Finite Group Databasebeta

← #729338

Subgroups — #729338

Order 512.

2820 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×#2580C211yesmaximal, characteristic, complemented
2C2×#2396C211yesmaximal, characteristic, complemented
3C2×#2355C211yesmaximal, characteristic, complemented
4C2×#2324C211yesmaximal, characteristic
5#17023C212yesmaximal, complemented
6#16887C212yesmaximal, complemented
7#15452C214yesmaximal, complemented
8#15339C214yesmaximal, complemented
9#15221C212yesmaximal, complemented
10#15219C212yesmaximal
11#15175C212yesmaximal, complemented
12#15084C212yesmaximal, complemented
13#14340C211yesmaximal, characteristic, complemented
14#14022C212yesmaximal, complemented
15#13974C212yesmaximal, complemented
16#13955C211yesmaximal, characteristic, complemented
17#13906C211yesmaximal, characteristic, complemented
18C22×#121C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2580—28no—
23#2396—28no—
24C2×#400C2211yescharacteristic
25C2×#399C2212yes—
26C2×#399C2211yescharacteristic
27C2×#398C2212yes—
28C2×#396C2212yes—
29C2×#395C2211yescharacteristic
30C2×#393C2212yes—
31C2×#393C2211yescharacteristic
32#2355—28no—
33#2324—28no—
34C2×#391C2212yes—
35C2×#390C2211yescharacteristic
36C2×#390C2212yes—
37C2×#390C2212yes—
38C2×#390C2212yes—
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#388C2212yes—
42C2×#388C2214yes—
43C2×#388C2214yes—
44C2×#387C2214yes—
45C2×#386C2212yes—
46C2×#385C2212yes—
47C2×#385C2214yes—
48C2×#384C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2212yes—
51C2×#380C2212yes—
52C2×#380C2211yescharacteristic
53C2×#380C2214yes—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2212yes—
58C2×C4×C4⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2212yes—
60#1445C2214yes—
61#1444C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2214yes—
77#1438C2214yes—
78#1438C2214yes—
79#1437C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85#1434C2212yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C23×D4C2312yes—
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×#121—24no—
93C2×#121—24no—
94C2×#121—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8D418yescomplemented
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×#117—24no—
102C2×#117—24no—
103C2×#117—24no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2314yes—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#400—24no—
141#399—28no—
142#399—24no—
143#398—28no—
144#396—28no—
145#395—24no—
146#393—28no—
147#393—24no—
148#391—28no—
149#390—28no—
150#390—24no—
151#390—28no—
152#390—28no—
153#389—24no—
154#389—28no—
155#388—28no—
156#388—48no—
157#388—48no—
158#387—216no—
159#386—28no—
160#385—216no—
161#385—28no—
162#384—24no—
163#382—28no—
164#382—24no—
165#380—28no—
166#380—24no—
167#380—48no—
168C4×C4⋊C4—28no—
169C4×C4⋊C4—28no—
170C4×C4⋊C4—28no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4D418yes—
173C4×C22⋊C4—28no—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2311yescharacteristic
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C25C2411yescharacteristic
199C25C2412yes—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8C2×D414yes—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4#4114yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—44no—
227C23×C4—44no—
228C23×C4#4114yes—
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2411yescharacteristic
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244#121—232no—
245C4×Q8—232no—
246C4×D4—416no—
247#117—416no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4—42no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D418yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D414yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C42—24no—
333C2×C42—24no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—24no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—28no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42—44no—
344C2×C42C2×D418yes—
345C2×C42C2×D418yes—
346C2×C42C2×D414yes—
347C2×C42C2×D414yes—
348C2×C42#4114yes—
349#95—24no—
350#95—24no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—22no—
355#95—22no—
356#95—24no—
357#95—24no—
358#95—24no—
359#95—28no—
360#95—24no—
361#95—28no—
362#95—24no—
363#95—28no—
364#95—28no—
365#95—28no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95C2×D418yes—
373C24—21no—
374C24—22no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24—22no—
379C24—22no—
380C24—21no—
381C24—22no—
382C24—24no—
383C24—21no—
384C24—21no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—24no—
389C24—24no—
390C24—24no—
391C24—22no—
392C24—22no—
393C24C2×#4114yes—
394C24C2×#4112yes—
395C24C2511yescharacteristic
396C2×Q8—28no—
397C2×Q8—48no—
398C2×Q8—44no—
399C2×Q8—44no—
400C2×Q8—48no—
401C2×D4—216no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—44no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—88no—
412C2×D4—88no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
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—28no—
450C22×C4—28no—
451C22×C4—28no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—416no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—44no—
492C22×C4—44no—
493C22×C4—48no—
494C22×C4—88no—
495C22×C4—84no—
496C22×C4—84no—
497C22×C4C22×D414yes—
498C22×C4C22×D414yes—
499C22×C4C22×D414yes—
500C22×C4C22×D414yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4H(5,𝔽2)14yes—
514C22×C42−1+412yes—
515C22×C42−1+412yes—
516C4⋊C4—28no—
517C4⋊C4—216no—
518C4⋊C4—28no—
519C4⋊C4—28no—
520C4⋊C4—48no—
521C4⋊C4—48no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—48no—
528C4⋊C4—48no—
529C4⋊C4—84no—
530C4⋊C4—88no—
531C4⋊C4—84no—
532C22⋊C4—216no—
533C22⋊C4—48no—
534C22⋊C4—416no—
535C22⋊C4—416no—
536C22⋊C4—48no—
537C22⋊C4—416no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—48no—
541C22⋊C4—84no—
542C22⋊C4—88no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—88no—
546C42—216no—
547C42—48no—
548C42—44no—
549C42—48no—
550C42—88no—
551C42#121116yes—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—28no—
561C23—28no—
562C23—41no—
563C23—42no—
564C23—41no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—41no—
569C23—42no—
570C23—41no—
571C23—42no—
572C23—42no—
573C23—41no—
574C23—42no—
575C23—41no—
576C23—42no—
577C23—44no—
578C23—44no—
579C23—42no—
580C23—41no—
581C23—42no—
582C23—42no—
583C23—41no—
584C23—44no—
585C23—44no—
586C23—42no—
587C23—44no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—44no—
592C23—42no—
593C23—44no—
594C23—42no—
595C23—44no—
596C23—42no—
597C23—44no—
598C23C23×D411yescharacteristic
599C23C23×D411yescharacteristic
600C23C22×#4111yescharacteristic
601C23C22×#4111yescharacteristic
602C23C22×#4111yescharacteristic
603C23C2×H(5,𝔽2)12yes—
604C23C2×2−1+411yescharacteristic
605C23C2×2−1+411yescharacteristic
606C23#58512yes—
607C23#58512yes—
608C23#58512yes—
609Q8—416no—
610D4—416no—
611D4—816no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—22no—
628C2×C4—22no—
629C2×C4—24no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—24no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—41no—
643C2×C4—41no—
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—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—416no—
659C2×C4—41no—
660C2×C4—41no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—41no—
669C2×C4—44no—
670C2×C4—41no—
671C2×C4—44no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—41no—
678C2×C4—42no—
679C2×C4—41no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—44no—
684C2×C4—42no—
685C2×C4—44no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—816no—
696C2×C4C2×#12114yes—
697C2×C4C2×#12114yes—
698C2×C4C2×#12114yes—
699C2×C4#52514yes—
700C2×C4#53614yes—
701C2×C4#54814yes—
702C2×C4#54814yes—
703C2×C4#55214yes—
704C2×C4#55214yes—
705C22—44no—
706C22—42no—
707C22—44no—
708C22—48no—
709C22—42no—
710C22—44no—
711C22—42no—
712C22—44no—
713C22—44no—
714C22—48no—
715C22—84no—
716C22—82no—
717C22—81no—
718C22—82no—
719C22—81no—
720C22—82no—
721C22—81no—
722C22—82no—
723C22—84no—
724C22—82no—
725C22—81no—
726C22—82no—
727C22C22×#12111yescharacteristic
728C22C2×#52511yescharacteristic
729C22C2×#53611yescharacteristic
730C22#344912yes—
731C22C2×#54811yescharacteristic
732C22C2×#54811yescharacteristic
733C22C2×#55211yescharacteristic
734C22C2×#55211yescharacteristic
735C22#348511yescharacteristic
736C22#348711yescharacteristic
737C22#348812yes—
738C22#349312yes—
739C22#349811yescharacteristic
740C22#350011yescharacteristic
741C22#350612yes—
742C22#350612yes—
743C22#351012yes—
744C22#351312yes—
745C22#352112yes—
746C22#355412yes—
747C22#355712yes—
748C22#355912yes—
749C22#356512yes—
750C4—24no—
751C4—24no—
752C4—42no—
753C4—44no—
754C4—44no—
755C4—44no—
756C4—44no—
757C4—44no—
758C4—84no—
759C4—84no—
760C4—81no—
761C4—81no—
762C4—82no—
763C4—82no—
764C4#253618yes—
765C2—82no—
766C2—84no—
767C2C2×#253611yescharacteristic
768C2#3515011yescharacteristic
769C2#3553311yescharacteristic
770C2#3584411yescharacteristic
771C2#3603311yescharacteristic
772C2#3911812yes—
773C2#3971812yes—
774C2#4265111yescharacteristic
775C2#4287511yescharacteristic
776C2#4513212yes—
777C2#4525412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2580C2semidirect
2. C2×#2396C2semidirect
3. C2×#2355C2semidirect
5. #17023C2semidirect
6. #16887C2semidirect
7. #15452C2semidirect
8. #15339C2semidirect
9. #15221C2semidirect
11. #15175C2semidirect
12. #15084C2semidirect
13. #14340C2semidirect
14. #14022C2semidirect
15. #13974C2semidirect
16. #13955C2semidirect
17. #13906C2semidirect
97. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25802
C2×#23962
C2×#23552
C2×#23242
#170232
#168872
#154522
#153392
#152212
#152192
#151752
#150842
#143402
#140222
#139742
#139552
#139062