Finite Group Databasebeta

← #765941

Subgroups — #765941

Order 512.

3722 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2485C212yesmaximal, complemented
2C2×#2409C212yesmaximal, complemented
3C2×#2392C211yesmaximal, characteristic
4C2×#2310C211yesmaximal, characteristic, complemented
5#15856C212yesmaximal, complemented
6#15833C212yesmaximal, complemented
7#15825C214yesmaximal, complemented
8#15561C212yesmaximal
9#15549C214yesmaximal, complemented
10#15539C212yesmaximal, complemented
11#15193C212yesmaximal, complemented
12#15177C212yesmaximal, complemented
13#13997C212yesmaximal, complemented
14#13997C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic, complemented
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#2485—216no—
24#2409—216no—
25#2392—28no—
26C2×#398C2211yescharacteristic
27C2×#397C2211yescharacteristic
28C2×#397C2214yes—
29C2×#396C2212yes—
30C2×#395C2214yes—
31C2×#394C2212yes—
32C2×#394C2214yes—
33C2×#394C2211yescharacteristic
34C2×#393C2214yes—
35C2×#392C2211yescharacteristic
36#2310—28no—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#388C2214yes—
46C2×#387C2214yes—
47C2×#386C2214yes—
48C2×#385C2214yes—
49C2×#381C2212yes—
50C2×#381C2212yes—
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2212yes—
58#1445C2214yes—
59#1445C2214yes—
60#1445C2214yes—
61#1445C2214yes—
62#1445C2214yes—
63#1444C2214yes—
64#1444C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1437C2212yes—
73#1437C2212yes—
74#1436C2212yes—
75#1436C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78C4⋊#95C2214yes—
79C23×Q8—28no—
80C23×Q8—24no—
81C23×Q8—28no—
82C23×Q8—42no—
83C23×Q8—42no—
84C23×Q8C2312yes—
85C23×Q8C2312yes—
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—44no—
96C2×C22⋊Q8—44no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4—24no—
101C2×C4×D4—42no—
102C2×C4×D4—42no—
103C2×#117—24no—
104C2×#117—28no—
105C2×#117—42no—
106C2×#117—42no—
107C2×#117D418yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2314yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2312yes—
138C22×C42C2312yes—
139#398—24no—
140#397—24no—
141#397—216no—
142#396—28no—
143#395—216no—
144#394—24no—
145#394—28no—
146#394—216no—
147#393—216no—
148#392—24no—
149#391—28no—
150#391—28no—
151#391—28no—
152#391—28no—
153#391—28no—
154#391—28no—
155#389—28no—
156#389—28no—
157#388—216no—
158#387—216no—
159#386—216no—
160#385—216no—
161#381—28no—
162#381—28no—
163#380—28no—
164#380—28no—
165#380D4116yescomplemented
166C4×C4⋊C4—24no—
167C4×C4⋊C4—42no—
168C4×C4⋊C4—42no—
169C4×C22⋊C4—28no—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C25C2412yes—
192C25C2411yescharacteristic
193C22×Q8—232no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—28no—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—432no—
206C22×Q8—432no—
207C22×Q8—416no—
208C22×Q8—416no—
209C22×Q8#4118yes—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—24no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—24no—
217C22×D4—41no—
218C22×D4—41no—
219C22×D4—42no—
220C22×D4—42no—
221C22×D4—42no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—28no—
227C23×C4—28no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4C2×D414yes—
238C23×C4C2×D418yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2414yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2414yes—
249C23×C4C2412yes—
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C22⋊Q8—432no—
253C22⋊Q8—432no—
254C4×D4—416no—
255C4×D4—432no—
256#117—416no—
257#117—416no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
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—22no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—41no—
281C2×C4⋊C4—41no—
282C2×C4⋊C4—41no—
283C2×C4⋊C4—41no—
284C2×C4⋊C4—41no—
285C2×C4⋊C4—41no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4C2×D418yes—
291C2×C4⋊C4C2×D418yes—
292C2×C4⋊C4C2×D414yes—
293C2×C4⋊C4C2×D414yes—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—42no—
319C2×C22⋊C4—41no—
320C2×C22⋊C4—41no—
321C2×C22⋊C4—41no—
322C2×C22⋊C4—41no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—42no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4C2×D414yes—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4C2×D4116yes—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—28no—
337C2×C42—24no—
338C2×C42—24no—
339C2×C42—24no—
340C2×C42—24no—
341C2×C42—42no—
342C2×C42—42no—
343C2×C42—44no—
344C2×C42—44no—
345C2×C42C2×D414yes—
346C2×C42C2×D414yes—
347#95—24no—
348#95—24no—
349#95—22no—
350#95—22no—
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#95—28no—
362#95—28no—
363#95—44no—
364#95—44no—
365#95—42no—
366#95—42no—
367#95C2×D4116yes—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24C22×D412yes—
390C24C22×D414yes—
391C242−1+414yes—
392C24C2511yescharacteristic
393C2×Q8—216no—
394C2×Q8—216no—
395C2×Q8—216no—
396C2×Q8—216no—
397C2×Q8—416no—
398C2×Q8—416no—
399C2×Q8—48no—
400C2×Q8—4128no—
401C2×Q8—48no—
402C2×Q8—48no—
403C2×Q8—48no—
404C2×Q8—48no—
405C2×Q8—48no—
406C2×D4—216no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—44no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—44no—
414C2×D4—48no—
415C2×D4—48no—
416C2×D4—44no—
417C2×D4—48no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—216no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—416no—
488C22×C4—416no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4#127116yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D418yes—
503C22×C4C22×D418yes—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C22×D414yes—
509C22×C4C2×#4118yes—
510C22×C4C2×#4114yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C4H(5,𝔽2)12yes—
513C22×C42−1+414yes—
514C4⋊C4—216no—
515C4⋊C4—216no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—48no—
519C4⋊C4—416no—
520C4⋊C4—416no—
521C4⋊C4—416no—
522C4⋊C4—44no—
523C4⋊C4—48no—
524C4⋊C4—48no—
525C4⋊C4—88no—
526C22⋊C4—416no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—48no—
530C22⋊C4—416no—
531C22⋊C4—416no—
532C22⋊C4—416no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—88no—
539C42—28no—
540C42—28no—
541C42—48no—
542C42—48no—
543C42—84no—
544C42—84no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—24no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—41no—
567C23—41no—
568C23—42no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—42no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23C2×#12714yes—
601C23C23×D411yescharacteristic
602C23C23×D412yes—
603C23C22×#4112yes—
604C23C2×H(5,𝔽2)11yescharacteristic
605C23C2×H(5,𝔽2)11yescharacteristic
606C23C2×2−1+412yes—
607C23C2×2−1+412yes—
608C23#58512yes—
609C23#58512yes—
610Q8—432no—
611Q8—432no—
612D4—432no—
613D4—416no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—232no—
619C2×C4—28no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—28no—
629C2×C4—28no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—24no—
636C2×C4—24no—
637C2×C4—28no—
638C2×C4—24no—
639C2×C4—432no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—41no—
653C2×C4—41no—
654C2×C4—41no—
655C2×C4—44no—
656C2×C4—41no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—41no—
664C2×C4—41no—
665C2×C4—41no—
666C2×C4—44no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—44no—
686C2×C4—42no—
687C2×C4—44no—
688C2×C4—42no—
689C2×C4—816no—
690C2×C4#52514yes—
691C2×C4C2×#12714yes—
692C2×C4C2×#12714yes—
693C2×C4#53414yes—
694C2×C4#54618yes—
695C2×C4#54618yes—
696C22—24no—
697C22—28no—
698C22—42no—
699C22—44no—
700C22—44no—
701C22—44no—
702C22—44no—
703C22—44no—
704C22—42no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—44no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—82no—
713C22—82no—
714C22—82no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22C2×#52511yescharacteristic
721C22C22×#12711yescharacteristic
722C22C2×#53411yescharacteristic
723C22C2×#54612yes—
724C22C2×#54612yes—
725C22#348512yes—
726C22#348612yes—
727C22#348912yes—
728C22#348912yes—
729C22#348912yes—
730C22#348912yes—
731C22#351612yes—
732C22#351812yes—
733C22#352312yes—
734C22#352412yes—
735C22#353811yescharacteristic
736C22#354011yescharacteristic
737C22#354112yes—
738C22#354712yes—
739C22#355011yescharacteristic
740C22#355111yescharacteristic
741C4—24no—
742C4—42no—
743C4—42no—
744C4—48no—
745C4—44no—
746C4—48no—
747C4—48no—
748C4—42no—
749C4—42no—
750C4—82no—
751C4—82no—
752C4—84no—
753C4#253918yes—
754C2—44no—
755C2—84no—
756C2C2×#253911yescharacteristic
757C2#3563412yes—
758C2#3566312yes—
759C2#3737411yescharacteristic
760C2#3737611yescharacteristic
761C2#4513612yes—
762C2#4529112yes—
763C2#4538512yes—
764C2#4541612yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2485C2semidirect
2. C2×#2409C2semidirect
4. C2×#2310C2semidirect
5. #15856C2semidirect
6. #15833C2semidirect
7. #15825C2semidirect
9. #15549C2semidirect
10. #15539C2semidirect
11. #15193C2semidirect
12. #15177C2semidirect
13. #13997C2semidirect
14. #13997C2semidirect
15. #13777C2semidirect
165. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24852
C2×#24092
C2×#23922
C2×#23102
#158562
#158332
#158252
#155612
#155492
#155392
#151932
#151772
#139972
#137772