Finite Group Databasebeta

← #642691

Subgroups — #642691

Order 512.

2546 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2C2×#2295C211yesmaximal, characteristic, complemented
3#16896C214yesmaximal, complemented
4#16566C212yesmaximal, complemented
5#16564C212yesmaximal, complemented
6#15204C212yesmaximal, complemented
7#15201C211yesmaximal, characteristic, complemented
8#15193C211yesmaximal, characteristic, complemented
9#15161C212yesmaximal, complemented
10#15157C211yesmaximal, characteristic
11#15148C211yesmaximal, characteristic
12#14923C214yesmaximal, complemented
13#14687C211yesmaximal, characteristic, complemented
14#14276C212yesmaximal, complemented
15#14273C212yesmaximal
16#14018C212yesmaximal
17#14017C212yesmaximal, complemented
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C2×#399C2212yes—
23C2×#393C2212yes—
24C4×#124—28no—
25#2295—28no—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2211yescharacteristic
30C2×#391C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2212yes—
39C2×#388C2212yes—
40C2×#387C2212yes—
41C2×#387C2212yes—
42C2×#385C2214yes—
43C2×#384C2212yes—
44C2×#382C2211yescharacteristic
45C2×#382C2212yes—
46C2×#382C2212yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C4×C22⋊C4C2212yes—
51C2×#376C2212yes—
52C2×C43C2211yescharacteristic
53#1446C2212yes—
54#1445C2214yes—
55#1445C2212yes—
56#1444C2214yes—
57#1444C2212yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2212yes—
73#1439C2214yes—
74#1439C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#124—24no—
90C2×#124—24no—
91C2×#124Q818yescomplemented
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—28no—
95C2×C4×Q8—44no—
96C2×C4×D4—24no—
97C2×C4×D4—44no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#399—44no—
133#393—44no—
134#391—24no—
135#391—24no—
136#391—24no—
137#391—28no—
138#391—44no—
139#390—24no—
140#390—28no—
141#390—44no—
142#389—24no—
143#389—24no—
144#389—44no—
145#389—44no—
146#388—28no—
147#388—28no—
148#387—28no—
149#387—28no—
150#385—48no—
151#384—28no—
152#382—24no—
153#382—44no—
154#382—44no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—28no—
159#376—28no—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C43D418yes—
188C25C2412yes—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—24no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—42no—
198C22×Q8C2×Q814yes—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4C2×Q814yes—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225#124—232no—
226C4×Q8—232no—
227C4×Q8—416no—
228C4×D4—416no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
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×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×Q818yes—
293C2×C22⋊C4C2×Q818yes—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—44no—
307C2×C42—42no—
308C2×C42—42no—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312C2×C42C2×D414yes—
313C2×C42C2×D414yes—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×Q814yes—
317C2×C42#4118yes—
318C2×C42#4114yes—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—44no—
336#95—44no—
337#95—42no—
338#95—44no—
339#95—42no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24C22×Q814yes—
357C24C2511yescharacteristic
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—216no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—44no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—44no—
367C2×Q8—48no—
368C2×D4—28no—
369C2×D4—216no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—48no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q812yes—
459C22×C4C22×Q812yes—
460C22×C4C22×Q812yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4114yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)14yes—
470C22×C42−1+412yes—
471C22×C42−1+412yes—
472C4⋊C4—28no—
473C4⋊C4—216no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C22⋊C4—216no—
488C22⋊C4—216no—
489C22⋊C4—48no—
490C22⋊C4—416no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C42—28no—
497C42—216no—
498C42—28no—
499C42—28no—
500C42—28no—
501C42—44no—
502C42—44no—
503C42—48no—
504C42—48no—
505C42C22⋊Q8116yes—
506C23—24no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—44no—
518C23—44no—
519C23—44no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23C23×D411yescharacteristic
530C23C23×Q811yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4112yes—
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×2−1+411yescharacteristic
537C23C2×2−1+411yescharacteristic
538C23#58512yes—
539C23#58512yes—
540Q8—416no—
541Q8—416no—
542D4—416no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—42no—
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—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—816no—
629C2×C4C2×C22⋊Q814yes—
630C2×C4C2×C22⋊Q814yes—
631C2×C4C2×C22⋊Q814yes—
632C2×C4#52514yes—
633C2×C4#53514yes—
634C2×C4#53714yes—
635C2×C4#53714yes—
636C2×C4#55418yes—
637C2×C4#55918yes—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—48no—
642C22—48no—
643C22—82no—
644C22—84no—
645C22—84no—
646C22—82no—
647C22—82no—
648C22—82no—
649C22C22×C22⋊Q811yescharacteristic
650C22C2×#52511yescharacteristic
651C22C2×#53511yescharacteristic
652C22C2×#53711yescharacteristic
653C22C2×#53711yescharacteristic
654C22#344911yescharacteristic
655C22#344911yescharacteristic
656C22#346012yes—
657C22#346112yes—
658C22C2×#55412yes—
659C22C2×#55912yes—
660C22#349412yes—
661C22#349512yes—
662C22#350311yescharacteristic
663C22#350311yescharacteristic
664C22#350412yes—
665C22#350811yescharacteristic
666C22#350911yescharacteristic
667C22#351012yes—
668C22#354612yes—
669C22#355412yes—
670C22#355812yes—
671C22#356112yes—
672C4—24no—
673C4—24no—
674C4—24no—
675C4—24no—
676C4—42no—
677C4—44no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—84no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4#258018yes—
688C2—84no—
689C2#3461411yescharacteristic
690C2C2×#258011yescharacteristic
691C2#3528011yescharacteristic
692C2#3530211yescharacteristic
693C2#3533411yescharacteristic
694C2#3754611yescharacteristic
695C2#3761611yescharacteristic
696C2#3967412yes—
697C2#3967512yes—
698C2#3981012yes—
699C2#3981112yes—
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×#2295C2semidirect
3. #16896C2semidirect
4. #16566C2semidirect
5. #16564C2semidirect
6. #15204C2semidirect
7. #15201C2semidirect
8. #15193C2semidirect
9. #15161C2semidirect
12. #14923C2semidirect
13. #14687C2semidirect
14. #14276C2semidirect
17. #14017C2semidirect
91. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
C2×#22952
#168962
#165662
#165642
#152042
#152012
#151932
#151612
#151572
#151482
#149232
#146872
#142762
#142732
#140182
#140172