Finite Group Databasebeta

← #763886

Subgroups — #763886

Order 512.

3494 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2535C211yesmaximal, characteristic
4C2×#2426C212yesmaximal, complemented
5C2×#2327C211yesmaximal, characteristic, complemented
6#15648C214yesmaximal, complemented
7#15640C214yesmaximal, complemented
8#15380C214yesmaximal, complemented
9#15335C214yesmaximal, complemented
10#15215C212yesmaximal, complemented
11#15179C212yesmaximal, complemented
12#14150C211yesmaximal, characteristic, complemented
13#13969C212yesmaximal, complemented
14#13847C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×C22⋊Q8C2212yes—
20C22×#121C2212yes—
21C22×C4×Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23C22×C4×D4C2211yescharacteristic
24#2547—28no—
25#2542—28no—
26#2535—28no—
27#2426—216no—
28C2×#399C2214yes—
29C2×#397C2214yes—
30C2×#397C2212yes—
31C2×#396C2214yes—
32C2×#394C2214yes—
33C2×#392C2212yes—
34#2327—28no—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#390C2212yes—
38C2×#390C2214yes—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#387C2214yes—
44C2×#387C2214yes—
45C2×#386C2214yes—
46C2×#384C2212yes—
47C2×#382C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55C2×#376C2211yescharacteristic
56#1445C2212yes—
57#1444C2212yes—
58#1442C2212yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2214yes—
70#1438C2212yes—
71#1438C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C22⋊#95C2214yes—
80C23×Q8C2312yes—
81C23×D4C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C23×D4C2314yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8D418yescomplemented
89C2×#124—28no—
90C2×#124—28no—
91C2×#124—44no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—44no—
97C2×C22⋊Q8—44no—
98C2×#121—28no—
99C2×#121—28no—
100C2×#121—44no—
101C2×C4×Q8—28no—
102C2×C4×Q8—28no—
103C2×C4×Q8—44no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×C4×D4—24no—
107C2×C4×D4—24no—
108C2×C4×D4—24no—
109C2×C4×D4—24no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C22⋊C4C2312yes—
136C22×C22⋊C4C2314yes—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139C22×C42C2314yes—
140C22×C42C2311yescharacteristic
141C22×C42C2312yes—
142C22×C42C2311yescharacteristic
143#399—216no—
144#397—216no—
145#397—28no—
146#396—216no—
147#394—216no—
148#392—28no—
149#391—28no—
150#391—28no—
151#390—28no—
152#390—24no—
153#390—216no—
154#390D418yescomplemented
155#389—28no—
156#388—216no—
157#387—216no—
158#387—216no—
159#386—216no—
160#384—28no—
161#382D418yes—
162#380—28no—
163#380—28no—
164#380—28no—
165C4×C4⋊C4—24no—
166C4×C4⋊C4—24no—
167C4×C22⋊C4—28no—
168C4×C22⋊C4—44no—
169#376—24no—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C25C2412yes—
191C25C2412yes—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8C2×D418yes—
199C22×D4—24no—
200C22×D4—28no—
201C22×D4—28no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—28no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—22no—
212C22×D4—42no—
213C22×D4—44no—
214C22×D4—44no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4C2×D414yes—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2414yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2414yes—
239C23×C4C2412yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C4⋊Q8—232no—
244#124—432no—
245C22⋊Q8—432no—
246C22⋊Q8—432no—
247#121—432no—
248C4×Q8—432no—
249C4×D4—416no—
250C4×D4—416no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
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—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4C2×D418yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C4⋊C4#4114yes—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4—42no—
317C2×C22⋊C4—42no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
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×D4116yes—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—28no—
333C2×C42—28no—
334C2×C42—24no—
335C2×C42—24no—
336C2×C42—22no—
337C2×C42—24no—
338C2×C42—22no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42—44no—
343C2×C42C2×D414yes—
344C2×C42C2×D414yes—
345C2×C42C2×D414yes—
346C2×C42#4114yes—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—24no—
351#95—24no—
352#95—24no—
353#95—28no—
354#95—28no—
355#95—28no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95—44no—
363#95—42no—
364#95—42no—
365#95C2×D418yes—
366#95C2×D418yes—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—22no—
380C24—22no—
381C24—24no—
382C24—22no—
383C24—22no—
384C24—24no—
385C24—22no—
386C24—24no—
387C24—24no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24C22×D414yes—
392C242−1+414yes—
393C24C2511yescharacteristic
394C2×Q8—216no—
395C2×Q8—216no—
396C2×Q8—216no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—28no—
404C2×D4—28no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—44no—
408C2×D4—416no—
409C2×D4—416no—
410C2×D4—416no—
411C2×D4—44no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—88no—
417C2×D4—88no—
418C2×D4—88no—
419C2×D4—88no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
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—24no—
476C22×C4—24no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—48no—
486C22×C4—48no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—48no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—88no—
494C22×C4—88no—
495C22×C4C22×D418yes—
496C22×C4C22×D414yes—
497C22×C4C22×D418yes—
498C22×C4C22×D414yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C2×#4114yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4112yes—
511C22×C4H(5,𝔽2)12yes—
512C22×C4H(5,𝔽2)12yes—
513C4⋊C4—216no—
514C4⋊C4—216no—
515C4⋊C4—216no—
516C4⋊C4—28no—
517C4⋊C4—28no—
518C4⋊C4—416no—
519C4⋊C4—416no—
520C4⋊C4—416no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—88no—
526C22⋊C4—416no—
527C22⋊C4—416no—
528C22⋊C4—416no—
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—
539C22⋊C4—88no—
540C42—28no—
541C42—416no—
542C42—48no—
543C42—48no—
544C42—44no—
545C42#121116yes—
546C23—24no—
547C23—24no—
548C23—24no—
549C23—28no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—28no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—44no—
583C23—42no—
584C23—44no—
585C23—42no—
586C23—44no—
587C23—42no—
588C23—42no—
589C23—44no—
590C23—42no—
591C23—44no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—44no—
596C23—42no—
597C23—42no—
598C23—44no—
599C23—44no—
600C23—42no—
601C23—42no—
602C23—44no—
603C23—42no—
604C23C23×D411yescharacteristic
605C23C23×D411yescharacteristic
606C23C23×D411yescharacteristic
607C23C22×#4111yescharacteristic
608C23C22×#4111yescharacteristic
609C23C2×H(5,𝔽2)11yescharacteristic
610C23C2×H(5,𝔽2)11yescharacteristic
611C23C2×2−1+412yes—
612C23#58512yes—
613C23#58512yes—
614C23#58512yes—
615Q8—432no—
616D4—416no—
617D4—416no—
618D4—832no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—28no—
628C2×C4—24no—
629C2×C4—28no—
630C2×C4—28no—
631C2×C4—24no—
632C2×C4—24no—
633C2×C4—28no—
634C2×C4—24no—
635C2×C4—28no—
636C2×C4—24no—
637C2×C4—28no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
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—42no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—44no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—816no—
692C2×C4—816no—
693C2×C4C2×#12114yes—
694C2×C4C2×#12114yes—
695C2×C4C2×#12114yes—
696C2×C4#53214yes—
697C2×C4#54014yes—
698C2×C4D4214yes—
699C2×C4D4214yes—
700C2×C4#54614yes—
701C2×C4#55014yes—
702C22—44no—
703C22—48no—
704C22—44no—
705C22—44no—
706C22—44no—
707C22—44no—
708C22—48no—
709C22—44no—
710C22—44no—
711C22—44no—
712C22—44no—
713C22—44no—
714C22—82no—
715C22—84no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22—82no—
724C22—84no—
725C22—82no—
726C22—82no—
727C22—82no—
728C22C22×#12111yescharacteristic
729C22C2×#53211yescharacteristic
730C22C2×#54011yescharacteristic
731C22C2×D4211yescharacteristic
732C22C2×D4211yescharacteristic
733C22C2×#54611yescharacteristic
734C22C2×#55011yescharacteristic
735C22#348512yes—
736C22#348611yescharacteristic
737C22#348711yescharacteristic
738C22#348711yescharacteristic
739C22#348711yescharacteristic
740C22#348912yes—
741C22#348912yes—
742C22#349512yes—
743C22#350012yes—
744C22#351012yes—
745C22#351612yes—
746C22#352112yes—
747C22#353912yes—
748C22#354112yes—
749C22#356112yes—
750C22#356212yes—
751C4—24no—
752C4—24no—
753C4—44no—
754C4—44no—
755C4—48no—
756C4—48no—
757C4—44no—
758C4—82no—
759C4—82no—
760C4—82no—
761C4—82no—
762C4—82no—
763C4#242818yes—
764C2—84no—
765C2—84no—
766C2C2×#242811yescharacteristic
767C2#3563612yes—
768C2#3566111yescharacteristic
769C2#3567711yescharacteristic
770C2#3740912yes—
771C2#4513412yes—
772C2#4519612yes—
773C2#4526711yescharacteristic
774C2#4526811yescharacteristic
775C2#4526911yescharacteristic
776C2#4538711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2535C2
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×#2547C2semidirect
2. C2×#2542C2semidirect
4. C2×#2426C2semidirect
5. C2×#2327C2semidirect
6. #15648C2semidirect
7. #15640C2semidirect
8. #15380C2semidirect
9. #15335C2semidirect
10. #15215C2semidirect
11. #15179C2semidirect
12. #14150C2semidirect
13. #13969C2semidirect
14. #13847C2semidirect
15. #13785C2semidirect
88. C2×C4⋊Q8D4semidirect
154. #390D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×#25422
C2×#25352
C2×#24262
C2×#23272
#156482
#156402
#153802
#153352
#152152
#151792
#141502
#139692
#138472
#137852