Finite Group Databasebeta

← #732770

Subgroups — #732770

Order 512.

2714 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2567C211yesmaximal, characteristic, complemented
2C2×#2592C211yesmaximal, characteristic
3C2×#2640C211yesmaximal, characteristic
4C2×#2367C211yesmaximal, characteristic
5#17060C212yesmaximal, complemented
6#16805C212yesmaximal, complemented
7#15632C214yesmaximal, complemented
8#15426C214yesmaximal, complemented
9#15186C212yesmaximal, complemented
10#15150C212yesmaximal, complemented
11#15138C212yesmaximal
12#15050C212yesmaximal, complemented
13#14155C211yesmaximal, characteristic, complemented
14#13935C212yesmaximal, complemented
15#13920C212yesmaximal
16#13850C211yesmaximal, characteristic, complemented
17#13783C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2640—28no—
24#2592—28no—
25#2567—28no—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C2×#397C2214yes—
30C2×#395C2214yes—
31C2×#393C2214yes—
32C2×#393C2212yes—
33#2367—28no—
34C2×#391C2212yes—
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#390C2211yescharacteristic
41C2×#390C2212yes—
42C2×#390C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#389C2212yes—
46C2×#388C2212yes—
47C2×#388C2214yes—
48C2×#388C2212yes—
49C2×#385C2212yes—
50C2×#384C2212yes—
51C2×#380C2214yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2212yes—
56C2×#376C2211yescharacteristic
57#1446C2211yescharacteristic
58#1446C2211yescharacteristic
59#1445C2211yescharacteristic
60#1445C2211yescharacteristic
61#1445C2214yes—
62#1445C2212yes—
63#1445C2212yes—
64#1443C2214yes—
65#1443C2214yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1440C2214yes—
72#1440C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1436C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2214yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2311yescharacteristic
89C23×Q8C2312yes—
90C23×D4C2311yescharacteristic
91C24×C4C2312yes—
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—24no—
94C2×C4⋊Q8—24no—
95C2×#124—24no—
96C2×#124—24no—
97C2×#124D418yes—
98C2×C4×Q8—24no—
99C2×C4×Q8—28no—
100C2×C4×Q8—24no—
101C2×C4×Q8—24no—
102C2×C4×Q8—24no—
103C2×C4×D4—24no—
104C2×C4×D4—28no—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#397—28no—
136#397—28no—
137#397—28no—
138#397—48no—
139#395—216no—
140#393—28no—
141#393—216no—
142#391—28no—
143#391—28no—
144#391—24no—
145#391—28no—
146#391—44no—
147#391—44no—
148#390—28no—
149#390—28no—
150#390—24no—
151#389—28no—
152#389—28no—
153#389—28no—
154#388—216no—
155#388—28no—
156#388—28no—
157#385—28no—
158#384—28no—
159#380—48no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4D418yes—
163C4×C22⋊C4—28no—
164#376—24no—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C25C2412yes—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—24no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8—42no—
211C22×Q8—42no—
212C22×Q8C2×D414yes—
213C22×D4—22no—
214C22×D4—24no—
215C22×D4—22no—
216C22×D4—24no—
217C22×D4C2×D414yes—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—28no—
221C23×C4—44no—
222C23×C4—42no—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C4⋊Q8—416no—
240#124—232no—
241C4×Q8—416no—
242C4×Q8—416no—
243C4×D4—416no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
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—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—28no—
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—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4#4118yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4C2×D418yes—
304C2×C22⋊C4C2×D418yes—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—44no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42C2×D418yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42#4114yes—
325#95—24no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—28no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—42no—
347#95—42no—
348#95—44no—
349#95—42no—
350#95—42no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95C2×D418yes—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—24no—
362C24—22no—
363C24—24no—
364C24—22no—
365C24C22×D414yes—
366C24C2511yescharacteristic
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8—216no—
370C2×Q8—48no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—44no—
376C2×Q8—48no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—48no—
380C2×Q8—84no—
381C2×Q8—84no—
382C2×Q8—84no—
383C2×Q8—84no—
384C2×D4—28no—
385C2×D4—216no—
386C2×D4—44no—
387C2×D4—44no—
388C2×D4—48no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—48no—
466C22×C4—416no—
467C22×C4—48no—
468C22×C4—84no—
469C22×C4—88no—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C2×#4112yes—
480C22×C4C2×#4112yes—
481C22×C4C2×#4118yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4114yes—
484C22×C4H(5,𝔽2)12yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C42−1+414yes—
487C22×C42−1+414yes—
488C4⋊C4—28no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C4⋊C4#124116yes—
501C22⋊C4—216no—
502C22⋊C4—216no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—48no—
511C42—48no—
512C42—44no—
513C42—48no—
514C42—44no—
515C42—48no—
516C42—84no—
517C42#121116yes—
518C23—24no—
519C23—28no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23C23×D411yescharacteristic
542C23C23×D411yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4112yes—
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×2−1+412yes—
548C23C2×2−1+412yes—
549C23#58511yescharacteristic
550C23#58511yescharacteristic
551C23#58512yes—
552Q8—416no—
553Q8—416no—
554Q8—816no—
555D4—416no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—816no—
643C2×C4C2×#12114yes—
644C2×C4C2×#12114yes—
645C2×C4C2×#12114yes—
646C2×C4C2×#12418yes—
647C2×C4C2×#12414yes—
648C2×C4#53614yes—
649C2×C4#54718yes—
650C2×C4#55618yes—
651C22—44no—
652C22—44no—
653C22—48no—
654C22—44no—
655C22—48no—
656C22—82no—
657C22—82no—
658C22—84no—
659C22—84no—
660C22—82no—
661C22—82no—
662C22C22×#12111yescharacteristic
663C22C22×#12411yescharacteristic
664C22C2×#53611yescharacteristic
665C22#346011yescharacteristic
666C22#346011yescharacteristic
667C22C2×#54712yes—
668C22C2×#55612yes—
669C22#348511yescharacteristic
670C22#348611yescharacteristic
671C22#348811yescharacteristic
672C22#348811yescharacteristic
673C22#348912yes—
674C22#349512yes—
675C22#349811yescharacteristic
676C22#350011yescharacteristic
677C22#351312yes—
678C22#351812yes—
679C22#352112yes—
680C22#352212yes—
681C22#355012yes—
682C22#355412yes—
683C22#355512yes—
684C22#355712yes—
685C4—28no—
686C4—24no—
687C4—24no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—44no—
692C4—44no—
693C4—81no—
694C4—81no—
695C4—82no—
696C4—82no—
697C4—84no—
698C4—82no—
699C4—84no—
700C4#254718yes—
701C2—84no—
702C2C2×#254711yescharacteristic
703C2#3525611yescharacteristic
704C2#3563311yescharacteristic
705C2#3584711yescharacteristic
706C2#3594811yescharacteristic
707C2#3823712yes—
708C2#3824012yes—
709C2#4267811yescharacteristic
710C2#4280911yescharacteristic
711C2#4505112yes—
712C2#4522712yes—
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×#2567C2semidirect
5. #17060C2semidirect
6. #16805C2semidirect
7. #15632C2semidirect
8. #15426C2semidirect
9. #15186C2semidirect
10. #15150C2semidirect
12. #15050C2semidirect
13. #14155C2semidirect
14. #13935C2semidirect
16. #13850C2semidirect
17. #13783C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#25922
C2×#26402
C2×#23672
#170602
#168052
#156322
#154262
#151862
#151502
#151382
#150502
#141552
#139352
#139202
#138502
#137832