Finite Group Databasebeta

← #4930411

Subgroups — #4930411

Order 512.

2448 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2592C211yesmaximal, characteristic
3#15047C212yesmaximal, complemented
4#15006C214yesmaximal, complemented
5#14972C212yesmaximal, complemented
6#14956C214yesmaximal, complemented
7#14933C212yesmaximal, complemented
8#14933C212yesmaximal, complemented
9#14273C211yesmaximal, characteristic
10#14153C211yesmaximal, characteristic
11#14142C212yesmaximal, complemented
12#14141C212yesmaximal
13#14001C212yesmaximal
14#14000C212yesmaximal, complemented
15#13926C211yesmaximal, characteristic, complemented
16#13850C211yesmaximal, characteristic, complemented
17#13750C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2592—28no—
22#2552—28no—
23C2×#398C2214yes—
24C2×#397C2212yes—
25C2×#393C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#384C2211yescharacteristic
43C2×#384C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×#376C2211yescharacteristic
50#1446C2212yes—
51#1445C2212yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1441C2211yescharacteristic
57#1441C2211yescharacteristic
58#1441C2211yescharacteristic
59#1441C2211yescharacteristic
60#1441C2212yes—
61#1441C2212yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79#1434C2214yes—
80C4×#95C2212yes—
81C4×#95C2212yes—
82C4×#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2214yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C2×#125—28no—
89C2×#125—48no—
90C2×#124—24no—
91C2×#124—24no—
92C2×#124—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#398—216no—
130#397—28no—
131#393—28no—
132#391—24no—
133#391—24no—
134#391—28no—
135#391—28no—
136#391—44no—
137#390—24no—
138#390—24no—
139#390—28no—
140#390—28no—
141#389—28no—
142#389—28no—
143#388—28no—
144#388—28no—
145#388—48no—
146#388—48no—
147#388—48no—
148#384—28no—
149#384—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4D418yes—
155#376—24no—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2311yescharacteristic
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C25C2412yes—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—41no—
197C22×Q8—41no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219#125—432no—
220#124—416no—
221C4×Q8—416no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
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—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4C2×D414yes—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4#4118yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—22no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—28no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42—44no—
308C2×C42C2×D418yes—
309C2×C42C2×D414yes—
310C2×C42#4118yes—
311C2×C42#4114yes—
312C2×C42#4114yes—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—22no—
320#95—22no—
321#95—24no—
322#95—28no—
323#95—28no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—42no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95C2×D418yes—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—24no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24C2511yescharacteristic
354C2×Q8—28no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—44no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×D4—44no—
367C2×D4—48no—
368C2×D4—48no—
369C2×D4—44no—
370C2×D4—44no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4118yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C42−1+414yes—
468C22×C42−1+412yes—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—216no—
472C4⋊C4—28no—
473C4⋊C4—28no—
474C4⋊C4—416no—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C4⋊C4—88no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—84no—
491C22⋊C4—84no—
492C22⋊C4—88no—
493C42—216no—
494C42—48no—
495C42—44no—
496C42—44no—
497C42—48no—
498C42—84no—
499C42—88no—
500C42—88no—
501C42#124116yes—
502C23—42no—
503C23—44no—
504C23—44no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23C23×D411yescharacteristic
526C23C22×#4111yescharacteristic
527C23C22×#4111yescharacteristic
528C23C22×#4112yes—
529C23C22×#4112yes—
530C23C2×H(5,𝔽2)12yes—
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×2−1+412yes—
533C23C2×2−1+411yescharacteristic
534C23#58511yescharacteristic
535C23#58511yescharacteristic
536Q8—416no—
537Q8—88no—
538D4—88no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4C2×#12414yes—
631C2×C4C2×#12414yes—
632C2×C4C2×#12414yes—
633C2×C4#52918yes—
634C2×C4#52918yes—
635C2×C4#53214yes—
636C2×C4#53214yes—
637C22—82no—
638C22—82no—
639C22—84no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—84no—
645C22—82no—
646C22—84no—
647C22—84no—
648C22C22×#12411yescharacteristic
649C22C2×#52912yes—
650C22C2×#52912yes—
651C22C2×#53211yescharacteristic
652C22C2×#53211yescharacteristic
653C22#348411yescharacteristic
654C22#348611yescharacteristic
655C22#348811yescharacteristic
656C22#348911yescharacteristic
657C22#349211yescharacteristic
658C22#349311yescharacteristic
659C22#349411yescharacteristic
660C22#349511yescharacteristic
661C22#350612yes—
662C22#350712yes—
663C22#350812yes—
664C22#350912yes—
665C22#351112yes—
666C22#351112yes—
667C22#351312yes—
668C22#351612yes—
669C22#352012yes—
670C22#352212yes—
671C4—24no—
672C4—24no—
673C4—44no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—42no—
679C4—42no—
680C4—48no—
681C4—44no—
682C4—81no—
683C4—81no—
684C4—82no—
685C4—84no—
686C4#286118yes—
687C2—162no—
688C2#3554211yescharacteristic
689C2#3556112yes—
690C2#3559211yescharacteristic
691C2#3561012yes—
692C2#3574512yes—
693C2#3583812yes—
694C2#3590911yescharacteristic
695C2#3591211yescharacteristic
696C2#3593011yescharacteristic
697C2#3593811yescharacteristic
698C2C2×#286111yescharacteristic
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
3. #15047C2semidirect
4. #15006C2semidirect
5. #14972C2semidirect
6. #14956C2semidirect
7. #14933C2semidirect
8. #14933C2semidirect
11. #14142C2semidirect
14. #14000C2semidirect
15. #13926C2semidirect
16. #13850C2semidirect
17. #13750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#25922
#150472
#150062
#149722
#149562
#149332
#142732
#141532
#141422
#141412
#140012
#140002
#139262
#138502
#137502