Finite Group Databasebeta

← #1393992

Subgroups — #1393992

Order 512.

2686 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2452C211yesmaximal, characteristic
2C2×#2625C212yesmaximal, complemented
3C2×#2354C211yesmaximal, characteristic
4#16399C212yesmaximal, complemented
5#16365C212yesmaximal, complemented
6#16314C214yesmaximal, complemented
7#16278C214yesmaximal, complemented
8#15156C211yesmaximal, characteristic, complemented
9#15151C211yesmaximal, characteristic, complemented
10#15145C211yesmaximal, characteristic, complemented
11#15138C212yesmaximal
12#15136C212yesmaximal
13#14906C212yesmaximal, complemented
14#14053C212yesmaximal, complemented
15#14005C212yesmaximal, complemented
16#13920C211yesmaximal, characteristic
17C4×#400C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22#2625—216no—
23#2452—28no—
24C2×#400C2212yes—
25C2×#400C2211yescharacteristic
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#397C2214yes—
31C2×#397C2214yes—
32C2×#397C2214yes—
33C2×#396C2212yes—
34C2×#393C2212yes—
35#2354—28no—
36C2×#391C2214yes—
37C2×#390C2211yescharacteristic
38C2×#390C2212yes—
39C2×#390C2211yescharacteristic
40C2×#390C2212yes—
41C2×#389C2211yescharacteristic
42C2×#389C2214yes—
43C2×#389C2212yes—
44C2×#389C2211yescharacteristic
45C2×#389C2212yes—
46C2×#388C2214yes—
47C2×#385C2212yes—
48C2×#384C2211yescharacteristic
49C2×#384C2212yes—
50C2×#380C2212yes—
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55#1446C2214yes—
56#1445C2212yes—
57#1445C2211yescharacteristic
58#1445C2212yes—
59#1444C2214yes—
60#1444C2214yes—
61#1443C2212yes—
62#1443C2214yes—
63#1442C2211yescharacteristic
64#1442C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2211yescharacteristic
69#1440C2214yes—
70#1440C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4⋊Q8—24no—
90C2×C4⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—44no—
94C2×C4×D4—28no—
95C2×C4×D4D418yes—
96C2×#117—28no—
97C2×#117—28no—
98C2×#117—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2314yes—
131C22×C42C2311yescharacteristic
132#400—28no—
133#400D418yescomplemented
134#398—28no—
135#398—28no—
136#398—44no—
137#398—44no—
138#397—216no—
139#397—216no—
140#397—48no—
141#396—28no—
142#393—28no—
143#391—216no—
144#390—28no—
145#390—24no—
146#390—24no—
147#390—44no—
148#389—24no—
149#389—28no—
150#389—24no—
151#389—216no—
152#389—44no—
153#388—216no—
154#385—28no—
155#384—24no—
156#384—44no—
157#380—28no—
158#380—28no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—24no—
161C4×C4⋊C4—24no—
162C2×#95C2311yescharacteristic
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2311yescharacteristic
191C25C2412yes—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—28no—
195C22×Q8—42no—
196C22×Q8—44no—
197C22×D4—24no—
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4C2×D414yes—
202C23×C4—24no—
203C23×C4—28no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4C2×D418yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C4⋊Q8—416no—
224C22⋊Q8—432no—
225C4×D4—232no—
226#117—432no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4#4118yes—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
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⋊C4C2×D418yes—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—22no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—44no—
317C2×C42C2×D414yes—
318#95—28no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—42no—
337#95—42no—
338#95—42no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—42no—
344#95C2×D414yes—
345#95C2×D418yes—
346#95C2×D418yes—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24C22×D414yes—
358C24C2511yescharacteristic
359C2×Q8—48no—
360C2×Q8—416no—
361C2×Q8—48no—
362C2×Q8—84no—
363C2×Q8—88no—
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—28no—
367C2×D4—48no—
368C2×D4—48no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—216no—
411C22×C4—216no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D414yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C2×#4118yes—
463C22×C4C2×#4114yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+414yes—
468C22×C42−1+414yes—
469C22×C42−1+412yes—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—416no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—84no—
484C4⋊C4—88no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C4⋊C4#123116yes—
488C22⋊C4—216no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—84no—
496C22⋊C4—88no—
497C42—28no—
498C42—416no—
499C42—44no—
500C42—48no—
501C42—44no—
502C42—48no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C22×#4112yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×2−1+412yes—
533C23C2×2−1+411yescharacteristic
534C23C2×2−1+412yes—
535C23#58512yes—
536C23#58511yescharacteristic
537Q8—816no—
538D4—416no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—432no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4C2×#12318yes—
630C2×C4C2×#12314yes—
631C2×C4#53614yes—
632C2×C4#53714yes—
633C2×C4#54118yes—
634C2×C4#54218yes—
635C2×C4D4214yes—
636C22—44no—
637C22—44no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—84no—
644C22—84no—
645C22—84no—
646C22—84no—
647C22C22×#12311yescharacteristic
648C22C2×#53611yescharacteristic
649C22C2×#53711yescharacteristic
650C22C2×#54112yes—
651C22C2×#54212yes—
652C22C2×D4211yescharacteristic
653C22D4×#4112yes—
654C22#348511yescharacteristic
655C22#348811yescharacteristic
656C22#349711yescharacteristic
657C22#349811yescharacteristic
658C22#349911yescharacteristic
659C22#349912yes—
660C22#351312yes—
661C22#351612yes—
662C22#352212yes—
663C22#354611yescharacteristic
664C22#354712yes—
665C22#354811yescharacteristic
666C22#355212yes—
667C22#355512yes—
668C22#356112yes—
669C22#356512yes—
670C4—24no—
671C4—28no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—42no—
676C4—44no—
677C4—44no—
678C4—42no—
679C4—81no—
680C4—84no—
681C4—82no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4#262518yescomplemented
686C2—84no—
687C2C2×#262511yescharacteristic
688C2#3599611yescharacteristic
689C2D4×#12311yescharacteristic
690C2#3934111yescharacteristic
691C2#3936011yescharacteristic
692C2#3939912yes—
693C2#3940612yes—
694C2#4257412yes—
695C2#4262612yes—
696C2#4264911yescharacteristic
697C2#4268111yescharacteristic
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. #16399C2semidirect
5. #16365C2semidirect
6. #16314C2semidirect
7. #16278C2semidirect
8. #15156C2semidirect
9. #15151C2semidirect
10. #15145C2semidirect
13. #14906C2semidirect
14. #14053C2semidirect
15. #14005C2semidirect
17. C4×#400C2semidirect
133. #400D4semidirect
685. C4#2625semidirect

Maximal subgroups

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

subgroupindex
C2×#24522
C2×#26252
C2×#23542
#163992
#163652
#163142
#162782
#151562
#151512
#151452
#151382
#151362
#149062
#140532
#140052
#139202
C4×#4002