Finite Group Databasebeta

← #625868

Subgroups — #625868

Order 512.

3340 conjugacy classes of subgroups.

The subgroup lattice has 709 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2500C211yesmaximal, characteristic
2C2×#2499C211yesmaximal, characteristic, complemented
3C2×#2502C211yesmaximal, characteristic, complemented
4C2×#2501C211yesmaximal, characteristic
5C2×#2620C211yesmaximal, characteristic
6C2×#2317C211yesmaximal, characteristic, complemented
7#15765C218yesmaximal, complemented
8#15384C218yesmaximal, complemented
9#15244C211yesmaximal, characteristic, complemented
10#15240C211yesmaximal, characteristic, complemented
11#15199C211yesmaximal, characteristic, complemented
12#15155C211yesmaximal, characteristic, complemented
13#14708C211yesmaximal, characteristic, complemented
14#14692C211yesmaximal, characteristic, complemented
15#14354C211yesmaximal, characteristic, complemented
16#14273C211yesmaximal, characteristic, complemented
17#13761C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×#123C2214yes—
20C22×C22⋊Q8C2211yescharacteristic
21C22×#121C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25#2620—28no—
26#2502—28no—
27#2501—28no—
28#2500—28no—
29#2499—28no—
30C2×#399C2214yes—
31C2×#399C2218yes—
32C2×#398C2214yes—
33C2×#397C2214yes—
34C2×#396C2214yes—
35C2×#394C2214yes—
36C2×#394C2214yes—
37#2317—28no—
38C2×#391C2211yescharacteristic
39C2×#391C2211yescharacteristic
40C2×#391C2211yescharacteristic
41C2×#391C2211yescharacteristic
42C2×#391C2211yescharacteristic
43C2×#390C2211yescharacteristic
44C2×#390C2211yescharacteristic
45C2×#390C2211yescharacteristic
46C2×#390C2211yescharacteristic
47C2×#390C2211yescharacteristic
48C2×#389C2214yes—
49C2×#388C2214yes—
50C2×#387C2214yes—
51C2×#385C2214yes—
52C2×#385C2214yes—
53C2×#384C2211yescharacteristic
54C2×#384C2211yescharacteristic
55C2×#384C2211yescharacteristic
56C2×#384C2211yescharacteristic
57C2×#384C2211yescharacteristic
58C2×#384C2211yescharacteristic
59C2×#384C2211yescharacteristic
60C2×#384C2211yescharacteristic
61C2×#384C2211yescharacteristic
62C2×#384C2218yes—
63C2×#382C2211yescharacteristic
64C2×#382C2211yescharacteristic
65C2×#382C2211yescharacteristic
66C2×#382C2211yescharacteristic
67C2×C4×C22⋊C4C2211yescharacteristic
68C2×C4×C22⋊C4C2211yescharacteristic
69C2×#376C2211yescharacteristic
70C2×C43C2211yescharacteristic
71#1445C2218yes—
72#1444C2218yes—
73#1444C2218yes—
74#1439C2218yes—
75#1439C2218yes—
76#1437C2214yes—
77#1437C2218yes—
78#1436C2214yes—
79C4×#95C2211yescharacteristic
80C4×#95C2211yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2312yes—
86C2×#124—24no—
87C2×#124—24no—
88C2×#124—24no—
89C2×#123—216no—
90C2×#123—416no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C2×#117—24no—
98C2×#117—24no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117D418yes—
103C22×C4⋊C4—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2314yes—
122C22×C4⋊C4C2314yes—
123C22×C4⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2314yes—
136C22×C42C2312yes—
137C22×C42C2314yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#399—216no—
145#399—416no—
146#398—216no—
147#397—216no—
148#396—216no—
149#394—216no—
150#394—216no—
151#391—24no—
152#391—24no—
153#391—24no—
154#391—24no—
155#391—24no—
156#390—24no—
157#390—24no—
158#390—24no—
159#390—24no—
160#390—24no—
161#389—216no—
162#388—216no—
163#387—216no—
164#385—216no—
165#385—216no—
166#384—24no—
167#384—24no—
168#384—24no—
169#384—24no—
170#384—24no—
171#384—24no—
172#384—24no—
173#384—24no—
174#384—24no—
175#384—416no—
176#382—24no—
177#382—24no—
178#382—24no—
179#382—24no—
180C4×C22⋊C4D418yes—
181C4×C22⋊C4D418yes—
182#376—24no—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C43—24no—
198C25C2412yes—
199C25C2411yescharacteristic
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—22no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—48no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—48no—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244#124—416no—
245#123—464no—
246C22⋊Q8—416no—
247#121—416no—
248#117—232no—
249#117—416no—
250C2×C4⋊C4—24no—
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—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—216no—
280C2×C4⋊C4—216no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—48no—
288C2×C4⋊C4—416no—
289C2×C4⋊C4—48no—
290C2×C4⋊C4—48no—
291C2×C4⋊C4—48no—
292C2×C4⋊C4—48no—
293C2×C4⋊C4—48no—
294C2×C4⋊C4—48no—
295C2×C4⋊C4C2×D418yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—216no—
316C2×C22⋊C4—216no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—48no—
320C2×C22⋊C4—48no—
321C2×C22⋊C4—48no—
322C2×C22⋊C4C2×D418yes—
323C2×C22⋊C4C2×D418yes—
324C2×C22⋊C4C2×D418yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—28no—
328C2×C42—22no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—28no—
333C2×C42—28no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344C2×C42C2×D414yes—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347C2×C42#4114yes—
348C2×C42#4114yes—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95C2×D418yes—
362#95C2×D418yes—
363C24—21no—
364C24—21no—
365C24—24no—
366C24—24no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—28no—
375C24—28no—
376C24—22no—
377C24—22no—
378C24C22×D412yes—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C2511yescharacteristic
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×D4—416no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—88no—
399C2×D4—88no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
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—28no—
454C22×C4—28no—
455C22×C4—28no—
456C22×C4—28no—
457C22×C4—28no—
458C22×C4—28no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4#127116yes—
466C22×C4C22×D414yes—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D414yes—
474C22×C4C22×D414yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C2×#4114yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4112yes—
482C22×C4H(5,𝔽2)18yes—
483C22×C42−1+418yes—
484C22×C42−1+412yes—
485C22×C42−1+412yes—
486C4⋊C4—216no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—416no—
497C4⋊C4—416no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C4⋊C4—88no—
501C4⋊C4—88no—
502C4⋊C4—88no—
503C4⋊C4—88no—
504C22⋊C4—216no—
505C22⋊C4—216no—
506C22⋊C4—216no—
507C22⋊C4—416no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—48no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—88no—
515C22⋊C4—88no—
516C42—232no—
517C42—28no—
518C42—28no—
519C42—28no—
520C42—28no—
521C42—28no—
522C42—48no—
523C42#123116yescomplemented
524C23—22no—
525C23—24no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—48no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—48no—
555C23—48no—
556C23—48no—
557C23C2×#12714yes—
558C23C23×D411yescharacteristic
559C23C23×D411yescharacteristic
560C23C23×D411yescharacteristic
561C23C22×#4111yescharacteristic
562C23C22×#4111yescharacteristic
563C23C2×H(5,𝔽2)14yes—
564C23C2×2−1+411yescharacteristic
565C23C2×2−1+411yescharacteristic
566C23C2×2−1+414yes—
567Q8—88no—
568D4—816no—
569D4—88no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—232no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
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—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—816no—
636C2×C4C2×#12318yes—
637C2×C4C2×#12314yes—
638C2×C4C2×#12714yes—
639C2×C4C2×#12714yes—
640C2×C4#53518yes—
641C2×C4#53618yes—
642C2×C4#53614yes—
643C2×C4#54714yes—
644C2×C4#54714yes—
645C2×C4#54714yes—
646C2×C4#54714yes—
647C22—24no—
648C22—24no—
649C22—24no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—84no—
657C22—84no—
658C22—84no—
659C22—82no—
660C22—82no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—82no—
665C22C22×#12311yescharacteristic
666C22C22×#12711yescharacteristic
667C22C2×#53512yes—
668C22C2×#53612yes—
669C22C2×#53611yescharacteristic
670C22C2×#54711yescharacteristic
671C22C2×#54711yescharacteristic
672C22C2×#54711yescharacteristic
673C22C2×#54711yescharacteristic
674C22#349214yes—
675C22#349812yes—
676C22#349812yes—
677C22#349912yes—
678C22#349912yes—
679C22#351114yes—
680C22#354414yes—
681C22#355214yes—
682C4—24no—
683C4—28no—
684C4—28no—
685C4—28no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4#250118yes—
697C2—44no—
698C2—162no—
699C2C2×#250111yescharacteristic
700C2#3527411yescharacteristic
701C2#3531711yescharacteristic
702C2#3553012yes—
703C2#3730312yes—
704C2#3786212yes—
705C2#3786312yes—
706C2#3948812yes—
707C2#3948912yes—
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
2. C2×#2499C2semidirect
3. C2×#2502C2semidirect
6. C2×#2317C2semidirect
7. #15765C2semidirect
8. #15384C2semidirect
9. #15244C2semidirect
10. #15240C2semidirect
11. #15199C2semidirect
12. #15155C2semidirect
13. #14708C2semidirect
14. #14692C2semidirect
15. #14354C2semidirect
16. #14273C2semidirect
17. #13761C2semidirect
523. C42#123semidirect

Maximal subgroups

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

subgroupindex
C2×#25002
C2×#24992
C2×#25022
C2×#25012
C2×#26202
C2×#23172
#157652
#153842
#152442
#152402
#151992
#151552
#147082
#146922
#143542
#142732
#137612