Finite Group Databasebeta

← #734988

Subgroups — #734988

Order 512.

3078 conjugacy classes of subgroups.

The subgroup lattice has 760 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2C2×#2580C211yesmaximal, characteristic, complemented
3C2×#2398C211yesmaximal, characteristic, complemented
4C2×#2355C211yesmaximal, characteristic, complemented
5#16652C214yesmaximal, complemented
6#15525C214yesmaximal, complemented
7#15338C214yesmaximal, complemented
8#15220C212yesmaximal, complemented
9#15219C212yesmaximal, complemented
10#14778C214yesmaximal, complemented
11#14276C211yesmaximal, characteristic, complemented
12#13998C212yesmaximal, complemented
13#13974C212yesmaximal, complemented
14#13905C211yesmaximal, characteristic, complemented
15#13785C211yesmaximal, characteristic, complemented
16C22×#121C2212yes—
17C22×#121C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2580—28no—
22#2537—28no—
23#2398—28no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#397C2212yes—
27C2×#394C2214yes—
28C2×#394C2212yes—
29C2×#393C2212yes—
30C2×#393C2212yes—
31#2355—28no—
32C2×#391C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#390C2211yescharacteristic
36C2×#390C2214yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#388C2214yes—
40C2×#387C2214yes—
41C2×#385C2214yes—
42C2×#384C2211yescharacteristic
43C2×#382C2212yes—
44C2×#382C2211yescharacteristic
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×#376C2211yescharacteristic
55#1445C2212yes—
56#1443C2214yes—
57#1443C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2212yes—
70#1438C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78C22⋊#95C2214yes—
79C23×Q8C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2314yes—
83C24×C4C2312yes—
84C24×C4C2312yes—
85C2×#121—28no—
86C2×#121—28no—
87C2×#121—28no—
88C2×#121—48no—
89C2×#121—44no—
90C2×C4×Q8—24no—
91C2×C4×Q8—24no—
92C2×C4×Q8D418yescomplemented
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131#400—28no—
132#399—28no—
133#397—28no—
134#394—28no—
135#394—216no—
136#393—28no—
137#393—28no—
138#391—28no—
139#390—24no—
140#390—24no—
141#390—28no—
142#390—48no—
143#389—28no—
144#389—44no—
145#388—48no—
146#387—216no—
147#385—216no—
148#384D418yes—
149#382—28no—
150#382—24no—
151#380—28no—
152#380—28no—
153#380—28no—
154#380—44no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—24no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—28no—
160#376—24no—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C25C2412yes—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—42no—
187C22×Q8—42no—
188C22×Q8C2×D414yes—
189C22×D4—24no—
190C22×D4—28no—
191C22×D4—28no—
192C22×D4—22no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—28no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—42no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4—44no—
207C22×D4#4114yes—
208C23×C4—24no—
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—44no—
219C23×C4—44no—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233#121—432no—
234#121—432no—
235C4×Q8—232no—
236C4×D4—416no—
237C4×D4—416no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4C2×D418yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2×D418yes—
269C2×C4⋊C4C2×D418yes—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4#4114yes—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—28no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42C2×D418yes—
328C2×C42C2×D414yes—
329C2×C42C2×D414yes—
330C2×C42#4114yes—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—28no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—24no—
342#95—28no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—24no—
369C24—22no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—22no—
376C24C2×#4114yes—
377C24H(5,𝔽2)14yes—
378C24C2511yescharacteristic
379C2×Q8—28no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—48no—
384C2×D4—28no—
385C2×D4—28no—
386C2×D4—28no—
387C2×D4—28no—
388C2×D4—48no—
389C2×D4—48no—
390C2×D4—44no—
391C2×D4—416no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—44no—
395C2×D4—44no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—44no—
399C2×D4—88no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
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—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—88no—
478C22×C4—88no—
479C22×C4C22×D418yes—
480C22×C4C22×D414yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4C2×#4112yes—
493C22×C4C2×#4112yes—
494C22×C4H(5,𝔽2)12yes—
495C22×C42−1+412yes—
496C4⋊C4—28no—
497C4⋊C4—216no—
498C4⋊C4—216no—
499C4⋊C4—28no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—48no—
506C4⋊C4—88no—
507C4⋊C4—88no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—416no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—88no—
519C22⋊C4—88no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C42—216no—
523C42—48no—
524C42—44no—
525C42—44no—
526C42—88no—
527C42#121116yes—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—24no—
539C23—24no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
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—44no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23—44no—
567C23—42no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—44no—
575C23—42no—
576C23—44no—
577C23—42no—
578C23—44no—
579C23—42no—
580C23—42no—
581C23—44no—
582C23—42no—
583C23—42no—
584C23—42no—
585C23—44no—
586C23C23×D411yescharacteristic
587C23C23×D411yescharacteristic
588C23C22×#4111yescharacteristic
589C23C22×#4111yescharacteristic
590C23C22×#4111yescharacteristic
591C23C2×H(5,𝔽2)11yescharacteristic
592C23C2×H(5,𝔽2)12yes—
593C23C2×2−1+411yescharacteristic
594C23#58512yes—
595C23#58512yes—
596C23#58512yes—
597Q8—416no—
598D4—416no—
599D4—416no—
600D4—832no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—28no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—28no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—28no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—816no—
674C2×C4—816no—
675C2×C4C2×#12114yes—
676C2×C4C2×#12114yes—
677C2×C4C2×#12114yes—
678C2×C4#52914yes—
679C2×C4#54214yes—
680C2×C4#54614yes—
681C2×C4#54814yes—
682C2×C4#54814yes—
683C2×C4#55014yes—
684C22—48no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—48no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—44no—
696C22—82no—
697C22—82no—
698C22—82no—
699C22—82no—
700C22—82no—
701C22—84no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22—82no—
708C22—84no—
709C22—82no—
710C22C22×#12111yescharacteristic
711C22C2×#52911yescharacteristic
712C22#344912yes—
713C22C2×#54211yescharacteristic
714C22C2×#54611yescharacteristic
715C22C2×#54811yescharacteristic
716C22C2×#54811yescharacteristic
717C22C2×#55011yescharacteristic
718C22#348411yescharacteristic
719C22#348511yescharacteristic
720C22#348711yescharacteristic
721C22#348711yescharacteristic
722C22#348812yes—
723C22#349312yes—
724C22#349412yes—
725C22#351312yes—
726C22#351312yes—
727C22#352112yes—
728C22#352412yes—
729C22#355812yes—
730C22#355912yes—
731C22#356212yes—
732C22#356512yes—
733C4—24no—
734C4—24no—
735C4—44no—
736C4—48no—
737C4—44no—
738C4—44no—
739C4—82no—
740C4—82no—
741C4—82no—
742C4—82no—
743C4—84no—
744C4—82no—
745C4#242618yes—
746C2—84no—
747C2—84no—
748C2C2×#242611yescharacteristic
749C2#3515011yescharacteristic
750C2#3556311yescharacteristic
751C2#3584912yes—
752C2#3911812yes—
753C2#3913012yes—
754C2#4269312yes—
755C2#4514011yescharacteristic
756C2#4526911yescharacteristic
757C2#4527011yescharacteristic
758C2#4527111yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2537C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#2580C2semidirect
3. C2×#2398C2semidirect
4. C2×#2355C2semidirect
5. #16652C2semidirect
6. #15525C2semidirect
7. #15338C2semidirect
8. #15220C2semidirect
9. #15219C2semidirect
10. #14778C2semidirect
11. #14276C2semidirect
12. #13998C2semidirect
13. #13974C2semidirect
14. #13905C2semidirect
15. #13785C2semidirect
92. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
C2×#25802
C2×#23982
C2×#23552
#166522
#155252
#153382
#152202
#152192
#147782
#142762
#139982
#139742
#139052
#137852