Finite Group Databasebeta

← #4380491

Subgroups — #4380491

Order 512.

3930 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#543C211yesmaximal, characteristic
2C2×#2482C211yesmaximal, characteristic
3C2×#2499C211yesmaximal, characteristic, complemented
4C2×#2625C211yesmaximal, characteristic, complemented
5#16262C214yesmaximal, complemented
6#15985C214yesmaximal, complemented
7#15670C212yesmaximal, complemented
8#15466C212yesmaximal, complemented
9#15465C212yesmaximal, complemented
10#15430C212yesmaximal, complemented
11#15354C214yesmaximal, complemented
12#14727C214yesmaximal
13#14660C211yesmaximal, characteristic, complemented
14#14292C211yesmaximal, characteristic, complemented
15#13903C211yesmaximal, characteristic
16C2×#543—28no—
17C2×#543—28no—
18C2×#543—28no—
19C22×C22⋊Q8C2212yes—
20C22×C22⋊Q8C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C22⋊Q8C2211yescharacteristic
25C22×C22⋊Q8C2211yescharacteristic
26C22×#117C2211yescharacteristic
27C23×C22⋊C4C2212yes—
28C23×C22⋊C4C2211yescharacteristic
29#2625—28no—
30#2499—28no—
31#2482—28no—
32C2×#400C2212yescomplemented
33C2×#399C2212yes—
34C2×#398C2212yescomplemented
35C2×#397C2214yes—
36C2×#397C2212yes—
37C2×#397C2214yes—
38C2×#395C2214yescomplemented
39C2×#391C2212yescomplemented
40C2×#391C2211yescharacteristic, complemented
41C2×#391C2211yescharacteristic, complemented
42C2×#389C2212yescomplemented
43C2×#388C2214yes—
44C2×#387C2214yescomplemented
45C2×#385C2212yes—
46C2×#385C2214yes—
47C2×#384C2211yescharacteristic, complemented
48C2×#382C2211yescharacteristic, complemented
49C2×#381C2211yescharacteristic
50C2×#381C2212yes—
51C2×#380C2211yescharacteristic
52C2×#379C2212yes—
53C2×#379C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55#1446C2214yescomplemented
56#1446C2212yescomplemented
57#1446C2212yescomplemented
58#1445C2212yescomplemented
59#1445C2212yescomplemented
60#1444C2214yescomplemented
61#1444C2214yescomplemented
62#1442C2214yescomplemented
63#1442C2214yescomplemented
64#1442C2214yescomplemented
65#1441C2214yescomplemented
66#1441C2214yescomplemented
67#1440C2214yes—
68#1440C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1437C2212yes—
75#1437C2212yes—
76#1436C2212yescomplemented
77#1436C2212yescomplemented
78C22⋊#95C2214yes—
79C22⋊#95C2214yes—
80C22⋊#95C2214yes—
81C26C2311yescharacteristic
82C23×Q8C2312yes—
83C23×Q8C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89#543—2128no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—28no—
103C2×C22⋊Q8—28no—
104C2×C22⋊Q8—28no—
105C2×C22⋊Q8—28no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—28no—
108C2×C22⋊Q8—28no—
109C2×C22⋊Q8D418yes—
110C2×C22⋊Q8D418yes—
111C2×#117—24no—
112C2×#117—24no—
113C2×#117—24no—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C4⋊C4C2314yes—
125C22×C4⋊C4C2314yes—
126C22×C4⋊C4C2312yes—
127C22×C22⋊C4—28no—
128C22×C22⋊C4—24no—
129C22×C22⋊C4—24no—
130C22×C22⋊C4—28no—
131C22×C22⋊C4—28no—
132C22×C22⋊C4D418yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C22⋊C4C2314yes—
139C22×C22⋊C4C2312yes—
140C22×C22⋊C4C2312yes—
141C22×C22⋊C4C2312yes—
142C22×C22⋊C4C2312yes—
143C22×C22⋊C4C2312yes—
144C22×C22⋊C4C2314yes—
145C22×C22⋊C4C2314yes—
146C22×C22⋊C4C2312yes—
147C22×C42C2312yes—
148C22×C42C2311yescharacteristic
149C22×C42C2311yescharacteristic
150#400—28no—
151#399—28no—
152#398—28no—
153#397—216no—
154#397—28no—
155#397—216no—
156#395—48no—
157#391—28no—
158#391—24no—
159#391—24no—
160#389—28no—
161#388—216no—
162#387—216no—
163#385—28no—
164#385—48no—
165#384—24no—
166#382—24no—
167#381—24no—
168#381—44no—
169#380—24no—
170#379—28no—
171#379—44no—
172C4×C22⋊C4—24no—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C25—22no—
195C25—22no—
196C25—28no—
197C25—42no—
198C25—42no—
199C25—42no—
200C25—42no—
201C25C2×D414yes—
202C25C2411yescharacteristic
203C25C2412yes—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—22no—
208C22×Q8—22no—
209C22×Q8—22no—
210C22×Q8—22no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8—42no—
214C22×Q8—42no—
215C22×Q8—44no—
216C22×Q8—44no—
217C22×Q8C2×D414yes—
218C22×Q8C2×D414yes—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C22⋊Q8—264no—
253C22⋊Q8—264no—
254C22⋊Q8—264no—
255C22⋊Q8—264no—
256C22⋊Q8—264no—
257C22⋊Q8—232no—
258C22⋊Q8—232no—
259#117—416no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
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—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—28no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—42no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2×D418yes—
297C2×C4⋊C4C2×D414yes—
298C2×C4⋊C4C2×D418yes—
299C2×C22⋊C4—264no—
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—232no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4—24no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—28no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—28no—
326C2×C22⋊C4—28no—
327C2×C22⋊C4—24no—
328C2×C22⋊C4—28no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—42no—
331C2×C22⋊C4—42no—
332C2×C22⋊C4—42no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—42no—
337C2×C22⋊C4—42no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4C2×D418yes—
340C2×C22⋊C4C2×D418yes—
341C2×C22⋊C4C2×D418yes—
342C2×C22⋊C4C2×D418yes—
343C2×C42—24no—
344C2×C42—24no—
345C2×C42—24no—
346C2×C42—22no—
347C2×C42—22no—
348C2×C42—42no—
349C2×C42—42no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—44no—
364#95—44no—
365#95—44no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371C24—216no—
372C24—216no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—24no—
378C24—24no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—24no—
383C24—22no—
384C24—24no—
385C24—28no—
386C24—216no—
387C24—48no—
388C24—48no—
389C24—82no—
390C24—82no—
391C24—84no—
392C24—84no—
393C24—82no—
394C24—82no—
395C24—88no—
396C24—88no—
397C24—88no—
398C24—88no—
399C24C22×D412yes—
400C24C22×D412yes—
401C24C22×D412yes—
402C24C22×D414yes—
403C242−1+418yes—
404C24C2511yescharacteristic
405C2×Q8—216no—
406C2×Q8—28no—
407C2×Q8—28no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8—84no—
412C2×Q8—84no—
413C2×Q8—82no—
414C2×Q8—82no—
415C2×Q8—84no—
416C2×Q8—88no—
417C2×Q8—88no—
418C2×Q8—84no—
419C2×Q8—84no—
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—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—216no—
449C22×C4—216no—
450C22×C4—28no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—84no—
486C22×C4—84no—
487C22×C4—88no—
488C22×C4—88no—
489C22×C4—84no—
490C22×C4—84no—
491C22×C4—88no—
492C22×C4—84no—
493C22×C4—88no—
494C22×C4—82no—
495C22×C4—82no—
496C22×C4#120116yes—
497C22×C4C22×D414yes—
498C22×C4C22×D414yes—
499C22×C4C22×D414yes—
500C22×C4C22×D414yes—
501C22×C4C22×D414yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4H(5,𝔽2)14yes—
509C4⋊C4—216no—
510C4⋊C4—216no—
511C4⋊C4—216no—
512C4⋊C4—216no—
513C4⋊C4—28no—
514C4⋊C4—28no—
515C4⋊C4—216no—
516C4⋊C4—416no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—88no—
521C4⋊C4—88no—
522C22⋊C4—216no—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C22⋊C4—216no—
527C22⋊C4—216no—
528C22⋊C4—48no—
529C22⋊C4—416no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—416no—
533C22⋊C4—88no—
534C22⋊C4—84no—
535C22⋊C4—88no—
536C42—44no—
537C42—44no—
538C42—48no—
539C23—232no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
544C23—22no—
545C23—22no—
546C23—24no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—28no—
551C23—28no—
552C23—24no—
553C23—28no—
554C23—24no—
555C23—22no—
556C23—22no—
557C23—28no—
558C23—28no—
559C23—28no—
560C23—28no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—832no—
576C23—88no—
577C23—88no—
578C23—816no—
579C23—816no—
580C23—816no—
581C23—816no—
582C23C2×#12014yes—
583C23#53618yes—
584C23C23×D411yescharacteristic
585C23C23×D411yescharacteristic
586C23C23×D411yescharacteristic
587C23C2×H(5,𝔽2)12yes—
588C23C2×2−1+414yes—
589C23#58512yes—
590C23#58512yes—
591C23#58512yes—
592Q8—88no—
593Q8—88no—
594Q8—816no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—416no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—88no—
664C2×C4—816no—
665C2×C4—816no—
666C2×C4C2×#12014yes—
667C2×C4C2×#12014yes—
668C22—216no—
669C22—24no—
670C22—24no—
671C22—28no—
672C22—24no—
673C22—48no—
674C22—48no—
675C22—44no—
676C22—44no—
677C22—48no—
678C22—44no—
679C22—48no—
680C22—44no—
681C22—44no—
682C22—48no—
683C22—44no—
684C22—48no—
685C22—832no—
686C22C22×#12011yescharacteristic
687C22C2×#53612yes—
688C22#348512yes—
689C22#348511yescharacteristic
690C22#348511yescharacteristic
691C22#348512yes—
692C22#348612yes—
693C22#348612yes—
694C22#348812yes—
695C22#349612yes—
696C22#349811yescharacteristic
697C22#350011yescharacteristic
698C22#353514yes—
699C22#356114yes—
700C22#356414yes—
701C22#356414yes—
702C4—24no—
703C4—24no—
704C4—44no—
705C4—44no—
706C4—44no—
707C4—44no—
708C4—44no—
709C4—44no—
710C4—84no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—84no—
715C2—48no—
716C2—44no—
717C2#3524211yescharacteristic
718C2#3544611yescharacteristic
719C2#3551311yescharacteristic
720C2#4051114yes—
721C2#4681512yes—
722C2#4681612yes—
723C2#4877812yes—
724C2#4877912yes—
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
3. C2×#2499C2semidirect
4. C2×#2625C2semidirect
5. #16262C2semidirect
6. #15985C2semidirect
7. #15670C2semidirect
8. #15466C2semidirect
9. #15465C2semidirect
10. #15430C2semidirect
11. #15354C2semidirect
13. #14660C2semidirect
14. #14292C2semidirect
32. C2×#400C22semidirect
34. C2×#398C22semidirect
38. C2×#395C22semidirect
39. C2×#391C22semidirect
40. C2×#391C22semidirect
41. C2×#391C22semidirect
42. C2×#389C22semidirect
44. C2×#387C22semidirect
47. C2×#384C22semidirect
48. C2×#382C22semidirect
55. #1446C22semidirect
56. #1446C22semidirect
57. #1446C22semidirect
58. #1445C22semidirect
59. #1445C22semidirect
60. #1444C22semidirect
61. #1444C22semidirect
62. #1442C22semidirect
63. #1442C22semidirect
64. #1442C22semidirect
65. #1441C22semidirect
66. #1441C22semidirect
76. #1436C22semidirect
77. #1436C22semidirect

Maximal subgroups

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

subgroupindex
C22×#5432
C2×#24822
C2×#24992
C2×#26252
#162622
#159852
#156702
#154662
#154652
#154302
#153542
#147272
#146602
#142922
#139032