Finite Group Databasebeta

← #4389662

Subgroups — #4389662

Order 512.

3500 conjugacy classes of subgroups.

The subgroup lattice has 773 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2816C211yesmaximal, characteristic, complemented
2C2×#2539C211yesmaximal, characteristic
3C2×#2482C211yesmaximal, characteristic, complemented
4C2×#2499C211yesmaximal, characteristic, complemented
5C2×#2624C211yesmaximal, characteristic, complemented
6C2×#2626C211yesmaximal, characteristic, complemented
7#16814C212yesmaximal, complemented
8#16692C212yesmaximal, complemented
9#15776C214yesmaximal, complemented
10#15765C214yesmaximal, complemented
11#15384C214yesmaximal, complemented
12#15283C214yesmaximal, complemented
13#15167C212yesmaximal, complemented
14#15155C212yesmaximal, complemented
15#13780C211yesmaximal, characteristic, complemented
16C22×#123C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C22×#121C2211yescharacteristic
19C22×#120C2212yes—
20C22×#117C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2816—28no—
23#2626—28no—
24#2624—28no—
25#2539—28no—
26#2499—28no—
27#2482—28no—
28C2×#399C2212yes—
29C2×#399C2212yes—
30C2×#399C2212yes—
31C2×#399C2214yes—
32C2×#399C2212yescomplemented
33C2×#399C2212yescomplemented
34C2×#399C2214yescomplemented
35C2×#398C2212yescomplemented
36C2×#397C2214yes—
37C2×#397C2212yes—
38C2×#397C2212yescomplemented
39C2×#396C2214yes—
40C2×#394C2214yes—
41C2×#394C2214yes—
42C2×#394C2212yescomplemented
43C2×#393C2212yes—
44C2×#393C2212yescomplemented
45C2×#392C2212yes—
46C2×#391C2211yescharacteristic, complemented
47C2×#391C2211yescharacteristic, complemented
48C2×#390C2212yes—
49C2×#390C2212yes—
50C2×#390C2214yes—
51C2×#390C2212yescomplemented
52C2×#389C2212yescomplemented
53C2×#388C2214yes—
54C2×#387C2214yescomplemented
55C2×#385C2212yes—
56C2×#385C2212yes—
57C2×#384C2211yescharacteristic, complemented
58C2×#384C2214yescomplemented
59C2×#382C2211yescharacteristic, complemented
60C2×#381C2211yescharacteristic
61C2×#380C2211yescharacteristic
62C2×#379C2212yes—
63C2×#379C2212yes—
64C2×C4×C22⋊C4C2211yescharacteristic
65#1445C2214yescomplemented
66#1444C2214yescomplemented
67#1444C2214yescomplemented
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yescomplemented
71#1439C2214yescomplemented
72#1438C2214yes—
73#1438C2214yes—
74#1438C2214yes—
75#1437C2214yes—
76#1437C2214yes—
77#1437C2214yescomplemented
78#1436C2214yescomplemented
79#1434C2212yescomplemented
80C4×#95C2212yescomplemented
81C26C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×D4C2312yes—
84C23×D4C2311yescharacteristic
85C23×D4C2312yes—
86C23×D4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×#123—28no—
90C2×#123—48no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×#121—24no—
95C2×#121—24no—
96C2×#121—24no—
97C2×#120—28no—
98C2×#120—48no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—24no—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—24no—
116C22×C22⋊C4—24no—
117C22×C22⋊C4—24no—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C22⋊C4C2314yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2312yes—
137C22×C22⋊C4C2312yes—
138C22×C42C2312yes—
139C22×C42C2314yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#399—28no—
143#399—28no—
144#399—28no—
145#399—28no—
146#399—28no—
147#399—48no—
148#399—48no—
149#398—28no—
150#397—216no—
151#397—28no—
152#397—28no—
153#396—216no—
154#394—216no—
155#394—216no—
156#394—28no—
157#393—28no—
158#393—28no—
159#392—28no—
160#391—24no—
161#391—24no—
162#390—28no—
163#390—28no—
164#390—28no—
165#390—48no—
166#389—28no—
167#388—216no—
168#387—216no—
169#385—28no—
170#385—28no—
171#384—24no—
172#384—48no—
173#382—24no—
174#381D418yescomplemented
175#380D418yescomplemented
176#379—28no—
177#379—28no—
178C4×C22⋊C4D418yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2312yes—
193C25—22no—
194C25—24no—
195C25—22no—
196C25—22no—
197C25—42no—
198C25—44no—
199C25—42no—
200C25—42no—
201C25C2412yes—
202C25C2412yes—
203C25C2411yescharacteristic
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—22no—
215C22×D4—24no—
216C22×D4—24no—
217C22×D4—22no—
218C22×D4—22no—
219C22×D4—28no—
220C22×D4—44no—
221C22×D4—44no—
222C22×D4—44no—
223C22×D4—44no—
224C22×D4—42no—
225C22×D4—42no—
226C22×D4—42no—
227C22×D4—42no—
228C22×D4—44no—
229C23×C4—22no—
230C23×C4—22no—
231C23×C4—22no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—44no—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2414yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250#123—432no—
251C22⋊Q8—416no—
252#121—416no—
253#120—432no—
254#117—416no—
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—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—28no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D418yes—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
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—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—28no—
315C2×C22⋊C4—28no—
316C2×C22⋊C4—28no—
317C2×C22⋊C4—28no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—28no—
320C2×C22⋊C4—28no—
321C2×C22⋊C4—28no—
322C2×C22⋊C4—28no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—42no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—416no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4—44no—
332C2×C22⋊C4—44no—
333C2×C22⋊C4—44no—
334C2×C22⋊C4—44no—
335C2×C22⋊C4—44no—
336C2×C22⋊C4—44no—
337C2×C22⋊C4—44no—
338C2×C22⋊C4—44no—
339C2×C22⋊C4—44no—
340C2×C22⋊C4—44no—
341C2×C22⋊C4—44no—
342C2×C22⋊C4—44no—
343C2×C22⋊C4—44no—
344C2×C22⋊C4C2×D418yes—
345C2×C22⋊C4C2×D418yes—
346C2×C22⋊C4C2×D418yes—
347C2×C42—24no—
348C2×C42—24no—
349C2×C42—24no—
350C2×C42—28no—
351C2×C42—42no—
352C2×C42—42no—
353C2×C42—44no—
354C2×C42—44no—
355C2×C42C2×D414yes—
356C2×C42C2×D414yes—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—28no—
365#95—44no—
366#95—44no—
367#95C2×D418yes—
368#95C2×D418yes—
369#95C2×D418yes—
370#95C2×D418yes—
371C24—24no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—28no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—24no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—24no—
385C24—24no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—24no—
391C24—24no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24—22no—
396C24—24no—
397C24—24no—
398C24—48no—
399C24—48no—
400C24—44no—
401C24—48no—
402C24—48no—
403C24—44no—
404C24—84no—
405C24—82no—
406C24—82no—
407C24—84no—
408C24—84no—
409C24—88no—
410C24—88no—
411C24—88no—
412C24—88no—
413C24C22×D412yes—
414C24C22×D412yes—
415C24C22×D412yes—
416C242−1+414yes—
417C24C2511yescharacteristic
418C2×Q8—44no—
419C2×Q8—44no—
420C2×Q8—44no—
421C2×Q8—84no—
422C2×Q8—84no—
423C2×D4—48no—
424C2×D4—416no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—48no—
430C2×D4—48no—
431C2×D4—44no—
432C2×D4—48no—
433C2×D4—84no—
434C2×D4—84no—
435C2×D4—82no—
436C2×D4—82no—
437C2×D4—88no—
438C2×D4—84no—
439C2×D4—88no—
440C2×D4—88no—
441C2×D4—88no—
442C2×D4—88no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
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—24no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—44no—
502C22×C4—84no—
503C22×C4—84no—
504C22×C4#127116yes—
505C22×C4C22×D414yes—
506C22×C4C22×D418yes—
507C22×C4C22×D418yes—
508C22×C4C22×D412yes—
509C22×C4C22×D412yes—
510C22×C4C22×D412yes—
511C22×C4C22×D412yes—
512C22×C4C22×D412yes—
513C22×C4C22×D412yes—
514C22×C4C22×D414yes—
515C22×C4H(5,𝔽2)18yes—
516C4⋊C4—48no—
517C4⋊C4—48no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—416no—
524C4⋊C4—48no—
525C4⋊C4—88no—
526C4⋊C4—88no—
527C4⋊C4—88no—
528C4⋊C4—88no—
529C22⋊C4—216no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—48no—
541C22⋊C4—48no—
542C22⋊C4—88no—
543C22⋊C4—88no—
544C22⋊C4—88no—
545C22⋊C4—88no—
546C22⋊C4—88no—
547C22⋊C4—88no—
548C22⋊C4—88no—
549C42—44no—
550C42—44no—
551C42—416no—
552C42—48no—
553C23—24no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—28no—
568C23—28no—
569C23—416no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—44no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—44no—
582C23—42no—
583C23—42no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—44no—
597C23—44no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—88no—
616C23—88no—
617C23—816no—
618C23—816no—
619C23—816no—
620C23—832no—
621C23—816no—
622C23C2×#12714yes—
623C23C23×D411yescharacteristic
624C23C23×D411yescharacteristic
625C23C23×D411yescharacteristic
626C23C2×H(5,𝔽2)14yes—
627C23C2×2−1+412yes—
628C23#58512yes—
629C23#58512yes—
630C23#58512yes—
631Q8—88no—
632D4—816no—
633D4—88no—
634D4—816no—
635D4—816no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—24no—
639C2×C4—24no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—28no—
647C2×C4—28no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—28no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—24no—
655C2×C4—28no—
656C2×C4—28no—
657C2×C4—416no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
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—44no—
679C2×C4—44no—
680C2×C4—42no—
681C2×C4—44no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—88no—
699C2×C4C2×#12714yes—
700C2×C4C2×#12714yes—
701C2×C4#53518yes—
702C22—24no—
703C22—24no—
704C22—24no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—48no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—48no—
714C22—48no—
715C22—832no—
716C22—84no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—84no—
721C22—82no—
722C22—82no—
723C22—84no—
724C22—82no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—84no—
730C22—84no—
731C22—82no—
732C22—84no—
733C22C22×#12711yescharacteristic
734C22C2×#53512yes—
735C22#348511yescharacteristic
736C22#348512yes—
737C22#348511yescharacteristic
738C22#348511yescharacteristic
739C22#348512yes—
740C22#348611yescharacteristic
741C22#348912yes—
742C22#348912yes—
743C22#349812yes—
744C22#349812yes—
745C22#353314yes—
746C22#353514yes—
747C22#356114yes—
748C22#356214yes—
749C4—24no—
750C4—24no—
751C4—28no—
752C4—48no—
753C4—44no—
754C4—48no—
755C4—82no—
756C4—82no—
757C4—82no—
758C4—82no—
759C4—82no—
760C4—82no—
761C2—44no—
762C2—84no—
763C2—162no—
764C2#3527411yescharacteristic
765C2#3546311yescharacteristic
766C2#3563411yescharacteristic
767C2#4140712yes—
768C2#4268012yes—
769C2#4681612yes—
770C2#4721312yes—
771C2#4883114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2539C2
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×#2816C2semidirect
3. C2×#2482C2semidirect
4. C2×#2499C2semidirect
5. C2×#2624C2semidirect
6. C2×#2626C2semidirect
7. #16814C2semidirect
8. #16692C2semidirect
9. #15776C2semidirect
10. #15765C2semidirect
11. #15384C2semidirect
12. #15283C2semidirect
13. #15167C2semidirect
14. #15155C2semidirect
15. #13780C2semidirect
32. C2×#399C22semidirect
33. C2×#399C22semidirect
34. C2×#399C22semidirect
35. C2×#398C22semidirect
38. C2×#397C22semidirect
42. C2×#394C22semidirect
44. C2×#393C22semidirect
46. C2×#391C22semidirect
47. C2×#391C22semidirect
51. C2×#390C22semidirect
52. C2×#389C22semidirect
54. C2×#387C22semidirect
57. C2×#384C22semidirect
58. C2×#384C22semidirect
59. C2×#382C22semidirect
65. #1445C22semidirect
66. #1444C22semidirect
67. #1444C22semidirect
70. #1439C22semidirect
71. #1439C22semidirect
77. #1437C22semidirect
78. #1436C22semidirect
79. #1434C22semidirect
80. C4×#95C22semidirect
174. #381D4semidirect
175. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28162
C2×#25392
C2×#24822
C2×#24992
C2×#26242
C2×#26262
#168142
#166922
#157762
#157652
#153842
#152832
#151672
#151552
#137802