Finite Group Databasebeta

← #4389632

Subgroups — #4389632

Order 512.

3334 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2432C211yesmaximal, characteristic
2C2×#2616C211yesmaximal, characteristic
3C2×#2659C211yesmaximal, characteristic, complemented
4C2×#2332C211yesmaximal, characteristic, complemented
5#16806C212yesmaximal, complemented
6#16505C212yesmaximal, complemented
7#15769C214yesmaximal, complemented
8#15303C214yesmaximal, complemented
9#15302C214yesmaximal, complemented
10#15276C214yesmaximal, complemented
11#15163C212yesmaximal, complemented
12#15128C212yesmaximal, complemented
13#14660C211yesmaximal, characteristic, complemented
14#13965C211yesmaximal, characteristic, complemented
15#13956C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C4⋊Q8C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22#2659—28no—
23#2616—28no—
24#2432—28no—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#396C2214yes—
28C2×#396C2214yes—
29C2×#395C2212yes—
30C2×#395C2212yes—
31C2×#395C2212yes—
32C2×#394C2212yes—
33C2×#394C2214yes—
34C2×#394C2214yes—
35#2332—28no—
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#389C2214yes—
40C2×#388C2212yes—
41C2×#388C2212yes—
42C2×#388C2214yes—
43C2×#387C2212yes—
44C2×#387C2212yes—
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#385C2214yes—
48C2×#385C2214yes—
49C2×#381C2211yescharacteristic
50C2×#380C2214yes—
51C2×#380C2211yescharacteristic
52C2×#380C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C22×#95C2211yescharacteristic
56#1446C2212yes—
57#1446C2212yes—
58#1445C2212yes—
59#1444C2214yes—
60#1444C2214yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1440C2212yes—
68#1440C2212yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2212yes—
74#1437C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2214yes—
77C4⋊#95C2212yes—
78C22⋊#95C2214yes—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—24no—
82C22×#41—44no—
83C22×#41—44no—
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2312yes—
88C23×D4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2311yescharacteristic
92C2×C4⋊Q8—24no—
93C2×C4⋊Q8—44no—
94C2×#124—24no—
95C2×#124—44no—
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—28no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—44no—
103C2×C22⋊Q8—44no—
104C2×C22⋊Q8D418yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#398—28no—
135#398—28no—
136#396—48no—
137#396—48no—
138#395—28no—
139#395—28no—
140#395—28no—
141#394—28no—
142#394—216no—
143#394—216no—
144#391—24no—
145#391—28no—
146#390—24no—
147#389—48no—
148#388—28no—
149#388—28no—
150#388—48no—
151#387—28no—
152#387—28no—
153#386—44no—
154#386—44no—
155#385—216no—
156#385—48no—
157#381—24no—
158#380—24no—
159#380—28no—
160#380—216no—
161C4×C4⋊C4—28no—
162C4×C22⋊C4D418yes—
163C2×#95—24no—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C25C2412yes—
187C25C2411yescharacteristic
188C2×#41—432no—
189C2×#41—416no—
190C2×#41—432no—
191C2×#41—416no—
192C2×#41—416no—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—24no—
202C22×Q8—42no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8C2×D414yes—
208C22×D4—24no—
209C22×D4—24no—
210C22×D4—24no—
211C22×D4—24no—
212C22×D4—24no—
213C22×D4—22no—
214C22×D4—22no—
215C22×D4—22no—
216C22×D4—24no—
217C22×D4—42no—
218C22×D4—42no—
219C22×D4—42no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—28no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4#4114yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C4⋊Q8—416no—
251#124—416no—
252C22⋊Q8—232no—
253C22⋊Q8—432no—
254C22⋊Q8—416no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—28no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—28no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—28no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—44no—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4C2×D418yes—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—28no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—28no—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4C2×D418yes—
325C2×C22⋊C4C2×D418yes—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—24no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—44no—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—24no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—28no—
349#95—28no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95C2×D418yes—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24C22×D412yes—
373C24C22×D412yes—
374C24C2×#4112yes—
375C24H(5,𝔽2)14yes—
376C24C2511yescharacteristic
377#41—4128no—
378C2×Q8—28no—
379C2×Q8—28no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—44no—
387C2×Q8—44no—
388C2×Q8—82no—
389C2×Q8—82no—
390C2×Q8—84no—
391C2×Q8—84no—
392C2×Q8—84no—
393C2×D4—28no—
394C2×D4—216no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—44no—
400C2×D4—48no—
401C2×D4—48no—
402C2×D4—48no—
403C2×D4—48no—
404C2×D4—44no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—216no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—216no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—48no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4—84no—
479C22×C4—84no—
480C22×C4#121116yes—
481C22×C4C22×D414yes—
482C22×C4C22×D414yes—
483C22×C4C22×D414yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D414yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C4H(5,𝔽2)14yes—
493C22×C42−1+418yes—
494C22×C42−1+414yes—
495C4⋊C4—28no—
496C4⋊C4—216no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—416no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—48no—
504C4⋊C4—84no—
505C4⋊C4—88no—
506C4⋊C4—88no—
507C4⋊C4—88no—
508C22⋊C4—216no—
509C22⋊C4—216no—
510C22⋊C4—416no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—48no—
514C22⋊C4—416no—
515C22⋊C4—88no—
516C22⋊C4—88no—
517C22⋊C4—82no—
518C22⋊C4—82no—
519C22⋊C4—84no—
520C42—44no—
521C42—44no—
522C42—48no—
523C42—84no—
524C23—24no—
525C23—22no—
526C23—22no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—24no—
532C23—22no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—28no—
539C23—28no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23C2×#12114yes—
563C23C23×D411yescharacteristic
564C23C23×D411yescharacteristic
565C23C22×#4111yescharacteristic
566C23C2×H(5,𝔽2)12yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)12yes—
569C23C2×2−1+412yes—
570C23C2×2−1+414yes—
571Q8—416no—
572Q8—88no—
573Q8—88no—
574D4—416no—
575D4—432no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—416no—
603C2×C4—432no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—816no—
649C2×C4C2×#12114yes—
650C2×C4C2×#12114yes—
651C2×C4#53718yes—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—48no—
664C22—48no—
665C22—82no—
666C22—84no—
667C22—82no—
668C22—82no—
669C22—84no—
670C22—82no—
671C22C22×#12111yescharacteristic
672C22C2×#53712yes—
673C22#348412yes—
674C22#348411yescharacteristic
675C22#348411yescharacteristic
676C22#348612yes—
677C22#349012yes—
678C22#349811yescharacteristic
679C22#349912yes—
680C22#349911yescharacteristic
681C22#350912yes—
682C22#351112yes—
683C22#353614yes—
684C22#353614yes—
685C22#354414yes—
686C22#355214yes—
687C4—24no—
688C4—24no—
689C4—28no—
690C4—44no—
691C4—44no—
692C4—48no—
693C4—44no—
694C4—44no—
695C4—82no—
696C4—84no—
697C4—82no—
698C4—82no—
699C4—82no—
700C2—44no—
701C2—84no—
702C2#3527311yescharacteristic
703C2#3534811yescharacteristic
704C2#3566711yescharacteristic
705C2#4049312yes—
706C2#4242812yes—
707C2#4635612yes—
708C2#4642612yes—
709C2#4801414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2659C2semidirect
4. C2×#2332C2semidirect
5. #16806C2semidirect
6. #16505C2semidirect
7. #15769C2semidirect
8. #15303C2semidirect
9. #15302C2semidirect
10. #15276C2semidirect
11. #15163C2semidirect
12. #15128C2semidirect
13. #14660C2semidirect
14. #13965C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24322
C2×#26162
C2×#26592
C2×#23322
#168062
#165052
#157692
#153032
#153022
#152762
#151632
#151282
#146602
#139652
#139562