Finite Group Databasebeta

← #719806

Subgroups — #719806

Order 512.

2714 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2448C212yesmaximal
2C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×#2326C211yesmaximal, characteristic
4#17062C212yesmaximal, complemented
5#17060C212yesmaximal, complemented
6#15294C214yesmaximal, complemented
7#15290C214yesmaximal, complemented
8#15138C212yesmaximal
9#15133C212yesmaximal, complemented
10#15078C212yesmaximal, complemented
11#15078C212yesmaximal, complemented
12#14277C211yesmaximal, characteristic, complemented
13#13942C211yesmaximal, characteristic, complemented
14#13941C211yesmaximal, characteristic
15#13931C212yesmaximal, complemented
16#13906C211yesmaximal, characteristic
17#13848C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2448—216no—
23C2×#399C2214yes—
24C2×#398C2214yes—
25C2×#397C2214yes—
26C2×#397C2214yes—
27Q8×C4⋊C4—28no—
28#2326—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#382C2211yescharacteristic
47C2×#380C2214yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×#376C2211yescharacteristic
53#1445C2214yes—
54#1445C2214yes—
55#1444C2211yescharacteristic
56#1444C2211yescharacteristic
57#1444C2212yes—
58#1444C2212yes—
59#1442C2211yescharacteristic
60#1442C2211yescharacteristic
61#1442C2212yes—
62#1442C2212yes—
63#1441C2212yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1440C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2214yes—
81#1434C2214yes—
82#1434C2212yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C4×#95C2212yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#124—24no—
90C2×#124—24no—
91C2×#124D418yes—
92C2×C4×Q8—28no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8D418yes—
98C2×C4×D4—24no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#399—216no—
135#398—48no—
136#397—216no—
137#397—48no—
138#391—24no—
139#391—28no—
140#390—28no—
141#390—28no—
142#390—24no—
143#389—28no—
144#389—216no—
145#389—28no—
146#388—216no—
147#388—216no—
148#388—28no—
149#388—28no—
150#385—28no—
151#385—28no—
152#384—24no—
153#384—28no—
154#384—24no—
155#382—24no—
156#380—48no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—24no—
161#376—24no—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C25C2412yes—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8C2×D414yes—
197C22×Q8C2×D414yes—
198C22×Q8#4114yes—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4C2×D414yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#124—232no—
228C4×Q8—232no—
229C4×Q8—432no—
230C4×D4—416no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4C2×D418yes—
300C2×C22⋊C4C2×D418yes—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—28no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—28no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—28no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42C2×D418yes—
318C2×C42C2×D414yes—
319C2×C42C2×D414yes—
320C2×C42#4114yes—
321C2×C42#4114yes—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24C22×D414yes—
362C24C2511yescharacteristic
363C2×Q8—28no—
364C2×Q8—28no—
365C2×Q8—28no—
366C2×Q8—28no—
367C2×Q8—28no—
368C2×Q8—28no—
369C2×Q8#121116yes—
370C2×D4—28no—
371C2×D4—28no—
372C2×D4—28no—
373C2×D4—44no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—44no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—416no—
453C22×C4—48no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C42−1+414yes—
476C22×C42−1+412yes—
477C22×C42−1+412yes—
478C4⋊C4—28no—
479C4⋊C4—216no—
480C4⋊C4—28no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—416no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—84no—
491C4⋊C4—88no—
492C4⋊C4—84no—
493C4⋊C4—88no—
494C4⋊C4—84no—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—84no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—28no—
505C42—216no—
506C42—416no—
507C42—48no—
508C42—44no—
509C42—48no—
510C42#123116yes—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—28no—
515C23—28no—
516C23—42no—
517C23—42no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23C23×D411yescharacteristic
535C23C23×D411yescharacteristic
536C23C22×#4111yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4111yescharacteristic
539C23C2×H(5,𝔽2)12yes—
540C23C2×2−1+411yescharacteristic
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+412yes—
543C23#58512yes—
544C23#58512yes—
545Q8—232no—
546D4—416no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—816no—
634C2×C4C2×#12118yes—
635C2×C4C2×#12114yes—
636C2×C4C2×#12314yes—
637C2×C4C2×#12314yes—
638C2×C4C2×#12314yes—
639C2×C4#53614yes—
640C2×C4#54714yes—
641C2×C4#54714yes—
642C2×C4#55214yes—
643C2×C4#55214yes—
644C22—48no—
645C22—44no—
646C22—48no—
647C22—44no—
648C22—44no—
649C22—82no—
650C22—84no—
651C22—82no—
652C22—82no—
653C22—84no—
654C22—82no—
655C22C22×#12111yescharacteristic
656C22C22×#12311yescharacteristic
657C22C2×#53611yescharacteristic
658C22#346112yes—
659C22C2×#54711yescharacteristic
660C22C2×#54711yescharacteristic
661C22C2×#55211yescharacteristic
662C22C2×#55211yescharacteristic
663C22#348611yescharacteristic
664C22#348711yescharacteristic
665C22#348812yes—
666C22#348912yes—
667C22#349412yes—
668C22#349811yescharacteristic
669C22#350011yescharacteristic
670C22#350612yes—
671C22#350712yes—
672C22#351612yes—
673C22#351712yes—
674C22#354412yes—
675C22#355012yes—
676C22#355712yes—
677C22#356512yes—
678C4—24no—
679C4—28no—
680C4—24no—
681C4—44no—
682C4—42no—
683C4—44no—
684C4—42no—
685C4—44no—
686C4—81no—
687C4—81no—
688C4—82no—
689C4—84no—
690C4—84no—
691C4—82no—
692C4—82no—
693C4#248618yes—
694C2—84no—
695C2C2×#248611yescharacteristic
696C2#3518211yescharacteristic
697C2#3553111yescharacteristic
698C2#3573711yescharacteristic
699C2#3602911yescharacteristic
700C2#3776612yes—
701C2#3823112yes—
702C2#4278911yescharacteristic
703C2#4287511yescharacteristic
704C2#4496412yes—
705C2#4496712yes—
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×Q8×C4⋊C4C2semidirect
4. #17062C2semidirect
5. #17060C2semidirect
6. #15294C2semidirect
7. #15290C2semidirect
9. #15133C2semidirect
10. #15078C2semidirect
11. #15078C2semidirect
12. #14277C2semidirect
13. #13942C2semidirect
15. #13931C2semidirect
17. #13848C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24482
C2×Q8×C4⋊C42
C2×#23262
#170622
#170602
#152942
#152902
#151382
#151332
#150782
#142772
#139422
#139412
#139312
#139062
#138482