Finite Group Databasebeta

← #739849

Subgroups — #739849

Order 512.

2698 conjugacy classes of subgroups.

The subgroup lattice has 750 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2398C211yesmaximal, characteristic
4C2×#2355C211yesmaximal, characteristic, complemented
5#16800C212yesmaximal
6#16705C212yesmaximal, complemented
7#15737C214yesmaximal, complemented
8#15322C214yesmaximal, complemented
9#15220C212yesmaximal, complemented
10#15163C212yesmaximal
11#15131C212yesmaximal, complemented
12#15085C212yesmaximal, complemented
13#14282C211yesmaximal, characteristic, complemented
14#13974C212yesmaximal, complemented
15#13973C212yesmaximal, complemented
16#13905C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2563—28no—
23#2546—28no—
24#2398—28no—
25C2×#400C2211yescharacteristic
26C2×#400C2211yescharacteristic
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#397C2212yes—
30C2×#396C2214yes—
31C2×#394C2211yescharacteristic
32C2×#394C2211yescharacteristic
33C2×#393C2214yes—
34C2×#393C2212yes—
35#2355—28no—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#390C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#387C2212yes—
44C2×#387C2212yes—
45C2×#384C2211yescharacteristic
46C2×#382C2212yes—
47C2×#382C2211yescharacteristic
48C2×#382C2211yescharacteristic
49C2×#380C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2212yes—
59C2×C4×C22⋊C4C2212yes—
60C2×#376C2211yescharacteristic
61#1446C2214yes—
62#1445C2212yes—
63#1445C2212yes—
64#1445C2212yes—
65#1443C2214yes—
66#1443C2214yes—
67#1442C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2214yes—
71#1440C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1438C2212yes—
77#1437C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85#1434C2214yes—
86#1434C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C23×D4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8—44no—
96C2×C4⋊Q8—44no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8D418yes—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2311yescharacteristic
136#400—24no—
137#400—24no—
138#398—28no—
139#398—28no—
140#397—28no—
141#396—48no—
142#394—24no—
143#394—24no—
144#393—28no—
145#393—216no—
146#391—28no—
147#391—28no—
148#390—28no—
149#389—28no—
150#389—28no—
151#389—28no—
152#388—48no—
153#387—28no—
154#387—28no—
155#384—24no—
156#382—24no—
157#382—28no—
158#382—24no—
159#380—28no—
160#380—28no—
161#380—28no—
162#380—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—24no—
167C4×C4⋊C4D418yes—
168C4×C22⋊C4—28no—
169C4×C22⋊C4—28no—
170#376—24no—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C25C2412yes—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—41no—
208C22×Q8—41no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8—42no—
213C22×Q8C2×D414yes—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—22no—
217C22×D4—22no—
218C22×D4—42no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—24no—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2412yes—
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C4⋊Q8—432no—
243C4⋊Q8—416no—
244C4×Q8—232no—
245C4×D4—416no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—28no—
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×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4#4118yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
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×C42—22no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—44no—
330C2×C42—42no—
331C2×C42—44no—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D418yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337C2×C42#4114yes—
338#95—24no—
339#95—22no—
340#95—22no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—24no—
353#95—24no—
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—44no—
364#95—44no—
365#95C2×D418yes—
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—
380C242−1+414yes—
381C24C2511yescharacteristic
382C2×Q8—28no—
383C2×Q8—28no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—44no—
389C2×Q8—48no—
390C2×Q8—48no—
391C2×Q8—48no—
392C2×Q8—84no—
393C2×Q8—84no—
394C2×Q8—84no—
395C2×Q8—84no—
396C2×D4—28no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4—44no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
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—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—48no—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D414yes—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C2×#4118yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4112yes—
500C22×C4C2×#4114yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C42−1+414yes—
504C4⋊C4—28no—
505C4⋊C4—28no—
506C4⋊C4—28no—
507C4⋊C4—28no—
508C4⋊C4—28no—
509C4⋊C4—216no—
510C4⋊C4—48no—
511C4⋊C4—416no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—48no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—416no—
518C4⋊C4—84no—
519C4⋊C4—88no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—416no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C42—28no—
530C42—28no—
531C42—48no—
532C42—48no—
533C42—48no—
534C42—48no—
535C42—84no—
536C42#121116yes—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—24no—
542C23—24no—
543C23—24no—
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—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23C23×D411yescharacteristic
573C23C23×D411yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4112yes—
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×H(5,𝔽2)11yescharacteristic
578C23C2×2−1+412yes—
579C23C2×2−1+412yes—
580C23#58511yescharacteristic
581C23#58511yescharacteristic
582C23#58512yes—
583Q8—416no—
584Q8—816no—
585D4—416no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—28no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
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—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—816no—
675C2×C4C2×#12114yes—
676C2×C4C2×#12114yes—
677C2×C4C2×#12114yes—
678C2×C4#53214yes—
679C2×C4#53514yes—
680C2×C4#53614yes—
681C2×C4#54818yes—
682C2×C4#55618yes—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22—82no—
696C22—82no—
697C22—82no—
698C22C22×#12111yescharacteristic
699C22C2×#53211yescharacteristic
700C22C2×#53511yescharacteristic
701C22C2×#53611yescharacteristic
702C22#344911yescharacteristic
703C22C2×#54812yes—
704C22C2×#55612yes—
705C22#348511yescharacteristic
706C22#348611yescharacteristic
707C22#348811yescharacteristic
708C22#348811yescharacteristic
709C22#348912yes—
710C22#349411yescharacteristic
711C22#349411yescharacteristic
712C22#349512yes—
713C22#351412yes—
714C22#351612yes—
715C22#352012yes—
716C22#352412yes—
717C22#354712yes—
718C22#355212yes—
719C22#356312yes—
720C22#356512yes—
721C4—24no—
722C4—24no—
723C4—24no—
724C4—42no—
725C4—42no—
726C4—42no—
727C4—44no—
728C4—44no—
729C4—48no—
730C4—82no—
731C4—82no—
732C4—84no—
733C4—82no—
734C4—82no—
735C4—82no—
736C4#254518yes—
737C2—84no—
738C2C2×#254511yescharacteristic
739C2#3525211yescharacteristic
740C2#3563311yescharacteristic
741C2#3590911yescharacteristic
742C2#3605111yescharacteristic
743C2#3911912yes—
744C2#3912412yes—
745C2#4268111yescharacteristic
746C2#4281411yescharacteristic
747C2#4508512yes—
748C2#4521212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2546C2semidirect
4. C2×#2355C2semidirect
6. #16705C2semidirect
7. #15737C2semidirect
8. #15322C2semidirect
9. #15220C2semidirect
11. #15131C2semidirect
12. #15085C2semidirect
13. #14282C2semidirect
14. #13974C2semidirect
15. #13973C2semidirect
17. #13783C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25462
C2×#23982
C2×#23552
#168002
#167052
#157372
#153222
#152202
#151632
#151312
#150852
#142822
#139742
#139732
#139052
#137832