Finite Group Databasebeta

← #793938

Subgroups — #793938

Order 512.

2798 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2611C212yesmaximal, complemented
3C2×#2354C211yesmaximal, characteristic
4#16682C212yesmaximal, complemented
5#15408C214yesmaximal, complemented
6#15294C214yesmaximal, complemented
7#15155C211yesmaximal, characteristic
8#15152C211yesmaximal, characteristic, complemented
9#15147C211yesmaximal, characteristic
10#14910C212yesmaximal, complemented
11#14893C212yesmaximal, complemented
12#14029C212yesmaximal, complemented
13#14018C212yesmaximal
14#14001C212yesmaximal
15#13910C212yesmaximal, complemented
16C4×#398C211yesmaximal, characteristic, complemented
17#13810C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21#2611—216no—
22#2552—28no—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#399C2212yes—
26C2×#399C2214yes—
27C2×#398C2214yes—
28C2×#398C2211yescharacteristic
29C2×#397C2212yes—
30#2354—28no—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#388C2212yes—
40C2×#388C2212yes—
41C2×#388C2214yes—
42C2×#388C2214yes—
43C2×#387C2214yes—
44C2×#385C2214yes—
45C2×#384C2211yescharacteristic
46C2×#384C2212yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×#376C2212yes—
57#1445C2211yescharacteristic
58#1445C2212yes—
59#1445C2214yes—
60#1444C2211yescharacteristic
61#1444C2214yes—
62#1444C2214yes—
63#1442C2212yes—
64#1442C2212yes—
65#1441C2211yescharacteristic
66#1441C2212yes—
67#1441C2214yes—
68#1441C2212yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1436C2214yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82#1434C2214yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C4×#95C2212yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#125—28no—
89C2×#125—28no—
90C2×#125—44no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×C4×D4—24no—
94C2×#117—28no—
95C2×#117—28no—
96C2×#117—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#400—28no—
135#399—28no—
136#399—44no—
137#399—48no—
138#398—216no—
139#398D418yescomplemented
140#397—28no—
141#390—28no—
142#390—24no—
143#390—24no—
144#390—28no—
145#389—216no—
146#389—28no—
147#389—28no—
148#389—48no—
149#388—28no—
150#388—28no—
151#388—216no—
152#388—48no—
153#387—216no—
154#385—216no—
155#384—24no—
156#384—28no—
157#384—24no—
158#384—28no—
159#380—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C4×C4⋊C4D418yes—
166#376—28no—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C2×#95C2312yes—
193C25C2412yes—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4#4114yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#125—432no—
223C4×D4—232no—
224#117—432no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—41no—
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⋊C4—44no—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D418yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—28no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—24no—
322C2×C42—42no—
323C2×C42C2×D418yes—
324C2×C42C2×D414yes—
325C2×C42#4114yes—
326C2×C42#4118yes—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—22no—
342#95—24no—
343#95—24no—
344#95—44no—
345#95—42no—
346#95—42no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95C2×D414yes—
351#95C2×D418yes—
352#95C2×D418yes—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24C2×#4114yes—
364C24C2511yescharacteristic
365C2×D4—28no—
366C2×D4—216no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—48no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—216no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—21no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4114yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C42−1+414yes—
471C22×C42−1+412yes—
472C22×C42−1+414yes—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C4⋊C4—28no—
476C4⋊C4—28no—
477C4⋊C4—28no—
478C4⋊C4—28no—
479C4⋊C4—216no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—416no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—416no—
488C4⋊C4—48no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—88no—
502C42—28no—
503C42—28no—
504C42—216no—
505C42—216no—
506C42—48no—
507C42—48no—
508C42—44no—
509C42—416no—
510C23—24no—
511C23—24no—
512C23—28no—
513C23—28no—
514C23—24no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—42no—
525C23—44no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23C23×D411yescharacteristic
534C23C23×D411yescharacteristic
535C23C22×#4111yescharacteristic
536C23C22×#4111yescharacteristic
537C23C22×#4112yes—
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)12yes—
540C23C2×2−1+412yes—
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+412yes—
543C23#58511yescharacteristic
544D4—416no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—432no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4#52514yes—
635C2×C4#53214yes—
636C2×C4#53614yes—
637C2×C4D4214yes—
638C2×C4#54714yes—
639C2×C4#55018yes—
640C2×C4#55218yes—
641C2×C4#55218yes—
642C22—44no—
643C22—48no—
644C22—48no—
645C22—44no—
646C22—44no—
647C22—82no—
648C22—82no—
649C22—84no—
650C22—82no—
651C22—84no—
652C22—82no—
653C22C2×#52511yescharacteristic
654C22C2×#53211yescharacteristic
655C22C2×#53611yescharacteristic
656C22C2×D4211yescharacteristic
657C22C2×#54711yescharacteristic
658C22C2×#55012yes—
659C22D4×#4111yescharacteristic
660C22C2×#55212yes—
661C22C2×#55212yes—
662C22#348611yescharacteristic
663C22#348911yescharacteristic
664C22#349212yes—
665C22#349811yescharacteristic
666C22#349912yes—
667C22#350611yescharacteristic
668C22#350712yes—
669C22#350812yes—
670C22#350912yes—
671C22#352011yescharacteristic
672C22#352012yes—
673C22#354412yes—
674C22#354412yes—
675C22#354712yes—
676C4—24no—
677C4—24no—
678C4—28no—
679C4—44no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—42no—
684C4—44no—
685C4—42no—
686C4—42no—
687C4—82no—
688C4—84no—
689C4—84no—
690C4—81no—
691C4#261118yescomplemented
692C2—84no—
693C2C2×#261111yescharacteristic
694C2#3570111yescharacteristic
695C2#3593811yescharacteristic
696C2#3610011yescharacteristic
697C2#3612811yescharacteristic
698C2#3933711yescharacteristic
699C2#4296811yescharacteristic
700C2#4490512yes—
701C2#4508312yes—
702C2#5931212yes—
703C2#5944412yes—
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×#2552C2semidirect
2. C2×#2611C2semidirect
4. #16682C2semidirect
5. #15408C2semidirect
6. #15294C2semidirect
8. #15152C2semidirect
10. #14910C2semidirect
11. #14893C2semidirect
12. #14029C2semidirect
15. #13910C2semidirect
16. C4×#398C2semidirect
17. #13810C2semidirect
139. #398D4semidirect
691. C4#2611semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#26112
C2×#23542
#166822
#154082
#152942
#151552
#151522
#151472
#149102
#148932
#140292
#140182
#140012
#139102
C4×#3982
#138102