Finite Group Databasebeta

← #4416308

Subgroups — #4416308

Order 512.

2500 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C2×#2290C211yesmaximal, characteristic
3#17067C211yesmaximal, characteristic, complemented
4#17066C211yesmaximal, characteristic, complemented
5#16986C212yesmaximal, complemented
6#16884C212yesmaximal
7#16507C211yesmaximal, characteristic, complemented
8#16506C211yesmaximal, characteristic, complemented
9#16395C212yesmaximal
10#16393C212yesmaximal, complemented
11#15122C212yesmaximal, complemented
12#15108C212yesmaximal, complemented
13#15001C212yesmaximal, complemented
14#14969C212yesmaximal, complemented
15#14907C212yesmaximal, complemented
16#14723C212yesmaximal
17#14684C211yesmaximal, characteristic, complemented
18#14626C212yesmaximal, complemented
19#14625C212yesmaximal, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2212yes—
22C22×C4×D4C2211yescharacteristic
23C22×#117C2211yescharacteristic
24C23×C42C2211yescharacteristic
25#2857—28no—
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#395C2212yes—
29C2×#393C2212yes—
30C2×#391C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33#2290—28no—
34C2×#388C2211yescharacteristic
35C2×#388C2211yescharacteristic
36C2×#387C2211yescharacteristic
37C2×#387C2211yescharacteristic
38C2×#387C2214yes—
39C2×#386C2211yescharacteristic
40C2×#386C2211yescharacteristic
41C2×#385C2211yescharacteristic
42C2×#385C2211yescharacteristic
43C2×#385C2214yes—
44C2×#384C2211yescharacteristic
45C2×#381C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×#376C2211yescharacteristic
52#1446C2212yes—
53#1446C2212yes—
54#1445C2212yes—
55#1445C2214yes—
56#1445C2212yes—
57#1444C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1443C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2212yes—
65#1442C2212yes—
66#1442C2212yes—
67#1442C2214yes—
68#1442C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2212yes—
74#1440C2214yes—
75#1440C2214yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2212yes—
79#1439C2214yes—
80#1439C2212yes—
81#1439C2212yes—
82#1439C2212yes—
83#1438C2214yes—
84#1436C2214yes—
85#1436C2214yes—
86#1434C2212yes—
87#1434C2212yes—
88#1434C2212yes—
89#1434C2211yescharacteristic
90#1434C2211yescharacteristic
91C22⋊#95C2212yes—
92C22⋊#95C2212yes—
93C23×Q8C2312yes—
94C23×D4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C24×C4C2311yescharacteristic
99C2×C4⋊Q8—24no—
100C2×C4⋊Q8—44no—
101C2×C22⋊Q8—28no—
102C2×C22⋊Q8—48no—
103C2×C4×D4—24no—
104C2×C4×D4—24no—
105C2×C4×D4—24no—
106C2×#117—24no—
107C2×#117—44no—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2314yes—
133C22×C42—44no—
134C22×C42D418yes—
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142#398—44no—
143#397—44no—
144#395—28no—
145#393—28no—
146#391—28no—
147#389—28no—
148#389—28no—
149#388—24no—
150#388—24no—
151#387—24no—
152#387—24no—
153#387—48no—
154#386—24no—
155#386—24no—
156#385—24no—
157#385—24no—
158#385—48no—
159#384—24no—
160#381—44no—
161#380—28no—
162#380—28no—
163C4×C4⋊C4—24no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—24no—
166#376—24no—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C25C2411yescharacteristic
201C25C2411yescharacteristic
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—24no—
210C22×D4—22no—
211C22×D4—42no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2414yes—
236C23×C4C2414yes—
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2411yescharacteristic
247C4⋊Q8—416no—
248C22⋊Q8—432no—
249C4×D4—232no—
250#117—416no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4#4114yes—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4#4114yes—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—22no—
319C2×C42—24no—
320C2×C42—416no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—28no—
339#95—28no—
340#95—24no—
341#95—28no—
342#95—24no—
343#95—28no—
344#95—24no—
345#95—24no—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24C22×D412yes—
377C24C2×#4112yes—
378C242−1+414yes—
379C24C2511yescharacteristic
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×Q8—88no—
384C2×Q8—88no—
385C2×D4—28no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—44no—
389C2×D4—44no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—48no—
475C22×C4—416no—
476C22×C4—44no—
477C22×C4—416no—
478C22×C4—44no—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4112yes—
489C22×C4C2×#4112yes—
490C22×C4H(5,𝔽2)14yes—
491C22×C4H(5,𝔽2)14yes—
492C22×C42−1+414yes—
493C4⋊C4—28no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—44no—
502C4⋊C4—48no—
503C4⋊C4—416no—
504C4⋊C4—44no—
505C4⋊C4—88no—
506C22⋊C4—28no—
507C22⋊C4—28no—
508C22⋊C4—44no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4—416no—
512C22⋊C4—416no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—88no—
517C22⋊C4—88no—
518C42—28no—
519C42—28no—
520C42—28no—
521C42—44no—
522C42—44no—
523C42—44no—
524C42—44no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—22no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23#53614yes—
559C23C23×D411yescharacteristic
560C23C22×#4111yescharacteristic
561C23C22×#4111yescharacteristic
562C23C2×H(5,𝔽2)12yes—
563C23C2×H(5,𝔽2)12yes—
564C23C2×2−1+412yes—
565C23C2×2−1+412yes—
566C23#58512yes—
567C23#58512yes—
568Q8—816no—
569D4—416no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—416no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
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—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4#52514yes—
656C2×C4#53414yes—
657C2×C4#53514yes—
658C2×C4#53614yes—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22—48no—
667C22—48no—
668C22—42no—
669C22—42no—
670C22—48no—
671C22—42no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—84no—
676C22—81no—
677C22C2×#52511yescharacteristic
678C22C2×#53411yescharacteristic
679C22C2×#53511yescharacteristic
680C22C2×#53611yescharacteristic
681C22C2×#53611yescharacteristic
682C22#344911yescharacteristic
683C22#344911yescharacteristic
684C22#349512yes—
685C22#349512yes—
686C22#351512yes—
687C22#351812yes—
688C22#351912yes—
689C22#351912yes—
690C22#352914yes—
691C22#353514yes—
692C22#354114yes—
693C22#355214yes—
694C4—24no—
695C4—24no—
696C4—24no—
697C4—24no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—84no—
704C4—81no—
705C4—81no—
706C4—81no—
707C4—84no—
708C4—84no—
709C4—84no—
710C4—81no—
711C2—44no—
712C2—82no—
713C2#3507611yescharacteristic
714C2#3520011yescharacteristic
715C2#3532511yescharacteristic
716C2#3668711yescharacteristic
717C2#3744711yescharacteristic
718C2#3744911yescharacteristic
719C2#3761511yescharacteristic
720C2#4602714yes—
721C2#4602914yes—
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×#2857C2semidirect
3. #17067C2semidirect
4. #17066C2semidirect
5. #16986C2semidirect
7. #16507C2semidirect
8. #16506C2semidirect
10. #16393C2semidirect
11. #15122C2semidirect
12. #15108C2semidirect
13. #15001C2semidirect
14. #14969C2semidirect
15. #14907C2semidirect
17. #14684C2semidirect
18. #14626C2semidirect
19. #14625C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C2×#22902
#170672
#170662
#169862
#168842
#165072
#165062
#163952
#163932
#151222
#151082
#150012
#149692
#149072
#147232
#146842
#146262
#146252