Finite Group Databasebeta

← #770009

Subgroups — #770009

Order 512.

3238 conjugacy classes of subgroups.

The subgroup lattice has 797 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic, complemented
2C2×#2547C211yesmaximal, characteristic, complemented
3C2×#2397C211yesmaximal, characteristic
4C2×#2331C211yesmaximal, characteristic, complemented
5#15739C214yesmaximal, complemented
6#15640C214yesmaximal, complemented
7#15499C214yesmaximal, complemented
8#15317C214yesmaximal, complemented
9#15181C212yesmaximal, complemented
10#15129C212yesmaximal, complemented
11#13906C212yesmaximal, complemented
12#13905C212yesmaximal, complemented
13#13905C211yesmaximal, characteristic, complemented
14#13905C211yesmaximal, characteristic, complemented
15#13779C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C4×D4C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22#2563—28no—
23#2547—28no—
24#2397—28no—
25C2×#399C2212yes—
26C2×#399C2212yes—
27C2×#397C2214yes—
28C2×#394C2214yes—
29C2×#393C2214yes—
30C2×#392C2212yes—
31C2×#392C2212yes—
32#2331—28no—
33C2×#391C2214yes—
34C2×#391C2212yes—
35C2×#390C2212yes—
36C2×#390C2214yes—
37C2×#389C2212yes—
38C2×#387C2214yes—
39C2×#384C2211yescharacteristic
40C2×#382C2211yescharacteristic
41C2×#382C2211yescharacteristic
42C2×#382C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×#376C2211yescharacteristic
55#1445C2212yes—
56#1445C2214yes—
57#1445C2212yes—
58#1442C2214yes—
59#1441C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1438C2212yes—
67#1438C2214yes—
68#1438C2214yes—
69#1438C2212yes—
70#1437C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2211yescharacteristic
78C4⋊#95C2214yes—
79C23×Q8C2312yes—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C23×D4C2312yes—
83C23×D4C2312yes—
84C24×C4C2312yes—
85C24×C4C2312yes—
86C2×C4⋊Q8—24no—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8D418yescomplemented
89C2×#124—28no—
90C2×#124—48no—
91C2×C4×D4—24no—
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—24no—
95C2×C4×D4—28no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—28no—
100C2×C4×D4—28no—
101C2×C4×D4—44no—
102C2×C4×D4—44no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#399—28no—
137#399—28no—
138#397—216no—
139#394—48no—
140#393—216no—
141#392—28no—
142#392—28no—
143#391—28no—
144#391—216no—
145#390—28no—
146#390—48no—
147#389—28no—
148#387—216no—
149#384—24no—
150#382—28no—
151#382—24no—
152#382—24no—
153#380—28no—
154#380—28no—
155#380—28no—
156#380—28no—
157#380—44no—
158#380—44no—
159#380—44no—
160#380—44no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—24no—
164#376D418yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C25C2412yes—
185C25C2412yes—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—44no—
189C22×Q8—44no—
190C22×Q8C2×D418yes—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—24no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4—42no—
211C22×D4—42no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—42no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
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—24no—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2412yes—
241C23×C4C2411yescharacteristic
242C4⋊Q8—232no—
243#124—432no—
244C4×D4—432no—
245C4×D4—416no—
246C4×D4—416no—
247C4×D4—432no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
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—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4C2×D418yes—
286C2×C4⋊C4C2×D418yes—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
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—24no—
300C2×C22⋊C4—24no—
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—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—42no—
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×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C42—22no—
330C2×C42—22no—
331C2×C42—24no—
332C2×C42—24no—
333C2×C42—28no—
334C2×C42—44no—
335C2×C42—44no—
336C2×C42—44no—
337C2×C42—44no—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42#4114yes—
343C2×C42#4114yes—
344C2×C42#4114yes—
345#95—28no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—24no—
354#95—28no—
355#95—28no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362#95C2×D418yes—
363#95C2×D418yes—
364C24—22no—
365C24—22no—
366C24—22no—
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—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—22no—
387C24—22no—
388C24—22no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24H(5,𝔽2)14yes—
393C242−1+414yes—
394C24C2511yescharacteristic
395C2×Q8—216no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—88no—
399C2×Q8—88no—
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—216no—
403C2×D4—48no—
404C2×D4—48no—
405C2×D4—48no—
406C2×D4—48no—
407C2×D4—48no—
408C2×D4—48no—
409C2×D4—48no—
410C2×D4—48no—
411C2×D4—48no—
412C2×D4—48no—
413C2×D4—48no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—48no—
417C2×D4—48no—
418C2×D4—48no—
419C2×D4—48no—
420C2×D4—48no—
421C2×D4—44no—
422C2×D4—44no—
423C2×D4—48no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
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—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
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—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
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—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4—48no—
497C22×C4—48no—
498C22×C4—48no—
499C22×C4C22×D418yes—
500C22×C4C22×D414yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D414yes—
506C22×C4C22×D412yes—
507C22×C4C22×D412yes—
508C22×C4C2×#4112yes—
509C22×C4C2×#4112yes—
510C22×C4C2×#4114yes—
511C22×C4C2×#4114yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4C2×#4112yes—
515C22×C4H(5,𝔽2)12yes—
516C22×C4H(5,𝔽2)12yes—
517C4⋊C4—216no—
518C4⋊C4—216no—
519C4⋊C4—48no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—48no—
523C4⋊C4—48no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—88no—
529C4⋊C4—88no—
530C22⋊C4—416no—
531C22⋊C4—416no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—48no—
536C22⋊C4—48no—
537C22⋊C4—48no—
538C22⋊C4—88no—
539C22⋊C4—88no—
540C22⋊C4—88no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C22⋊C4—88no—
544C42—28no—
545C42—48no—
546C42—48no—
547C42—88no—
548C42#121116yes—
549C42#124116yescomplemented
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—24no—
563C23—24no—
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—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—
600C23—42no—
601C23—42no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23—42no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23—42no—
619C23—42no—
620C23C23×D411yescharacteristic
621C23C23×D411yescharacteristic
622C23C22×#4111yescharacteristic
623C23C22×#4111yescharacteristic
624C23C22×#4111yescharacteristic
625C23C2×H(5,𝔽2)11yescharacteristic
626C23C2×H(5,𝔽2)11yescharacteristic
627C23C2×H(5,𝔽2)12yes—
628C23C2×2−1+412yes—
629C23#58512yes—
630C23#58512yes—
631Q8—816no—
632D4—416no—
633D4—416no—
634D4—432no—
635D4—432no—
636C2×C4—22no—
637C2×C4—22no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—24no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—24no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—28no—
651C2×C4—24no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
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—44no—
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—42no—
681C2×C4—42no—
682C2×C4—42no—
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—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—816no—
706C2×C4—816no—
707C2×C4C2×#12114yes—
708C2×C4C2×#12114yes—
709C2×C4C2×#12114yes—
710C2×C4C2×#12418yes—
711C2×C4C2×#12414yes—
712C2×C4#53514yes—
713C2×C4#55014yes—
714C2×C4#55014yes—
715C2×C4#55914yes—
716C2×C4#55914yes—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—44no—
723C22—44no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22—44no—
728C22—44no—
729C22—44no—
730C22—44no—
731C22—82no—
732C22—82no—
733C22—82no—
734C22—82no—
735C22—82no—
736C22—82no—
737C22—82no—
738C22—82no—
739C22—82no—
740C22—82no—
741C22—82no—
742C22—82no—
743C22—82no—
744C22—82no—
745C22—82no—
746C22—82no—
747C22C22×#12111yescharacteristic
748C22C22×#12411yescharacteristic
749C22C2×#53511yescharacteristic
750C22C2×#55011yescharacteristic
751C22C2×#55011yescharacteristic
752C22C2×#55911yescharacteristic
753C22C2×#55911yescharacteristic
754C22#348411yescharacteristic
755C22#348511yescharacteristic
756C22#348611yescharacteristic
757C22#348711yescharacteristic
758C22#348812yes—
759C22#348912yes—
760C22#349212yes—
761C22#349312yes—
762C22#349412yes—
763C22#349412yes—
764C22#349512yes—
765C22#349512yes—
766C22#353612yes—
767C22#354012yes—
768C22#354112yes—
769C22#354212yes—
770C4—24no—
771C4—24no—
772C4—28no—
773C4—44no—
774C4—48no—
775C4—82no—
776C4—82no—
777C4—82no—
778C4—82no—
779C4—82no—
780C4—82no—
781C4—84no—
782C4#254818yes—
783C2—84no—
784C2—84no—
785C2C2×#254811yescharacteristic
786C2#3551411yescharacteristic
787C2#3554212yes—
788C2#3566111yescharacteristic
789C2#3727112yes—
790C2#4493311yescharacteristic
791C2#4498112yes—
792C2#4513811yescharacteristic
793C2#4535611yescharacteristic
794C2#4538811yescharacteristic
795C2#4543312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2397C2
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×#2547C2semidirect
4. C2×#2331C2semidirect
5. #15739C2semidirect
6. #15640C2semidirect
7. #15499C2semidirect
8. #15317C2semidirect
9. #15181C2semidirect
10. #15129C2semidirect
11. #13906C2semidirect
12. #13905C2semidirect
13. #13905C2semidirect
14. #13905C2semidirect
15. #13779C2semidirect
88. C2×C4⋊Q8D4semidirect
549. C42#124semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25472
C2×#23972
C2×#23312
#157392
#156402
#154992
#153172
#151812
#151292
#139062
#139052
#137792