Finite Group Databasebeta

← #1409868

Subgroups — #1409868

Order 512.

2524 conjugacy classes of subgroups.

The subgroup lattice has 739 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic, complemented
2C2×#2422C211yesmaximal, characteristic
3#16837C214yesmaximal, complemented
4#16815C212yesmaximal, complemented
5#16788C214yesmaximal, complemented
6#16360C212yesmaximal, complemented
7#15230C211yesmaximal, characteristic, complemented
8#15219C212yesmaximal
9#15156C211yesmaximal, characteristic, complemented
10#15147C212yesmaximal, complemented
11#15125C211yesmaximal, characteristic, complemented
12#14841C212yesmaximal, complemented
13#14287C212yesmaximal, complemented
14#13909C212yesmaximal, complemented
15#13875C212yesmaximal, complemented
16C4×#399C211yesmaximal, characteristic, complemented
17#13820C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#123C2212yes—
20C22×#121C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2563—28no—
23#2422—28no—
24C2×#400C2212yes—
25C2×#399C2211yescharacteristic
26C2×#397C2212yes—
27C2×#397C2214yes—
28C2×#394C2212yes—
29C2×#393C2214yes—
30C2×#393C2212yes—
31C2×#390C2211yescharacteristic
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#389C2211yescharacteristic
35C2×#385C2212yes—
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#382C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1446C2212yes—
50#1446C2214yes—
51#1445C2212yes—
52#1445C2211yescharacteristic
53#1444C2211yescharacteristic
54#1443C2214yes—
55#1443C2214yes—
56#1442C2212yes—
57#1441C2212yes—
58#1441C2214yes—
59#1441C2214yes—
60#1441C2212yes—
61#1441C2212yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2211yescharacteristic
70#1439C2214yes—
71#1438C2214yes—
72#1438C2214yes—
73#1437C2214yes—
74#1437C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2214yes—
83C22⋊#95C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—44no—
92C2×#123—28no—
93C2×#123—28no—
94C2×#123—44no—
95C2×#121—24no—
96C2×#121—44no—
97C2×C4×D4—44no—
98C2×C4×D4D418yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2314yes—
131#400—44no—
132#399—24no—
133#397—28no—
134#397—48no—
135#394—28no—
136#393—216no—
137#393—44no—
138#390—28no—
139#390—24no—
140#390—28no—
141#389—24no—
142#385—28no—
143#384—24no—
144#382—28no—
145#382—44no—
146#380—28no—
147#380—28no—
148#380—24no—
149#380—44no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—44no—
153C4×C22⋊C4—24no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—44no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C22×Q8—24no—
187C22×Q8—44no—
188C22×Q8—44no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—44no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4C2×D414yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4C2×D418yes—
211C23×C4C2414yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C4⋊Q8—416no—
227#123—432no—
228#121—416no—
229C4×D4—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D418yes—
303C2×C22⋊C4#4114yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42C2×D414yes—
316C2×C42#4118yes—
317#95—22no—
318#95—22no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—24no—
331#95—28no—
332#95—44no—
333#95—44no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345C24—21no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—21no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—24no—
362C24—21no—
363C24—24no—
364C24—22no—
365C24C22×D414yes—
366C24C2×#4112yes—
367C24C2511yescharacteristic
368C2×Q8—48no—
369C2×Q8—84no—
370C2×Q8—84no—
371C2×Q8—88no—
372C2×Q8—88no—
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—28no—
376C2×D4—48no—
377C2×D4—48no—
378C2×D4—48no—
379C2×D4—48no—
380C2×D4—84no—
381C2×D4—88no—
382C2×D4—88no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
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—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—216no—
414C22×C4—21no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4112yes—
476C22×C4C2×#4114yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C42−1+412yes—
481C22×C42−1+412yes—
482C4⋊C4—28no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—84no—
495C22⋊C4—28no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C42—28no—
510C42—216no—
511C42—44no—
512C42—48no—
513C42—88no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—44no—
531C23—42no—
532C23—44no—
533C23—44no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—41no—
541C23—44no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—41no—
549C23—42no—
550C23—41no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—44no—
555C23—42no—
556C23—44no—
557C23—42no—
558C23—44no—
559C23—42no—
560C23C23×D411yescharacteristic
561C23C22×#4112yes—
562C23C22×#4111yescharacteristic
563C23C2×H(5,𝔽2)12yes—
564C23C2×H(5,𝔽2)11yescharacteristic
565C23C2×H(5,𝔽2)11yescharacteristic
566C23C2×2−1+411yescharacteristic
567C23C2×2−1+411yescharacteristic
568C23#58512yes—
569C23#58512yes—
570C23#58511yescharacteristic
571Q8—816no—
572D4—416no—
573D4—816no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—41no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—816no—
658C2×C4—88no—
659C2×C4#53214yes—
660C2×C4#53414yes—
661C2×C4#53614yes—
662C2×C4#53918yes—
663C2×C4#54218yes—
664C2×C4#54614yes—
665C22—44no—
666C22—44no—
667C22—42no—
668C22—44no—
669C22—42no—
670C22—44no—
671C22—44no—
672C22—44no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—42no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22—84no—
682C22—82no—
683C22—84no—
684C22—82no—
685C22—84no—
686C22—84no—
687C22C2×#53211yescharacteristic
688C22C2×#53411yescharacteristic
689C22C2×#53611yescharacteristic
690C22C2×#53912yes—
691C22C2×#54212yes—
692C22C2×#54611yescharacteristic
693C22#347512yes—
694C22#348511yescharacteristic
695C22#348811yescharacteristic
696C22#349312yes—
697C22#349412yes—
698C22#350011yescharacteristic
699C22#350612yes—
700C22#350711yescharacteristic
701C22#351111yescharacteristic
702C22#351412yes—
703C22#354612yes—
704C22#354711yescharacteristic
705C22#354811yescharacteristic
706C22#355712yes—
707C22#355812yes—
708C22#356212yes—
709C22#356512yes—
710C4—24no—
711C4—42no—
712C4—42no—
713C4—42no—
714C4—44no—
715C4—44no—
716C4—44no—
717C4—81no—
718C4—82no—
719C4—82no—
720C4—84no—
721C4—82no—
722C4—84no—
723C4—82no—
724C4#266318yes—
725C2—84no—
726C2—82no—
727C2C2×#266311yescharacteristic
728C2#3606211yescharacteristic
729C2#3641511yescharacteristic
730C2#3890611yescharacteristic
731C2#3947011yescharacteristic
732C2#3990512yes—
733C2#3991212yes—
734C2#4246512yes—
735C2#4246712yes—
736C2#4264711yescharacteristic
737C2#4296711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2563C2semidirect
3. #16837C2semidirect
4. #16815C2semidirect
5. #16788C2semidirect
6. #16360C2semidirect
7. #15230C2semidirect
9. #15156C2semidirect
10. #15147C2semidirect
11. #15125C2semidirect
12. #14841C2semidirect
13. #14287C2semidirect
14. #13909C2semidirect
15. #13875C2semidirect
16. C4×#399C2semidirect
17. #13820C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#24222
#168372
#168152
#167882
#163602
#152302
#152192
#151562
#151472
#151252
#148412
#142872
#139092
#138752
C4×#3992
#138202