Finite Group Databasebeta

← #823782

Subgroups — #823782

Order 512.

3004 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2489C211yesmaximal, characteristic
2C2×#2453C212yesmaximal, complemented
3C2×#2429C212yesmaximal
4C2×#2365C211yesmaximal, characteristic, complemented
5#16964C212yesmaximal, complemented
6#15808C214yesmaximal
7#15804C214yesmaximal, complemented
8#15193C211yesmaximal, characteristic, complemented
9#15190C211yesmaximal, characteristic, complemented
10#15178C211yesmaximal, characteristic
11#15079C212yesmaximal
12#14870C212yesmaximal
13#14155C212yesmaximal, complemented
14#13997C212yesmaximal
15#13954C212yesmaximal, complemented
16C4×#399C211yesmaximal, characteristic
17#13821C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×#125C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C22×C4×Q8C2212yes—
24#2489—28no—
25#2453—216no—
26#2429—216no—
27C2×#400C2214yes—
28C2×#400C2212yes—
29C2×#399C2211yescharacteristic
30C2×#398C2214yes—
31C2×#398C2214yes—
32C2×#397C2212yes—
33C2×#396C2214yes—
34#2365—28no—
35C2×#391C2212yes—
36C2×#391C2211yescharacteristic
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#391C2211yescharacteristic
40C2×#391C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#389C2214yes—
45C2×#389C2212yes—
46C2×#388C2212yes—
47C2×#388C2214yes—
48C2×#387C2214yes—
49C2×#387C2214yes—
50C2×#382C2212yes—
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2211yescharacteristic
59C2×C4×C22⋊C4C2212yes—
60#1445C2212yes—
61#1445C2211yescharacteristic
62#1445C2212yes—
63#1445C2214yes—
64#1445C2214yes—
65#1445C2214yes—
66#1445C2214yes—
67#1444C2212yes—
68#1444C2211yescharacteristic
69#1444C2212yes—
70#1444C2212yes—
71#1444C2212yes—
72#1444C2214yes—
73#1444C2212yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2211yescharacteristic
77#1439C2212yes—
78#1439C2214yes—
79#1437C2214yes—
80#1436C2214yes—
81C4⋊#95C2214yes—
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2214yes—
84C4⋊#95C2214yes—
85#1434C2212yes—
86C4×#95C2211yescharacteristic
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—28no—
94C2×C4⋊Q8—44no—
95C2×#125—28no—
96C2×#125—24no—
97C2×C22⋊Q8—28no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—28no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—44no—
102C2×C4×Q8—28no—
103C2×C4×Q8—28no—
104C2×C4×Q8—28no—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2314yes—
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2311yescharacteristic
123C22×C4⋊C4C2311yescharacteristic
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2314yes—
138#400—216no—
139#400—28no—
140#399—24no—
141#398—216no—
142#398—216no—
143#397—28no—
144#396—216no—
145#391—28no—
146#391—28no—
147#391—28no—
148#391—24no—
149#391—28no—
150#391—24no—
151#389—216no—
152#389—216no—
153#389—28no—
154#389—28no—
155#389D4116yes—
156#388—28no—
157#388—216no—
158#387—216no—
159#387—216no—
160#382—28no—
161#380—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4—28no—
166C4×C4⋊C4—44no—
167C4×C4⋊C4D418yes—
168C4×C22⋊C4—28no—
169C4×C22⋊C4—24no—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2312yes—
192C2×#95C2311yescharacteristic
193C2×#95C2312yes—
194C2×#95C2314yes—
195C2×#95C2312yes—
196C2×#95C2312yes—
197C2×#95C2314yes—
198C2×#95C2311yescharacteristic
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—22no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—24no—
210C22×Q8—22no—
211C22×Q8#4118yes—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2414yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C4⋊Q8—232no—
234C4⋊Q8—432no—
235#125—416no—
236C22⋊Q8—432no—
237C22⋊Q8—416no—
238C4×Q8—432no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—42no—
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×D4116yes—
286C2×C4⋊C4C2×D414yes—
287C2×C4⋊C4C2×D418yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—28no—
319C2×C42—22no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—44no—
323C2×C42—44no—
324C2×C42C2×D414yes—
325C2×C42C2×D418yes—
326C2×C42C2×D418yes—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—28no—
340#95—28no—
341#95—28no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—22no—
350#95—24no—
351#95—42no—
352#95—44no—
353#95C2×D418yes—
354#95C2×D4116yes—
355#95#4114yes—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—21no—
366C242−1+412yes—
367C24C2511yescharacteristic
368C2×Q8—216no—
369C2×Q8—216no—
370C2×Q8—216no—
371C2×Q8—44no—
372C2×Q8—48no—
373C2×Q8—44no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—48no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D414yes—
463C22×C4C22×D418yes—
464C22×C4C22×D414yes—
465C22×C4C22×D418yes—
466C22×C4C22×D412yes—
467C22×C4C22×D414yes—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C2×#4118yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C42−1+414yes—
476C22×C42−1+412yes—
477C22×C42−1+414yes—
478C22×C42−1+412yes—
479C4⋊C4—28no—
480C4⋊C4—216no—
481C4⋊C4—216no—
482C4⋊C4—28no—
483C4⋊C4—216no—
484C4⋊C4—232no—
485C4⋊C4—48no—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—416no—
490C4⋊C4—416no—
491C4⋊C4—48no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—44no—
496C4⋊C4—48no—
497C4⋊C4#127116yes—
498C22⋊C4—216no—
499C22⋊C4—416no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—48no—
505C42—28no—
506C42—216no—
507C42—216no—
508C42—416no—
509C42—48no—
510C42—44no—
511C42—88no—
512C23—24no—
513C23—22no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—41no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—41no—
534C23—42no—
535C23C23×D411yescharacteristic
536C23C23×D412yes—
537C23C22×#4112yes—
538C23C22×#4111yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×2−1+411yescharacteristic
541C23C2×2−1+412yes—
542C23C2×2−1+411yescharacteristic
543C23C2×2−1+412yes—
544C23C2×2−1+411yescharacteristic
545C23#58511yescharacteristic
546Q8—432no—
547Q8—88no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—88no—
635C2×C4#52514yes—
636C2×C4C2×#12718yes—
637C2×C4C2×#12714yes—
638C2×C4#53714yes—
639C2×C4#54618yes—
640C2×C4#54714yes—
641C2×C4#54718yes—
642C2×C4#54818yes—
643C22—44no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—82no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—82no—
654C22C2×#52511yescharacteristic
655C22C22×#12711yescharacteristic
656C22C2×#53711yescharacteristic
657C22C2×#54612yes—
658C22C2×#54711yescharacteristic
659C22C2×#54712yes—
660C22C2×#54812yes—
661C22#348212yes—
662C22#348612yes—
663C22#348611yescharacteristic
664C22#348911yescharacteristic
665C22#348912yes—
666C22#349912yes—
667C22#349911yescharacteristic
668C22#351411yescharacteristic
669C22#351612yes—
670C22#352312yes—
671C22#352312yes—
672C22#352311yescharacteristic
673C22#355211yescharacteristic
674C22#355212yes—
675C22#355511yescharacteristic
676C22#355612yes—
677C4—24no—
678C4—28no—
679C4—28no—
680C4—44no—
681C4—44no—
682C4—48no—
683C4—44no—
684C4—42no—
685C4—48no—
686C4—81no—
687C4—82no—
688C4—82no—
689C4—81no—
690C4—81no—
691C4—82no—
692C4#254018yes—
693C2—82no—
694C2C2×#254011yescharacteristic
695C2#3592711yescharacteristic
696C2#3599711yescharacteristic
697C2#3617911yescharacteristic
698C2#3642311yescharacteristic
699C2#3892311yescharacteristic
700C2#4279111yescharacteristic
701C2#4532212yes—
702C2#4534312yes—
703C2#5906112yes—
704C2#5907112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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×#2453C2semidirect
4. C2×#2365C2semidirect
5. #16964C2semidirect
7. #15804C2semidirect
8. #15193C2semidirect
9. #15190C2semidirect
13. #14155C2semidirect
15. #13954C2semidirect
17. #13821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24892
C2×#24532
C2×#24292
C2×#23652
#169642
#158082
#158042
#151932
#151902
#151782
#150792
#148702
#141552
#139972
#139542
C4×#3992
#138212