Finite Group Databasebeta

← #4930458

Subgroups — #4930458

Order 512.

2844 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic, complemented
2C2×#2487C211yesmaximal, characteristic
3C2×#2415C211yesmaximal, characteristic
4C2×#2274C211yesmaximal, characteristic, complemented
5#15175C212yesmaximal, complemented
6#15156C212yesmaximal, complemented
7#15103C212yesmaximal, complemented
8#15090C212yesmaximal, complemented
9#14838C214yesmaximal
10#14837C214yesmaximal, complemented
11#14258C211yesmaximal, characteristic, complemented
12#14142C212yesmaximal, complemented
13#14006C212yesmaximal, complemented
14#13997C212yesmaximal, complemented
15#13951C211yesmaximal, characteristic, complemented
16#13875C212yesmaximal, complemented
17#13839C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24#2503—28no—
25#2487—28no—
26#2415—28no—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#400C2212yes—
30C2×#399C2212yes—
31C2×#397C2212yes—
32C2×#394C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2211yescharacteristic
40C2×#388C2212yes—
41C2×#386C2212yes—
42C2×#385C2212yes—
43C2×#382C2212yes—
44C2×#381C2214yes—
45C2×#380C2211yescharacteristic
46C2×#380C2212yes—
47C2×#380C2211yescharacteristic
48#2274—28no—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2214yes—
56#1445C2214yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1441C2211yescharacteristic
62#1441C2211yescharacteristic
63#1441C2212yes—
64#1441C2212yes—
65#1441C2214yes—
66#1440C2212yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2211yescharacteristic
70#1439C2211yescharacteristic
71#1439C2214yes—
72#1439C2214yes—
73#1438C2212yes—
74#1437C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82C4⋊#95C2214yes—
83#1434C2214yes—
84#1434C2214yes—
85#1434C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—24no—
96C2×C22⋊Q8—28no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×#117—24no—
101C2×#117—28no—
102C2×#117—28no—
103C2×#117D418yes—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4D418yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2314yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2314yes—
138C22×C42C2314yes—
139#400—28no—
140#400—28no—
141#400—28no—
142#399—28no—
143#397—28no—
144#394—28no—
145#391—24no—
146#391—28no—
147#391—28no—
148#389—24no—
149#389—28no—
150#389—28no—
151#389—28no—
152#388—28no—
153#386—28no—
154#385—28no—
155#382—28no—
156#381—216no—
157#380—24no—
158#380—24no—
159#380—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—28no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—216no—
166C4×C22⋊C4—28no—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C25C2411yescharacteristic
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—41no—
208C22×D4—41no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D418yes—
223C23×C4#4114yes—
224C23×C4C2411yescharacteristic
225C23×C4C2412yes—
226C23×C4C2414yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C22⋊Q8—416no—
240C22⋊Q8—416no—
241C4×D4—416no—
242#117—232no—
243#117—232no—
244C2×C4⋊C4—232no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D418yes—
278C2×C4⋊C4C2×D418yes—
279C2×C4⋊C4C2×D414yes—
280C2×C4⋊C4C2×D414yes—
281C2×C4⋊C4#4118yes—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
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⋊C4C2×D418yes—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—22no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—22no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—28no—
326C2×C42—28no—
327C2×C42—44no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42C2×D418yes—
332C2×C42#4118yes—
333#95—28no—
334#95—28no—
335#95—22no—
336#95—22no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—28no—
353#95—24no—
354#95—42no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—42no—
360#95—42no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—22no—
377C24—21no—
378C24—21no—
379C24C22×D412yes—
380C24C22×D412yes—
381C24C2×#4112yes—
382C242−1+412yes—
383C24C2511yescharacteristic
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—48no—
388C2×Q8—48no—
389C2×D4—28no—
390C2×D4—44no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—44no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—21no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—216no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—416no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—48no—
482C22×C4—48no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—88no—
486C22×C4#121116yes—
487C22×C4C22×D418yes—
488C22×C4C22×D414yes—
489C22×C4C22×D412yes—
490C22×C4C22×D412yes—
491C22×C4C22×D414yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2×#4112yes—
497C22×C4C2×#4118yes—
498C22×C4C2×#4112yes—
499C22×C4C2×#4114yes—
500C22×C4C2×#4114yes—
501C22×C42−1+412yes—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—28no—
507C4⋊C4—48no—
508C4⋊C4—44no—
509C4⋊C4—44no—
510C4⋊C4—48no—
511C4⋊C4—48no—
512C4⋊C4—48no—
513C4⋊C4—88no—
514C4⋊C4—88no—
515C4⋊C4#123116yes—
516C22⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—44no—
524C22⋊C4—48no—
525C22⋊C4—44no—
526C22⋊C4—48no—
527C22⋊C4—88no—
528C42—216no—
529C42—216no—
530C42—44no—
531C42—48no—
532C42—88no—
533C42—88no—
534C23—22no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—22no—
552C23—24no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—41no—
568C23—41no—
569C23—42no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23C2×#12114yes—
580C23C23×D411yescharacteristic
581C23C23×D411yescharacteristic
582C23C22×#4111yescharacteristic
583C23C22×#4112yes—
584C23C22×#4112yes—
585C23C2×2−1+411yescharacteristic
586C23C2×2−1+411yescharacteristic
587C23#58512yes—
588C23#58512yes—
589C23#58511yescharacteristic
590C23#58511yescharacteristic
591Q8—88no—
592D4—416no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—28no—
601C2×C4—232no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—24no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—41no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—816no—
679C2×C4—88no—
680C2×C4C2×#12114yes—
681C2×C4C2×#12114yes—
682C2×C4C2×#12318yes—
683C2×C4C2×#12314yes—
684C2×C4#52514yes—
685C2×C4#52918yes—
686C2×C4#53218yes—
687C22—24no—
688C22—24no—
689C22—24no—
690C22—44no—
691C22—44no—
692C22—42no—
693C22—42no—
694C22—44no—
695C22—44no—
696C22—42no—
697C22—42no—
698C22—44no—
699C22—42no—
700C22—42no—
701C22—44no—
702C22—42no—
703C22—81no—
704C22—81no—
705C22—82no—
706C22—81no—
707C22—82no—
708C22—81no—
709C22C22×#12111yescharacteristic
710C22C22×#12311yescharacteristic
711C22C2×#52511yescharacteristic
712C22C2×#52912yes—
713C22C2×#53212yes—
714C22#348512yes—
715C22#348611yescharacteristic
716C22#348711yescharacteristic
717C22#348911yescharacteristic
718C22#348911yescharacteristic
719C22#349312yes—
720C22#349511yescharacteristic
721C22#349512yes—
722C22#349511yescharacteristic
723C22#350011yescharacteristic
724C22#350011yescharacteristic
725C22#350712yes—
726C22#351012yes—
727C22#351212yes—
728C22#351512yes—
729C22#351612yes—
730C22#352112yes—
731C22#352412yes—
732C4—24no—
733C4—28no—
734C4—48no—
735C4—42no—
736C4—44no—
737C4—44no—
738C4—44no—
739C4—81no—
740C4—82no—
741C4—81no—
742C4—82no—
743C4—82no—
744C4—84no—
745C4—82no—
746C4#285418yes—
747C2—44no—
748C2—82no—
749C2#3561711yescharacteristic
750C2#3564812yes—
751C2#3565312yes—
752C2#3566111yescharacteristic
753C2#3575911yescharacteristic
754C2#3577312yes—
755C2#3598311yescharacteristic
756C2#3601512yes—
757C2#3606711yescharacteristic
758C2#3608011yescharacteristic
759C2C2×#285411yescharacteristic
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
1. C2×#2503C2semidirect
4. C2×#2274C2semidirect
5. #15175C2semidirect
6. #15156C2semidirect
7. #15103C2semidirect
8. #15090C2semidirect
10. #14837C2semidirect
11. #14258C2semidirect
12. #14142C2semidirect
13. #14006C2semidirect
14. #13997C2semidirect
15. #13951C2semidirect
16. #13875C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#24872
C2×#24152
C2×#22742
#151752
#151562
#151032
#150902
#148382
#148372
#142582
#141422
#140062
#139972
#139512
#138752
#138392