Finite Group Databasebeta

← #711886

Subgroups — #711886

Order 512.

3672 conjugacy classes of subgroups.

The subgroup lattice has 778 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#552C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic, complemented
3C2×#2395C211yesmaximal, characteristic
4C2×#2317C211yesmaximal, characteristic, complemented
5#16657C214yesmaximal
6#15928C212yesmaximal, complemented
7#15927C212yesmaximal, complemented
8#15881C212yesmaximal, complemented
9#15850C212yesmaximal, complemented
10#15218C212yesmaximal, complemented
11#15179C212yesmaximal, complemented
12#14892C214yesmaximal, complemented
13#14138C211yesmaximal, characteristic, complemented
14#14129C212yesmaximal, complemented
15#14017C212yesmaximal, complemented
16#13924C211yesmaximal, characteristic, complemented
17#13871C211yesmaximal, characteristic
18C2×#552—28no—
19C2×#552—28no—
20C2×#552—28no—
21C22×#126C2212yes—
22C22×#125C2212yes—
23C22×#125C2211yescharacteristic
24C22×#123C2212yes—
25C22×C22⋊Q8C2211yescharacteristic
26C22×C22⋊Q8C2211yescharacteristic
27C22×C4×Q8C2211yescharacteristic
28C22×C4×D4C2211yescharacteristic
29C22×#117C2211yescharacteristic
30C22×#117C2211yescharacteristic
31C22×#117C2212yes—
32C22×#117C2211yescharacteristic
33C23×C4⋊C4C2211yescharacteristic
34#2625—28no—
35#2395—28no—
36C2×#400C2211yescharacteristic
37C2×#400C2212yes—
38C2×#399C2211yescharacteristic
39C2×#398C2212yes—
40C2×#398C2211yescharacteristic
41C2×#397C2214yes—
42C2×#397C2212yes—
43C2×#397C2211yescharacteristic
44C2×#394C2212yes—
45#2317—28no—
46C2×#391C2212yes—
47C2×#391C2212yes—
48C2×#389C2212yes—
49C2×#389C2212yes—
50C2×#389C2212yes—
51C2×#385C2214yes—
52C2×#384C2212yes—
53C2×#382C2212yes—
54C2×#380C2212yes—
55C2×#380C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C22⋊C4C2212yes—
58C2×C4×C22⋊C4C2211yescharacteristic
59#1445C2212yes—
60#1445C2212yes—
61#1444C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1438C2212yes—
76#1437C2212yes—
77#1436C2214yes—
78#1436C2212yes—
79#1436C2212yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83#1436C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87C4⋊#95C2212yes—
88C22⋊#95C2214yes—
89C23×Q8C2311yescharacteristic
90C23×D4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94#552—2128no—
95C2×#126—28no—
96C2×#126—28no—
97C2×#126—28no—
98C2×#125—28no—
99C2×#125—28no—
100C2×#125—28no—
101C2×#125—24no—
102C2×#125—24no—
103C2×#125—24no—
104C2×#123—28no—
105C2×#123—28no—
106C2×#123—28no—
107C2×C22⋊Q8—24no—
108C2×C22⋊Q8—24no—
109C2×C22⋊Q8—24no—
110C2×C22⋊Q8—24no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8D418yes—
113C2×C4×Q8—24no—
114C2×C4×Q8—24no—
115C2×C4×Q8—24no—
116C2×C4×D4—24no—
117C2×C4×D4—24no—
118C2×C4×D4D418yes—
119C2×#117—24no—
120C2×#117—24no—
121C2×#117—28no—
122C2×#117—24no—
123C2×#117—24no—
124C2×#117—24no—
125C2×#117—24no—
126C2×#117—24no—
127C2×#117—28no—
128C2×#117—28no—
129C2×#117—28no—
130C22×C4⋊C4—24no—
131C22×C4⋊C4—24no—
132C22×C4⋊C4D418yes—
133C22×C4⋊C4C2311yescharacteristic
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2312yes—
137C22×C4⋊C4C2312yes—
138C22×C4⋊C4C2312yes—
139C22×C4⋊C4C2312yes—
140C22×C4⋊C4C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C4⋊C4C2312yes—
143C22×C4⋊C4C2311yescharacteristic
144C22×C4⋊C4C2311yescharacteristic
145C22×C4⋊C4C2312yes—
146C22×C4⋊C4C2311yescharacteristic
147C22×C4⋊C4C2312yes—
148C22×C4⋊C4C2312yes—
149C22×C4⋊C4C2314yes—
150C22×C4⋊C4C2314yes—
151C22×C4⋊C4C2314yes—
152C22×C22⋊C4C2312yes—
153C22×C22⋊C4C2312yes—
154C22×C22⋊C4C2312yes—
155C22×C22⋊C4C2312yes—
156C22×C22⋊C4C2312yes—
157C22×C22⋊C4C2312yes—
158C22×C22⋊C4C2311yescharacteristic
159C22×C22⋊C4C2312yes—
160C22×C22⋊C4C2311yescharacteristic
161C22×C22⋊C4C2312yes—
162C22×C22⋊C4C2312yes—
163C22×C22⋊C4C2314yes—
164C22×C42C2312yes—
165C22×C42C2312yes—
166C22×C42C2312yes—
167C22×C42C2312yes—
168C22×C42C2312yes—
169C22×C42C2311yescharacteristic
170#400—28no—
171#400—24no—
172#399—24no—
173#398—28no—
174#398—24no—
175#397—216no—
176#397—28no—
177#397—24no—
178#394—28no—
179#391—28no—
180#391—28no—
181#389—28no—
182#389—28no—
183#389—28no—
184#385—216no—
185#384—28no—
186#382—28no—
187#380—24no—
188#380—28no—
189C4×C4⋊C4—28no—
190C4×C22⋊C4—28no—
191C4×C22⋊C4—24no—
192C2×#95C2312yes—
193C2×#95C2312yes—
194C2×#95C2312yes—
195C2×#95C2314yes—
196C2×#95C2312yes—
197C2×#95C2311yescharacteristic
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2311yescharacteristic
201C2×#95C2312yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2314yes—
210C2×#95C2314yes—
211C2×#95C2314yes—
212C2×#95C2314yes—
213C2×#95C2314yes—
214C2×#95C2314yes—
215C25C2411yescharacteristic
216C25C2411yescharacteristic
217C22×Q8—22no—
218C22×Q8—22no—
219C22×Q8—24no—
220C22×Q8—24no—
221C22×Q8C2×D414yes—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—42no—
225C22×D4—42no—
226C22×D4C2×D414yes—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—22no—
232C23×C4—22no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—44no—
236C23×C4—44no—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×D414yes—
240C23×C4C2×D414yes—
241C23×C4C2×D418yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2414yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2411yescharacteristic
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2414yes—
254C23×C4C2412yes—
255C23×C4C2411yescharacteristic
256C23×C4C2411yescharacteristic
257#126—264no—
258#125—264no—
259#125—232no—
260#123—264no—
261C22⋊Q8—232no—
262C22⋊Q8—232no—
263C4×Q8—232no—
264C4×D4—232no—
265#117—232no—
266#117—232no—
267#117—264no—
268#117—416no—
269C2×C4⋊C4—232no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—24no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—22no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—24no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—24no—
294C2×C4⋊C4—24no—
295C2×C4⋊C4—24no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—24no—
306C2×C4⋊C4—24no—
307C2×C4⋊C4—24no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—28no—
310C2×C4⋊C4—28no—
311C2×C4⋊C4—42no—
312C2×C4⋊C4—42no—
313C2×C4⋊C4—42no—
314C2×C4⋊C4—44no—
315C2×C4⋊C4—44no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—44no—
319C2×C4⋊C4—44no—
320C2×C4⋊C4C2×D414yes—
321C2×C4⋊C4C2×D414yes—
322C2×C4⋊C4C2×D418yes—
323C2×C4⋊C4C2×D414yes—
324C2×C4⋊C4C2×D414yes—
325C2×C4⋊C4C2×D418yes—
326C2×C4⋊C4#4118yes—
327C2×C22⋊C4—22no—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—24no—
330C2×C22⋊C4—24no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—24no—
334C2×C22⋊C4—24no—
335C2×C22⋊C4—24no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—24no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—28no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—24no—
343C2×C22⋊C4—24no—
344C2×C22⋊C4—24no—
345C2×C22⋊C4—24no—
346C2×C22⋊C4—24no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—28no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4—42no—
353C2×C22⋊C4—42no—
354C2×C22⋊C4—42no—
355C2×C22⋊C4—44no—
356C2×C22⋊C4—44no—
357C2×C22⋊C4—44no—
358C2×C22⋊C4C2×D414yes—
359C2×C22⋊C4C2×D418yes—
360C2×C22⋊C4C2×D414yes—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—28no—
371C2×C42—24no—
372C2×C42—24no—
373C2×C42—24no—
374C2×C42—24no—
375C2×C42—24no—
376C2×C42—24no—
377C2×C42C2×D414yes—
378#95—24no—
379#95—22no—
380#95—24no—
381#95—22no—
382#95—24no—
383#95—24no—
384#95—24no—
385#95—28no—
386#95—28no—
387#95—28no—
388#95—28no—
389#95—28no—
390#95—28no—
391#95—28no—
392#95—28no—
393#95—28no—
394#95—44no—
395#95—44no—
396#95—44no—
397#95—44no—
398#95—44no—
399#95—44no—
400#95—44no—
401C24—22no—
402C24—22no—
403C24—22no—
404C24—22no—
405C24—22no—
406C24—22no—
407C24—21no—
408C24—21no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—22no—
414C24—22no—
415C24C22×D412yes—
416C24C22×D412yes—
417C24C22×D412yes—
418C24C22×D412yes—
419C24C2511yescharacteristic
420C2×Q8—28no—
421C2×Q8—216no—
422C2×Q8—44no—
423C2×Q8—44no—
424C2×Q8—48no—
425C2×D4—28no—
426C2×D4—44no—
427C2×D4—44no—
428C2×D4—48no—
429C2×D4—48no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—28no—
434C22×C4—216no—
435C22×C4—216no—
436C22×C4—216no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—28no—
478C22×C4—216no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—24no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—24no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—24no—
495C22×C4—22no—
496C22×C4—22no—
497C22×C4—24no—
498C22×C4—22no—
499C22×C4—24no—
500C22×C4—24no—
501C22×C4—22no—
502C22×C4—22no—
503C22×C4—22no—
504C22×C4—22no—
505C22×C4—22no—
506C22×C4—22no—
507C22×C4—416no—
508C22×C4—48no—
509C22×C4—48no—
510C22×C4—416no—
511C22×C4—44no—
512C22×C4—44no—
513C22×C4—48no—
514C22×C4#120116yes—
515C22×C4C22×D412yes—
516C22×C4C22×D414yes—
517C22×C4C22×D412yes—
518C22×C4C22×D414yes—
519C22×C4C22×D414yes—
520C22×C4C22×D412yes—
521C22×C4C22×D414yes—
522C22×C4C22×D412yes—
523C22×C4C22×D412yes—
524C22×C4C22×D412yes—
525C22×C4C22×D412yes—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C2×#4118yes—
529C22×C4C2×#4114yes—
530C22×C42−1+414yes—
531C22×C42−1+414yes—
532C4⋊C4—216no—
533C4⋊C4—216no—
534C4⋊C4—216no—
535C4⋊C4—216no—
536C4⋊C4—28no—
537C4⋊C4—28no—
538C4⋊C4—216no—
539C4⋊C4—28no—
540C4⋊C4—28no—
541C4⋊C4—216no—
542C4⋊C4—216no—
543C4⋊C4—48no—
544C4⋊C4—48no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—48no—
548C4⋊C4—88no—
549C4⋊C4—88no—
550C4⋊C4—88no—
551C22⋊C4—216no—
552C22⋊C4—216no—
553C22⋊C4—216no—
554C22⋊C4—28no—
555C22⋊C4—216no—
556C22⋊C4—28no—
557C22⋊C4—48no—
558C22⋊C4—48no—
559C22⋊C4—48no—
560C22⋊C4—48no—
561C22⋊C4—88no—
562C22⋊C4—84no—
563C42—216no—
564C42—216no—
565C42—216no—
566C42—28no—
567C42—48no—
568C42—48no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—24no—
574C23—24no—
575C23—24no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—22no—
584C23—24no—
585C23—24no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23—42no—
594C23—41no—
595C23—41no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—41no—
601C23—41no—
602C23—42no—
603C23—42no—
604C23—42no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—42no—
610C23C2×#12014yes—
611C23C23×D411yescharacteristic
612C23C23×D411yescharacteristic
613C23C23×D411yescharacteristic
614C23C22×#4112yes—
615C23C2×2−1+412yes—
616C23C2×2−1+412yes—
617C23#58512yes—
618C23#58512yes—
619C23#58512yes—
620Q8—416no—
621D4—416no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—24no—
625C2×C4—24no—
626C2×C4—232no—
627C2×C4—22no—
628C2×C4—24no—
629C2×C4—22no—
630C2×C4—24no—
631C2×C4—24no—
632C2×C4—22no—
633C2×C4—24no—
634C2×C4—28no—
635C2×C4—22no—
636C2×C4—22no—
637C2×C4—24no—
638C2×C4—28no—
639C2×C4—28no—
640C2×C4—28no—
641C2×C4—24no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—28no—
646C2×C4—24no—
647C2×C4—28no—
648C2×C4—24no—
649C2×C4—432no—
650C2×C4—41no—
651C2×C4—42no—
652C2×C4—41no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—42no—
667C2×C4—44no—
668C2×C4—42no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—41no—
672C2×C4—41no—
673C2×C4—44no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—416no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—44no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—44no—
690C2×C4—42no—
691C2×C4—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—44no—
697C2×C4—42no—
698C2×C4—42no—
699C2×C4—44no—
700C2×C4—42no—
701C2×C4—44no—
702C2×C4—42no—
703C2×C4—42no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4C2×#12014yes—
707C2×C4C2×#12014yes—
708C2×C4#52514yes—
709C2×C4#53614yes—
710C2×C4#54718yes—
711C2×C4#54718yes—
712C22—24no—
713C22—24no—
714C22—24no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—42no—
723C22—42no—
724C22—44no—
725C22—42no—
726C22—82no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22C22×#12011yescharacteristic
731C22C2×#52511yescharacteristic
732C22C2×#53611yescharacteristic
733C22#344911yescharacteristic
734C22#344911yescharacteristic
735C22C2×#54712yes—
736C22C2×#54712yes—
737C22#348512yes—
738C22#348712yes—
739C22#348912yes—
740C22#349712yes—
741C22#349911yescharacteristic
742C22#350011yescharacteristic
743C22#351512yes—
744C22#351612yes—
745C22#352112yes—
746C22#352412yes—
747C22#355512yes—
748C22#356312yes—
749C22#356512yes—
750C22#356512yes—
751C4—24no—
752C4—44no—
753C4—42no—
754C4—44no—
755C4—44no—
756C4—44no—
757C4—42no—
758C4—42no—
759C4—44no—
760C4—48no—
761C4—82no—
762C4—82no—
763C4—82no—
764C4—84no—
765C4#240618yes—
766C2—44no—
767C2—82no—
768C2C2×#240611yescharacteristic
769C2#3514911yescharacteristic
770C2#3566011yescharacteristic
771C2#3580212yes—
772C2#3804812yes—
773C2#3891512yes—
774C2#4264412yes—
775C2#4523812yes—
776C2#4523912yes—
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×#2625C2semidirect
4. C2×#2317C2semidirect
6. #15928C2semidirect
7. #15927C2semidirect
8. #15881C2semidirect
9. #15850C2semidirect
10. #15218C2semidirect
11. #15179C2semidirect
12. #14892C2semidirect
13. #14138C2semidirect
14. #14129C2semidirect
15. #14017C2semidirect
16. #13924C2semidirect

Maximal subgroups

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

subgroupindex
C22×#5522
C2×#26252
C2×#23952
C2×#23172
#166572
#159282
#159272
#158812
#158502
#152182
#151792
#148922
#141382
#141292
#140172
#139242
#138712