Finite Group Databasebeta

← #764292

Subgroups — #764292

Order 512.

3238 conjugacy classes of subgroups.

The subgroup lattice has 765 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic, complemented
2C2×#2542C211yesmaximal, characteristic, complemented
3C2×#2535C211yesmaximal, characteristic
4C2×#2331C211yesmaximal, characteristic, complemented
5#15886C214yesmaximal, complemented
6#15776C214yesmaximal, complemented
7#15280C214yesmaximal, complemented
8#15275C214yesmaximal, complemented
9#15131C212yesmaximal, complemented
10#15126C212yesmaximal, complemented
11#14150C211yesmaximal, characteristic, complemented
12#13998C212yesmaximal, complemented
13#13876C212yesmaximal, complemented
14#13847C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C22⋊Q8C2212yes—
19C22×#121C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2563—28no—
23#2542—28no—
24#2535—28no—
25C2×#399C2214yes—
26C2×#397C2212yes—
27C2×#396C2214yes—
28C2×#394C2214yes—
29C2×#393C2214yes—
30C2×#392C2212yes—
31#2331—28no—
32C2×#391C2214yes—
33C2×#390C2211yescharacteristic
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2214yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#388C2214yes—
41C2×#387C2214yes—
42C2×#386C2214yes—
43C2×#384C2212yes—
44C2×#382C2212yes—
45C2×#382C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×#376C2211yescharacteristic
55#1445C2212yes—
56#1444C2212yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2214yes—
69#1438C2212yes—
70#1438C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78C22⋊#95C2214yes—
79C23×Q8C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2314yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×C4⋊Q8—24no—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8D418yescomplemented
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—48no—
91C2×C22⋊Q8—48no—
92C2×#121—28no—
93C2×#121—28no—
94C2×#121—44no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134#399—216no—
135#397—28no—
136#396—48no—
137#394—48no—
138#393—216no—
139#392—28no—
140#391—216no—
141#390—24no—
142#390—28no—
143#390—28no—
144#390—24no—
145#390—48no—
146#389—28no—
147#389—44no—
148#388—48no—
149#387—216no—
150#386—216no—
151#384—28no—
152#382—28no—
153#382D418yes—
154#380—28no—
155#380—28no—
156#380—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—24no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—44no—
162#376—24no—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C25C2412yes—
184C25C2412yes—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8C2×D418yes—
190C22×D4—24no—
191C22×D4—28no—
192C22×D4—28no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—28no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—42no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4#4114yes—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C4⋊Q8—232no—
235C22⋊Q8—432no—
236C22⋊Q8—432no—
237#121—432no—
238C4×D4—416no—
239C4×D4—416no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
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—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D418yes—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
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—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—28no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42#4114yes—
335C2×C42#4114yes—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—44no—
349#95—42no—
350#95—42no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95C2×D418yes—
355#95C2×D418yes—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—24no—
379C24—22no—
380C24C2×#4114yes—
381C242−1+414yes—
382C24C2511yescharacteristic
383C2×Q8—216no—
384C2×Q8—48no—
385C2×Q8—48no—
386C2×Q8—88no—
387C2×Q8—88no—
388C2×D4—28no—
389C2×D4—28no—
390C2×D4—28no—
391C2×D4—28no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—44no—
395C2×D4—416no—
396C2×D4—416no—
397C2×D4—416no—
398C2×D4—44no—
399C2×D4—44no—
400C2×D4—44no—
401C2×D4—44no—
402C2×D4—44no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—88no—
406C2×D4—88no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
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—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4C22×D418yes—
483C22×C4C22×D418yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4112yes—
495C22×C4C2×#4112yes—
496C22×C4H(5,𝔽2)14yes—
497C22×C4H(5,𝔽2)14yes—
498C22×C4H(5,𝔽2)12yes—
499C22×C42−1+412yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—216no—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—416no—
506C4⋊C4—416no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—44no—
511C4⋊C4—88no—
512C4⋊C4—88no—
513C22⋊C4—216no—
514C22⋊C4—48no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—48no—
518C22⋊C4—416no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—88no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—88no—
527C42—28no—
528C42—48no—
529C42—44no—
530C42—48no—
531C42—88no—
532C42#121116yes—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—24no—
542C23—28no—
543C23—24no—
544C23—24no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—42no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—44no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—42no—
581C23—44no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—44no—
586C23—44no—
587C23—42no—
588C23—42no—
589C23—44no—
590C23—42no—
591C23C23×D411yescharacteristic
592C23C23×D411yescharacteristic
593C23C22×#4111yescharacteristic
594C23C22×#4111yescharacteristic
595C23C22×#4111yescharacteristic
596C23C2×H(5,𝔽2)11yescharacteristic
597C23C2×H(5,𝔽2)12yes—
598C23C2×H(5,𝔽2)12yes—
599C23C2×2−1+411yescharacteristic
600C23C2×2−1+412yes—
601C23#58512yes—
602Q8—816no—
603D4—416no—
604D4—416no—
605D4—832no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—28no—
615C2×C4—28no—
616C2×C4—24no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—28no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—28no—
627C2×C4—24no—
628C2×C4—24no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
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—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—44no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—816no—
679C2×C4—816no—
680C2×C4C2×#12114yes—
681C2×C4C2×#12114yes—
682C2×C4C2×#12114yes—
683C2×C4#52914yes—
684C2×C4#54214yes—
685C2×C4#54614yes—
686C2×C4#54814yes—
687C2×C4#54814yes—
688C2×C4#55014yes—
689C22—44no—
690C22—48no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—44no—
697C22—44no—
698C22—44no—
699C22—48no—
700C22—44no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—82no—
709C22—84no—
710C22—82no—
711C22—82no—
712C22—82no—
713C22—84no—
714C22—82no—
715C22C22×#12111yescharacteristic
716C22C2×#52911yescharacteristic
717C22C2×#54211yescharacteristic
718C22C2×#54611yescharacteristic
719C22C2×#54811yescharacteristic
720C22C2×#54811yescharacteristic
721C22C2×#55011yescharacteristic
722C22#348411yescharacteristic
723C22#348411yescharacteristic
724C22#348611yescharacteristic
725C22#348711yescharacteristic
726C22#348912yes—
727C22#349212yes—
728C22#349412yes—
729C22#349512yes—
730C22#349812yes—
731C22#350812yes—
732C22#351412yes—
733C22#352012yes—
734C22#353612yes—
735C22#354212yes—
736C22#354712yes—
737C22#355212yes—
738C4—24no—
739C4—24no—
740C4—48no—
741C4—44no—
742C4—44no—
743C4—44no—
744C4—48no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4—82no—
750C4#242618yes—
751C2—84no—
752C2—84no—
753C2C2×#242611yescharacteristic
754C2#3552811yescharacteristic
755C2#3555312yes—
756C2#3566111yescharacteristic
757C2#3733312yes—
758C2#4493311yescharacteristic
759C2#4493811yescharacteristic
760C2#4497312yes—
761C2#4507912yes—
762C2#4533711yescharacteristic
763C2#4535611yescharacteristic
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×#2563C2semidirect
2. C2×#2542C2semidirect
4. C2×#2331C2semidirect
5. #15886C2semidirect
6. #15776C2semidirect
7. #15280C2semidirect
8. #15275C2semidirect
9. #15131C2semidirect
10. #15126C2semidirect
11. #14150C2semidirect
12. #13998C2semidirect
13. #13876C2semidirect
14. #13847C2semidirect
15. #13785C2semidirect
87. C2×C4⋊Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25422
C2×#25352
C2×#23312
#158862
#157762
#152802
#152752
#151312
#151262
#141502
#139982
#138762
#138472
#137852