Finite Group Databasebeta

← #2737061

Subgroups — #2737061

Order 512.

2486 conjugacy classes of subgroups.

The subgroup lattice has 736 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2490C211yesmaximal, characteristic
3#17158C214yesmaximal, complemented
4#17148C214yesmaximal, complemented
5#17077C212yesmaximal, complemented
6#16704C212yesmaximal, complemented
7#16691C212yesmaximal, complemented
8#15243C211yesmaximal, characteristic, complemented
9#15238C212yesmaximal, complemented
10#15226C212yesmaximal, complemented
11#15215C212yesmaximal
12#15179C212yesmaximal, complemented
13#15178C211yesmaximal, characteristic, complemented
14#15157C211yesmaximal, characteristic, complemented
15#14625C212yesmaximal, complemented
16#13933C211yesmaximal, characteristic, complemented
17C4×#393C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#124C2212yes—
21C22×C22⋊Q8C2212yes—
22C22×#121C2211yescharacteristic
23C22×C4×Q8C2211yescharacteristic
24#2568—28no—
25#2490—28no—
26C2×#400C2214yes—
27C2×#397C2212yes—
28C2×#394C2212yes—
29C2×#393C2212yes—
30C2×#393C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#390C2212yes—
37C2×#390C2211yescharacteristic
38C2×#390C2211yescharacteristic
39C2×#390C2212yes—
40C2×#389C2212yes—
41C2×#388C2212yes—
42C2×#382C2212yes—
43C2×#380C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2212yes—
47#1446C2211yescharacteristic
48#1444C2212yes—
49#1444C2212yes—
50#1442C2212yes—
51#1442C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2211yescharacteristic
57#1442C2214yes—
58#1441C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2211yescharacteristic
69#1439C2212yes—
70#1439C2212yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2212yes—
75#1439C2214yes—
76#1438C2214yes—
77#1438C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—44no—
92C2×#125—24no—
93C2×#125—44no—
94C2×#124—44no—
95C2×#124—44no—
96C2×#124—44no—
97C2×C22⋊Q8—44no—
98C2×C22⋊Q8—44no—
99C2×C22⋊Q8—44no—
100C2×#121—24no—
101C2×#121—44no—
102C2×C4×Q8—24no—
103C2×C4×Q8—44no—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136#400—216no—
137#397—28no—
138#394—28no—
139#393—24no—
140#393—28no—
141#391—28no—
142#391—28no—
143#391—28no—
144#391—44no—
145#391—44no—
146#390—24no—
147#390—24no—
148#390—28no—
149#390—44no—
150#389—44no—
151#388—28no—
152#382—28no—
153#380—24no—
154C4×C4⋊C4D418yes—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—44no—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—42no—
197C22×Q8—42no—
198C22×Q8#4114yes—
199C22×D4—24no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—44no—
207C22×D4—42no—
208C22×D4—44no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4C2414yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C4⋊Q8—416no—
230#125—416no—
231#124—432no—
232C22⋊Q8—432no—
233#121—416no—
234C4×Q8—416no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
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—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—41no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
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⋊C4—44no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
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⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—44no—
314C2×C42C2×D414yes—
315C2×C42C2×D418yes—
316#95—22no—
317#95—22no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—42no—
337#95—42no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95C2×D418yes—
344C24—21no—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—22no—
352C24—21no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—22no—
358C24—24no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—24no—
364C24—22no—
365C242−1+412yes—
366C24C2511yescharacteristic
367C2×Q8—28no—
368C2×Q8—48no—
369C2×Q8—44no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—48no—
373C2×Q8—48no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C2×D4—84no—
387C2×D4—88no—
388C2×D4—88no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C2×#4114yes—
475C22×C4C2×#4112yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C4H(5,𝔽2)14yes—
478C22×C4H(5,𝔽2)12yes—
479C22×C42−1+412yes—
480C22×C42−1+412yes—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—416no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—88no—
493C22⋊C4—44no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—416no—
500C22⋊C4—44no—
501C22⋊C4—48no—
502C22⋊C4—84no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—84no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C42—28no—
509C42—48no—
510C42—48no—
511C42—44no—
512C42—48no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—42no—
553C23—44no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23C23×D411yescharacteristic
560C23C22×#4111yescharacteristic
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)12yes—
563C23C2×H(5,𝔽2)11yescharacteristic
564C23C2×2−1+411yescharacteristic
565C23C2×2−1+411yescharacteristic
566C23C2×2−1+411yescharacteristic
567C23#58512yes—
568C23#58511yescharacteristic
569C23#58512yes—
570Q8—416no—
571Q8—816no—
572D4—816no—
573D4—88no—
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—24no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—41no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—44no—
651C2×C4—42no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—88no—
656C2×C4#53514yes—
657C2×C4#53514yes—
658C2×C4#53714yes—
659C2×C4#54814yes—
660C2×C4H(3,𝔽4)18yes—
661C2×C4#56218yes—
662C22—44no—
663C22—44no—
664C22—42no—
665C22—42no—
666C22—42no—
667C22—82no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—82no—
672C22—81no—
673C22—84no—
674C22—82no—
675C22—82no—
676C22—82no—
677C22—84no—
678C22—84no—
679C22—82no—
680C22—84no—
681C22—82no—
682C22—82no—
683C22—82no—
684C22C2×#53511yescharacteristic
685C22C2×#53511yescharacteristic
686C22C2×#53711yescharacteristic
687C22C2×#54811yescharacteristic
688C22#348511yescharacteristic
689C22#348911yescharacteristic
690C22#350011yescharacteristic
691C22#350512yes—
692C22#351711yescharacteristic
693C22#351712yes—
694C22#352311yescharacteristic
695C22C2×H(3,𝔽4)12yes—
696C22C2×#56212yes—
697C22#354112yes—
698C22#354112yes—
699C22#355112yes—
700C22#355211yescharacteristic
701C22#355412yes—
702C22#355511yescharacteristic
703C22#355712yes—
704C22#356112yes—
705C22#356212yes—
706C22#356212yes—
707C4—24no—
708C4—44no—
709C4—44no—
710C4—42no—
711C4—44no—
712C4—82no—
713C4—84no—
714C4—81no—
715C4—82no—
716C4—81no—
717C4—84no—
718C4—82no—
719C4—82no—
720C4—81no—
721C4#281918yes—
722C2—82no—
723C2—162no—
724C2C2×#281911yescharacteristic
725C2#3912711yescharacteristic
726C2#3917411yescharacteristic
727C2#3972211yescharacteristic
728C2#4264511yescharacteristic
729C2#4268011yescharacteristic
730C2#4291011yescharacteristic
731C2#4570312yes—
732C2#4608412yes—
733C2#5313812yes—
734C2#5314412yes—
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×#2568C2semidirect
3. #17158C2semidirect
4. #17148C2semidirect
5. #17077C2semidirect
6. #16704C2semidirect
7. #16691C2semidirect
8. #15243C2semidirect
9. #15238C2semidirect
10. #15226C2semidirect
12. #15179C2semidirect
13. #15178C2semidirect
14. #15157C2semidirect
15. #14625C2semidirect
16. #13933C2semidirect
17. C4×#393C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#24902
#171582
#171482
#170772
#167042
#166912
#152432
#152382
#152262
#152152
#151792
#151782
#151572
#146252
#139332
C4×#3932