Finite Group Databasebeta

← #792817

Subgroups — #792817

Order 512.

2686 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2611C212yesmaximal
3C2×#2344C211yesmaximal, characteristic
4#16683C212yesmaximal, complemented
5#15414C214yesmaximal, complemented
6#15298C214yesmaximal
7#15156C211yesmaximal, characteristic, complemented
8#15150C211yesmaximal, characteristic
9#15145C211yesmaximal, characteristic, complemented
10#14907C212yesmaximal
11#14903C212yesmaximal, complemented
12#14151C212yesmaximal, complemented
13#13994C212yesmaximal
14#13884C212yesmaximal, complemented
15#13880C212yesmaximal
16C4×#399C211yesmaximal, characteristic
17#13810C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C22×#117C2212yes—
20C23×C4⋊C4C2211yescharacteristic
21#2611—216no—
22#2552—28no—
23C2×#400C2212yes—
24C2×#399C2211yescharacteristic
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#398C2212yes—
28C2×#397C2212yes—
29C2×#397C2212yes—
30#2344—28no—
31C2×#389C2211yescharacteristic
32C2×#389C2214yes—
33C2×#389C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#388C2214yes—
37C2×#387C2212yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#385C2214yes—
41C2×#385C2214yes—
42C2×#384C2211yescharacteristic
43C2×#384C2212yes—
44C2×#384C2212yes—
45C2×#381C2212yes—
46C2×#380C2212yes—
47C2×#380C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55C2×#376C2212yes—
56#1445C2211yescharacteristic
57#1445C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1444C2211yescharacteristic
61#1444C2214yes—
62#1444C2214yes—
63#1444C2214yes—
64#1443C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1440C2214yes—
72#1439C2211yescharacteristic
73#1439C2212yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2214yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2214yes—
84#1434C2214yes—
85C4×#95C2211yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#125—28no—
89C2×#125—28no—
90C2×#125—44no—
91C2×#117—28no—
92C2×#117—28no—
93C2×#117—44no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2312yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#400—28no—
131#399D418yes—
132#398—28no—
133#398—216no—
134#398—44no—
135#397—28no—
136#397—44no—
137#389—24no—
138#389—216no—
139#389—28no—
140#389—24no—
141#389—28no—
142#388—216no—
143#387—216no—
144#387—28no—
145#387—28no—
146#385—216no—
147#385—48no—
148#384—28no—
149#384—28no—
150#384—24no—
151#381—28no—
152#380—28no—
153#380—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4D418yes—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—28no—
161#376—28no—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2311yescharacteristic
191C2×#95C2312yes—
192C25C2411yescharacteristic
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4#4114yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221#125—432no—
222#117—432no—
223C2×C4⋊C4—232no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
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—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
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—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D418yes—
280C2×C4⋊C4#4114yes—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
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—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—41no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D418yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—28no—
313C2×C42—28no—
314C2×C42—42no—
315C2×C42C2×D418yes—
316C2×C42C2×D414yes—
317C2×C42#4118yes—
318#95—28no—
319#95—24no—
320#95—22no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—24no—
337#95—28no—
338#95—22no—
339#95—24no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—42no—
344#95—44no—
345#95—44no—
346#95C2×D414yes—
347#95C2×D418yes—
348C24—21no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—21no—
356C24C22×D412yes—
357C24C2×#4112yes—
358C24H(5,𝔽2)12yes—
359C24C2511yescharacteristic
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—216no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
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—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—28no—
414C22×C4—216no—
415C22×C4—21no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—48no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D414yes—
456C22×C4C22×D412yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+414yes—
466C22×C42−1+414yes—
467C22×C42−1+414yes—
468C4⋊C4—28no—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—216no—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—84no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C22⋊C4—216no—
489C22⋊C4—28no—
490C22⋊C4—216no—
491C22⋊C4—44no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C42—28no—
495C42—216no—
496C42—216no—
497C42—48no—
498C42—48no—
499C42—44no—
500C42—416no—
501C23—22no—
502C23—22no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—22no—
510C23—24no—
511C23—22no—
512C23—24no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—42no—
518C23—41no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23#55014yes—
524C23C23×D411yescharacteristic
525C23C23×D411yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4111yescharacteristic
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×2−1+412yes—
532C23C2×2−1+412yes—
533C23C2×2−1+412yes—
534C23#58511yescharacteristic
535C2×C4—22no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—432no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—416no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4#52514yes—
627C2×C4#53214yes—
628C2×C4#53614yes—
629C2×C4D4214yes—
630C2×C4#54718yes—
631C2×C4#55218yes—
632C2×C4#55218yes—
633C22—24no—
634C22—28no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22C2×#52511yescharacteristic
645C22C2×#53211yescharacteristic
646C22C2×#53611yescharacteristic
647C22C2×D4211yescharacteristic
648C22C2×#54712yes—
649C22C2×#55011yescharacteristic
650C22D4×#4111yescharacteristic
651C22C2×#55212yes—
652C22C2×#55212yes—
653C22#348611yescharacteristic
654C22#348811yescharacteristic
655C22#349212yes—
656C22#349512yes—
657C22#349911yescharacteristic
658C22#349912yes—
659C22#350611yescharacteristic
660C22#350912yes—
661C22#351112yes—
662C22#351911yescharacteristic
663C22#352312yes—
664C22#354412yes—
665C22#354412yes—
666C22#356312yes—
667C4—28no—
668C4—24no—
669C4—44no—
670C4—44no—
671C4—44no—
672C4—42no—
673C4—44no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—84no—
679C4—81no—
680C4—82no—
681C4—84no—
682C4#261118yescomplemented
683C2—44no—
684C2C2×#261111yescharacteristic
685C2#3574911yescharacteristic
686C2#3592611yescharacteristic
687C2#3610111yescharacteristic
688C2#3612711yescharacteristic
689C2#3933711yescharacteristic
690C2#4296411yescharacteristic
691C2#4491712yes—
692C2#4495612yes—
693C2#5929012yes—
694C2#5931212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
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
4. #16683C2semidirect
5. #15414C2semidirect
7. #15156C2semidirect
9. #15145C2semidirect
11. #14903C2semidirect
12. #14151C2semidirect
14. #13884C2semidirect
17. #13810C2semidirect
682. 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×#23442
#166832
#154142
#152982
#151562
#151502
#151452
#149072
#149032
#141512
#139942
#138842
#138802
C4×#3992
#138102