic
58C22×#95C2212yes—
59C22×#95C2212yes—
60C22×#95C2212yes—
61#1443C2214yescomplemented
62#1443C2214yescomplemented
63#1442C2214yescomplemented
64#1442C2214yescomplemented
65#1442C2214yescomplemented
66#1441C2214yescomplemented
67#1441C2214yescomplemented
68#1441C2214yescomplemented
69#1441C2214yescomplemented
70#1441C2214yescomplemented
71#1441C2214yescomplemented
72#1441C2214yescomplemented
73#1440C2214yes—
74#1439C2214yes—
75#1436C2214yescomplemented
76#1436C2214yescomplemented
77#1436C2214yescomplemented
78#1434C2214yescomplemented
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2214yes—
86C22⋊#95C2212yes—
87C26C2311yescharacteristic
88C24×C4—24no—
89C24×C4—28no—
90C24×C4—24no—
91C24×C4—24no—
92C24×C4—44no—
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2312yes—
96C24×C4C2312yes—
97C24×C4C2312yes—
98C24×C4C2312yes—
99C24×C4C2311yescharacteristic
100C24×C4C2312yes—
101C24×C4C2311yescharacteristic
102C2×#117—24no—
103C2×#117—44no—
104C22×C4⋊C4—28no—
105C22×C4⋊C4—28no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—28no—
118C22×C22⋊C4—24no—
119C22×C22⋊C4—24no—
120C22×C22⋊C4—24no—
121C22×C22⋊C4—44no—
122C22×C22⋊C4D418yes—
123C22×C22⋊C4D418yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2311yescharacteristic
136C22×C22⋊C4C2311yescharacteristic
137C22×C22⋊C4C2311yescharacteristic
138C22×C22⋊C4C2311yescharacteristic
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C42C2312yes—
144C22×C42C2312yes—
145#397—28no—
146#397—28no—
147#388—28no—
148#388—28no—
149#385—28no—
150#385—28no—
151#385—24no—
152#385—24no—
153#385—28no—
154#385—28no—
155#385—28no—
156#385—28no—
157#385—44no—
158#381—28no—
159#381—28no—
160#381—24no—
161#381—24no—
162#381—24no—
163#381—28no—
164#381—28no—
165#381—216no—
166#381—28no—
167#381—44no—
168#380—24no—
169#380—28no—
170#379—24no—
171#379—24no—
172C4×C22⋊C4—28no—
173C4×C22⋊C4—24no—
174C2×#95—28no—
175C2×#95—28no—
176C2×#95—28no—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2312yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C25—24no—
205C25—24no—
206C25—24no—
207C25—24no—
208C25—44no—
209C25C2×D414yes—
210C25C2×D414yes—
211C25#4114yes—
212C25C2411yescharacteristic
213C25C2411yescharacteristic
214C25C2411yescharacteristic
215C23×C4—232no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—28no—
234C23×C4—28no—
235C23×C4—28no—
236C23×C4—22no—
237C23×C4—22no—
238C23×C4—22no—
239C23×C4—24no—
240C23×C4—24no—
241C23×C4—22no—
242C23×C4—24no—
243C23×C4—24no—
244C23×C4—24no—
245C23×C4—24no—
246C23×C4—24no—
247C23×C4—24no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—44no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4—44no—
254C23×C4—44no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—432no—
258C23×C4—432no—
259C23×C4—416no—
260C23×C4—416no—
261C23×C4C2×D414yes—
262C23×C4C2×D418yes—
263C23×C4C2×D414yes—
264C23×C4#4118yes—
265C23×C4C2412yes—
266C23×C4C2412yes—
267C23×C4C2412yes—
268C23×C4C2412yes—
269C23×C4C2412yes—
270C23×C4C2412yes—
271C23×C4C2411yescharacteristic
272C23×C4C2411yescharacteristic
273C23×C4C2411yescharacteristic
274C23×C4C2412yes—
275C23×C4C2411yescharacteristic
276C23×C4C2412yes—
277C23×C4C2414yes—
278C23×C4C2414yes—
279#117—416no—
280C2×C4⋊C4—28no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—28no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—432no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C22⋊C4—232no—
299C2×C22⋊C4—232no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—22no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—24no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—24no—
326C2×C22⋊C4—24no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—24no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—416no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—416no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4C2×D414yes—
342C2×C22⋊C4C2×D414yes—
343C2×C22⋊C4C2×D414yes—
344C2×C22⋊C4C2×D414yes—
345C2×C22⋊C4C2×D414yes—
346C2×C22⋊C4C2×D414yes—
347C2×C22⋊C4C2×D418yes—
348C2×C22⋊C4#4118yes—
349C2×C42—24no—
350C2×C42—24no—
351C2×C42—44no—
352C2×C42—44no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—24no—
360#95—24no—
361#95—24no—
362#95—24no—
363#95—24no—
364#95—24no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—24no—
369#95—28no—
370#95—28no—
371#95—28no—
372#95—28no—
373#95—28no—
374#95—44no—
375#95—44no—
376#95—44no—
377#95—44no—
378#95—44no—
379#95—44no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—28no—
387C24—28no—
388C24—28no—
389C24—28no—
390C24—216no—
391C24—216no—
392C24—216no—
393C24—216no—
394C24—22no—
395C24—22no—
396C24—22no—
397C24—22no—
398C24—24no—
399C24—48no—
400C24—48no—
401C24—48no—
402C24—416no—
403C24—416no—
404C24—416no—
405C24—416no—
406C24#121116yes—
407C24C22×D412yes—
408C24C22×D412yes—
409C24C22×D412yes—
410C24C22×D412yes—
411C24C22×D412yes—
412C24C22×D412yes—
413C24C2×#4112yes—
414C24C2×#4112yes—
415C24C2×#4112yes—
416C24C2×#4114yes—
417C24C2×#4114yes—
418C24H(5,𝔽2)14yes—
419C242−1+414yes—
420C24C2511yescharacteristic
421C22×C4—216no—
422C22×C4—216no—
423C22×C4—216no—
424C22×C4—28no—
425C22×C4—216no—
426C22×C4—216no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—28no—
464C22×C4—216no—
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—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—416no—
499C22×C4—416no—
500C22×C4—416no—
501C22×C4—416no—
502C22×C4—416no—
503C22×C4—416no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—4128no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—48no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4—48no—
518C22×C4—48no—
519C22×C4—44no—
520C22×C4—44no—
521C22×C4—44no—
522C22×C4—44no—
523C22×C4—48no—
524C22×C4—48no—
525C22×C4—84no—
526C22×C4—84no—
527C22×C4—88no—
528C22×C4—88no—
529C22×C4—84no—
530C22×C4—84no—
531C22×C4C22×D414yes—
532C22×C4C22×D412yes—
533C22×C4C22×D412yes—
534C22×C4C22×D414yes—
535C22×C4C22×D412yes—
536C22×C4C22×D412yes—
537C22×C4C2×#4118yes—
538C22×C4C2×#4118yes—
539C4⋊C4—416no—
540C4⋊C4—416no—
541C4⋊C4—48no—
542C4⋊C4—84no—
543C4⋊C4—84no—
544C4⋊C4—84no—
545C4⋊C4—88no—
546C22⋊C4—216no—
547C22⋊C4—28no—
548C22⋊C4—28no—
549C22⋊C4—28no—
550C22⋊C4—28no—
551C22⋊C4—216no—
552C22⋊C4—28no—
553C22⋊C4—28no—
554C22⋊C4—48no—
555C22⋊C4—48no—
556C22⋊C4—48no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—48no—
562C22⋊C4—48no—
563C22⋊C4—84no—
564C22⋊C4—84no—
565C42—48no—
566C42—84no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—24no—
572C23—24no—
573C23—24no—
574C23—24no—
575C23—28no—
576C23—28no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—24no—
583C23—28no—
584C23—28no—
585C23—28no—
586C23—28no—
587C23—28no—
588C23—28no—
589C23—22no—
590C23—22no—
591C23—232no—
592C23—22no—
593C23—22no—
594C23—24no—
595C23—24no—
596C23—24no—
597C23—24no—
598C23—24no—
599C23—24no—
600C23—24no—
601C23—232no—
602C23—416no—
603C23—464no—
604C23—432no—
605C23—432no—
606C23—432no—
607C23—432no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23C2×#12114yes—
613C23C2×#12114yes—
614C23C2×#12114yes—
615C23#52514yes—
616C23#52514yes—
617C23#53414yes—
618C23#53614yes—
619C23#54618yes—
620C23#54718yes—
621C23#55118yes—
622C23#55218yes—
623C23C23×D411yescharacteristic
624C23C23×D411yescharacteristic
625C23C22×#4111yescharacteristic
626C23C22×#4112yes—
627C23C22×#4112yes—
628C23C2×H(5,𝔽2)12yes—
629C23C2×2−1+412yes—
630C23#58514yes—
631C2×C4—22no—
632C2×C4—22no—
633C2×C4—22no—
634C2×C4—22no—
635C2×C4—28no—
636C2×C4—28no—
637C2×C4—24no—
638C2×C4—24no—
639C2×C4—28no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—28no—
645C2×C4—28no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—28no—
651C2×C4—432no—
652C2×C4—432no—
653C2×C4—432no—
654C2×C4—44no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—41no—
658C2×C4—41no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—416no—
677C2×C4—432no—
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—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—42no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—42no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—88no—
709C2×C4—816no—
710C2×C4—88no—
711C2×C4—88no—
712C22—24no—
713C22—216no—
714C22—216no—
715C22—28no—
716C22—28no—
717C22—24no—
718C22—24no—
719C22—28no—
720C22—28no—
721C22—24no—
722C22—28no—
723C22—28no—
724C22—28no—
725C22—28no—
726C22—24no—
727C22—24no—
728C22—24no—
729C22—464no—
730C22—48no—
731C22—44no—
732C22—44no—
733C22—44no—
734C22—44no—
735C22#241318yes—
736C22#253418yes—
737C22C22×#12111yescharacteristic
738C22C2×#52511yescharacteristic
739C22C2×#52511yescharacteristic
740C22C2×#53411yescharacteristic
741C22C2×#53611yescharacteristic
742C22C2×#54612yes—
743C22C2×#54712yes—
744C22C2×#55112yes—
745C22C2×#55212yes—
746C22#347514yes—
747C22#348512yes—
748C22#348512yes—
749C22#349112yes—
750C22#349514yes—
751C22#351314yes—
752C22#353514yes—
753C4—42no—
754C4—42no—
755C4—48no—
756C4—48no—
757C4—42no—
758C4—42no—
759C4—44no—
760C4—44no—
761C4—82no—
762C4—82no—
763C4—82no—
764C4—82no—
765C4—82no—
766C4—82no—
767C2—28no—
768C2—28no—
769C2—44no—
770C2C2×#241311yescharacteristic
771C2C2×#253411yescharacteristic
772C2#3544711yescharacteristic
773C2#3615912yes—
774C2#3634112yes—
775C2#3783512yes—
776C2#3803812yes—
777C2#5919614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #15717C2semidirect
6. #15560C2semidirect
7. #15073C2semidirect
8. #14955C2semidirect
9. #14935C2semidirect
10. #14935C2semidirect
12. #14149C2semidirect
13. #13992C2semidirect
14. #13972C2semidirect
16. #13870C2semidirect
61. #1443C22semidirect
62. #1443C22semidirect
63. #1442C22semidirect
64. #1442C22semidirect
65. #1442C22semidirect
66. #1441C22semidirect
67. #1441C22semidirect
68. #1441C22semidirect
69. #1441C22semidirect
70. #1441C22semidirect
71. #1441C22semidirect
72. #1441C22semidirect
75. #1436C22semidirect
76. #1436C22semidirect
77. #1436C22semidirect
78. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C2×#23422
C22×#3852
C22×#3812
#157172
#155602
#150732
#149552
#149352
#146812
#141492
#139922
#139722
#138772
#138702
#138692