Finite Group Databasebeta

← #4307606

Subgroups — #4307606

Order 512.

2886 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2367C212yesmaximal
2C2×#2365C211yesmaximal, characteristic, complemented
3C2×#2357C211yesmaximal, characteristic
4#15243C211yesmaximal, characteristic
5#15237C212yesmaximal, complemented
6#15226C212yesmaximal
7#15204C212yesmaximal, complemented
8#15180C211yesmaximal, characteristic, complemented
9#15147C211yesmaximal, characteristic
10#15081C212yesmaximal, complemented
11#14988C214yesmaximal, complemented
12#14972C212yesmaximal, complemented
13#14965C214yesmaximal, complemented
14C4×#398C211yesmaximal, characteristic, complemented
15C4×#397C212yesmaximal, complemented
16#13818C211yesmaximal, characteristic, complemented
17#13817C212yesmaximal, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×Q8C2212yes—
21C22×C4×D4C2211yescharacteristic
22C2×#398C2211yescharacteristic
23C2×#397C2212yes—
24#2367—216no—
25#2365—28no—
26#2357—28no—
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#388C2214yes—
40C2×#387C2214yes—
41C2×#385C2214yes—
42C2×#382C2211yescharacteristic
43C2×#381C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2214yes—
53#1445C2211yescharacteristic
54#1444C2211yescharacteristic
55#1443C2214yes—
56#1443C2214yes—
57#1442C2212yes—
58#1442C2212yes—
59#1442C2212yes—
60#1441C2211yescharacteristic
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2214yes—
65#1441C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2212yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2214yes—
82C4×#95C2212yes—
83C4×#95C2214yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×#124—28no—
92C2×#124—28no—
93C2×#124—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—28no—
96C2×C4×Q8—28no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4D418yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#398—24no—
133#397—28no—
134#391—28no—
135#391—28no—
136#391—28no—
137#391—28no—
138#390—28no—
139#390—24no—
140#390—24no—
141#389—28no—
142#389—28no—
143#388—216no—
144#388—216no—
145#388—28no—
146#388—28no—
147#387—216no—
148#385—216no—
149#382—24no—
150#381—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—28no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4D418yes—
158C4×C22⋊C4—216no—
159C4×C22⋊C4—28no—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C25C2412yes—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8#4118yes—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4C2×D414yes—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4C2×D418yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C4⋊Q8—232no—
225#124—264no—
226C4×Q8—432no—
227C4×D4—232no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—41no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D414yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4#4114yes—
264C2×C4⋊C4#4114yes—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C2×D418yes—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—24no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D418yes—
312C2×C42#4118yes—
313#95—22no—
314#95—24no—
315#95—24no—
316#95—28no—
317#95—22no—
318#95—24no—
319#95—28no—
320#95—24no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—22no—
337#95—42no—
338#95—42no—
339#95—42no—
340#95—42no—
341#95—44no—
342#95C2×D418yes—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24C22×D414yes—
354C24C2511yescharacteristic
355C2×Q8—216no—
356C2×Q8—216no—
357C2×Q8—216no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—44no—
365C2×D4—48no—
366C2×D4—44no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—216no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—416no—
443C22×C4—84no—
444C22×C4—84no—
445C22×C4—88no—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C2×#4114yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4114yes—
458C22×C4C2×#4118yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4114yes—
464C22×C42−1+412yes—
465C22×C42−1+412yes—
466C4⋊C4—216no—
467C4⋊C4—216no—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—416no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—88no—
479C4⋊C4#121116yes—
480C22⋊C4—232no—
481C22⋊C4—232no—
482C22⋊C4—216no—
483C22⋊C4—48no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—84no—
487C22⋊C4—84no—
488C22⋊C4—88no—
489C42—28no—
490C42—28no—
491C42—216no—
492C42—48no—
493C42—416no—
494C42—48no—
495C42—44no—
496C42—88no—
497C42—84no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—28no—
502C23—24no—
503C23—42no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—44no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—42no—
519C23—44no—
520C23—44no—
521C23C23×D411yescharacteristic
522C23C23×D411yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4112yes—
525C23C22×#4112yes—
526C23C22×#4111yescharacteristic
527C23C2×2−1+411yescharacteristic
528C23C2×2−1+411yescharacteristic
529C23#58511yescharacteristic
530C23#58512yes—
531C23#58512yes—
532Q8—432no—
533D4—416no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—28no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
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—24no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—816no—
622C2×C4C2×#12118yes—
623C2×C4C2×#12114yes—
624C2×C4#52514yes—
625C2×C4#52518yes—
626C2×C4#52918yes—
627C2×C4#53214yes—
628C2×C4#54814yes—
629C22—48no—
630C22—48no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—82no—
635C22—84no—
636C22—84no—
637C22—82no—
638C22—82no—
639C22—82no—
640C22C22×#12111yescharacteristic
641C22C2×#52512yes—
642C22C2×#52511yescharacteristic
643C22C2×#52912yes—
644C22C2×#53211yescharacteristic
645C22C2×#54811yescharacteristic
646C22D4×#4111yescharacteristic
647C22#348012yes—
648C22#348012yes—
649C22#348211yescharacteristic
650C22#348512yes—
651C22#348511yescharacteristic
652C22#348911yescharacteristic
653C22#349911yescharacteristic
654C22#350011yescharacteristic
655C22#350712yes—
656C22#350712yes—
657C22#351012yes—
658C22#351612yes—
659C22#351612yes—
660C22#351811yescharacteristic
661C22#352112yes—
662C22#352412yes—
663C4—28no—
664C4—24no—
665C4—44no—
666C4—44no—
667C4—42no—
668C4—44no—
669C4—44no—
670C4—42no—
671C4—42no—
672C4—42no—
673C4—48no—
674C4—84no—
675C4—82no—
676C4—82no—
677C4—81no—
678C4#239218yes—
679C2—84no—
680C2C2×#239211yescharacteristic
681C2#3584612yes—
682C2#3595911yescharacteristic
683C2#3597412yes—
684C2#3598511yescharacteristic
685C2#3600911yescharacteristic
686C2#3643712yes—
687C2#3644411yescharacteristic
688C2#3648512yes—
689C2#3653811yescharacteristic
690C2#3658211yescharacteristic
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×#2365C2semidirect
5. #15237C2semidirect
7. #15204C2semidirect
8. #15180C2semidirect
10. #15081C2semidirect
11. #14988C2semidirect
12. #14972C2semidirect
13. #14965C2semidirect
14. C4×#398C2semidirect
15. C4×#397C2semidirect
16. #13818C2semidirect
17. #13817C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23672
C2×#23652
C2×#23572
#152432
#152372
#152262
#152042
#151802
#151472
#150812
#149882
#149722
#149652
C4×#3982
C4×#3972
#138182
#138172