Finite Group Databasebeta

← #670331

Subgroups — #670331

Order 512.

4478 conjugacy classes of subgroups.

The subgroup lattice has 647 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2502C211yesmaximal, characteristic, complemented
2C2×#2435C211yesmaximal, characteristic
3C2×#2345C211yesmaximal, characteristic, complemented
4C22×#390C211yesmaximal, characteristic
5#15661C218yesmaximal, complemented
6#15226C212yesmaximal, complemented
7#15134C212yesmaximal, complemented
8#14726C214yesmaximal, complemented
9#14702C211yesmaximal, characteristic, complemented
10#14153C212yesmaximal, complemented
11#14001C212yesmaximal, complemented
12#13971C214yesmaximal, complemented
13#13902C211yesmaximal, characteristic
14C4×#385C211yesmaximal, characteristic, complemented
15C24×D4C2211yescharacteristic
16C22×#124C2212yes—
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2214yes—
20C23×C42C2211yescharacteristic
21#2502—28no—
22#2435—28no—
23C2×#398C2214yescomplemented
24C2×#396C2218yescomplemented
25C2×#394C2214yes—
26#2345—28no—
27C2×#391C2211yescharacteristic, complemented
28C2×#391C2211yescharacteristic, complemented
29C2×#391C2212yescomplemented
30C2×#390—28no—
31C2×#390C2212yes—
32C2×#390C2212yes—
33C2×#390C2214yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#390C2212yes—
37C2×#389C2212yescomplemented
38C2×#388C2218yescomplemented
39C2×#387C2214yescomplemented
40C2×#385C2212yes—
41C2×#385C2214yes—
42C2×#384C2211yescharacteristic, complemented
43C2×#384C2211yescharacteristic, complemented
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46C2×#379C2214yes—
47C2×#379C2212yes—
48C2×#379C2212yes—
49C2×C4×C4⋊C4C2212yescomplemented
50C2×C4×C4⋊C4C2212yescomplemented
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2211yescharacteristic
54#1440C2218yescomplemented
55#1439C2218yescomplemented
56#1439C2218yescomplemented
57#1439C2218yescomplemented
58#1439C2218yescomplemented
59#1438C2214yes—
60#1437C2218yescomplemented
61#1436C2214yescomplemented
62#1436C2214yescomplemented
63C4⋊#95C2212yescomplemented
64C22⋊#95C2214yes—
65C22⋊#95C2212yes—
66C4×#95C2212yescomplemented
67C26C2312yes—
68C23×Q8C2311yescharacteristic
69C23×D4—28no—
70C23×D4—24no—
71C23×D4—24no—
72C23×D4—24no—
73C23×D4—44no—
74C23×D4C2312yes—
75C23×D4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2312yes—
79C2×#124—28no—
80C2×#124—28no—
81C2×#124—44no—
82C2×C22⋊Q8—24no—
83C2×C22⋊Q8—24no—
84C2×C22⋊Q8—24no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4D418yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2314yes—
98C22×C22⋊C4—216no—
99C22×C22⋊C4—416no—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2318yes—
105C22×C22⋊C4C2318yes—
106C22×C22⋊C4C2318yes—
107C22×C22⋊C4C2311yescharacteristic
108C22×C22⋊C4C2311yescharacteristic
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2318yes—
113C22×C42—24no—
114C22×C42—44no—
115C22×C42C2312yes—
116C22×C42C2311yescharacteristic
117C22×C42C2311yescharacteristic
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2314yes—
121#398—216no—
122#396—416no—
123#394—216no—
124#391—24no—
125#391—24no—
126#391—44no—
127#390—28no—
128#390—28no—
129#390—28no—
130#390—216no—
131#390—28no—
132#390—44no—
133#389—28no—
134#388—416no—
135#387—216no—
136#385—216no—
137#385—28no—
138#384—24no—
139#384—24no—
140#380—24no—
141#380—28no—
142#379—28no—
143#379—28no—
144#379—216no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—44no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—28no—
149C4×C22⋊C4D418yes—
150C2×#95C2318yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2318yes—
161C2×#95C2312yes—
162C25—24no—
163C25—28no—
164C25—24no—
165C25—24no—
166C25—28no—
167C25—28no—
168C25—48no—
169C25—44no—
170C25C2414yes—
171C25C2411yescharacteristic
172C22×Q8—22no—
173C22×Q8—22no—
174C22×Q8—22no—
175C22×Q8—24no—
176C22×Q8—24no—
177C22×D4—232no—
178C22×D4—28no—
179C22×D4—24no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—24no—
184C22×D4—24no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—24no—
188C22×D4—432no—
189C22×D4—432no—
190C22×D4—44no—
191C22×D4—416no—
192C22×D4—416no—
193C22×D4#4118yes—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—44no—
203C23×C4—48no—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4C2×D418yes—
207C23×C4#4114yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220#124—432no—
221C22⋊Q8—416no—
222C2×C4⋊C4—232no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—48no—
247C2×C4⋊C4C2×D414yes—
248C2×C4⋊C4C2×D414yes—
249C2×C4⋊C4C2×D418yes—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—216no—
262C2×C22⋊C4—216no—
263C2×C22⋊C4—216no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—216no—
267C2×C22⋊C4—216no—
268C2×C22⋊C4—216no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—216no—
272C2×C22⋊C4—48no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—48no—
277C2×C22⋊C4—464no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—48no—
281C2×C22⋊C4—48no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C22⋊C4#4118yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—44no—
297C2×C42—416no—
298C2×C42—48no—
299C2×C42C2×D418yes—
300#95—28no—
301#95—24no—
302#95—28no—
303#95—24no—
304#95—28no—
305#95—28no—
306#95—28no—
307#95—28no—
308#95—216no—
309#95—44no—
310#95—48no—
311#95C2×D418yes—
312C24—22no—
313C24—22no—
314C24—24no—
315C24—216no—
316C24—24no—
317C24—24no—
318C24—28no—
319C24—24no—
320C24—24no—
321C24—28no—
322C24—28no—
323C24—24no—
324C24—24no—
325C24—28no—
326C24—416no—
327C24—416no—
328C24—432no—
329C24—432no—
330C24—416no—
331C24—48no—
332C24—48no—
333C24—416no—
334C24—48no—
335C24—48no—
336C24—416no—
337C24—88no—
338C24—88no—
339C24—816no—
340C24—816no—
341C24C22×D412yes—
342C24C22×D412yes—
343C24C2×#4118yes—
344C24C2×#4112yes—
345C24C2×#4114yes—
346C242−1+414yes—
347C24C2511yescharacteristic
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×D4—216no—
354C2×D4—216no—
355C2×D4—216no—
356C2×D4—216no—
357C2×D4—416no—
358C2×D4—416no—
359C2×D4—48no—
360C2×D4—4128no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—48no—
364C2×D4—48no—
365C2×D4—48no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—28no—
369C22×C4—28no—
370C22×C4—216no—
371C22×C4—216no—
372C22×C4—216no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—28no—
413C22×C4—28no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—416no—
417C22×C4—416no—
418C22×C4—416no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—88no—
422C22×C4—88no—
423C22×C4#121116yes—
424C22×C4C22×D414yes—
425C22×C4C22×D414yes—
426C22×C4C22×D414yes—
427C22×C4C22×D414yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C2×#4114yes—
433C22×C4C2×#4114yes—
434C22×C4C2×#4112yes—
435C22×C4C2×#4112yes—
436C22×C4C2×#4114yes—
437C22×C4H(5,𝔽2)18yes—
438C22×C4H(5,𝔽2)14yes—
439C4⋊C4—216no—
440C4⋊C4—28no—
441C4⋊C4—28no—
442C4⋊C4—416no—
443C4⋊C4—44no—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—88no—
447C4⋊C4—88no—
448C4⋊C4—88no—
449C22⋊C4—28no—
450C22⋊C4—28no—
451C22⋊C4—216no—
452C22⋊C4—432no—
453C22⋊C4—432no—
454C22⋊C4—432no—
455C22⋊C4—416no—
456C22⋊C4—416no—
457C22⋊C4—48no—
458C22⋊C4—48no—
459C22⋊C4—88no—
460C22⋊C4—88no—
461C22⋊C4—816no—
462C42—216no—
463C42—48no—
464C42—44no—
465C42—44no—
466C42—48no—
467C42—88no—
468C23—22no—
469C23—22no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—24no—
474C23—24no—
475C23—24no—
476C23—28no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23—216no—
483C23—216no—
484C23—432no—
485C23—464no—
486C23—44no—
487C23—44no—
488C23—44no—
489C23—44no—
490C23—44no—
491C23—48no—
492C23—44no—
493C23—48no—
494C23—48no—
495C23—48no—
496C23—44no—
497C23—44no—
498C23—48no—
499C23—48no—
500C23—48no—
501C23—48no—
502C23—48no—
503C23—48no—
504C23—832no—
505C23—864no—
506C23—832no—
507C23—816no—
508C23—832no—
509C23—816no—
510C23C2×#12114yes—
511C23#53214yes—
512C23#53614yes—
513C23#54818yes—
514C23#55618yes—
515C23C23×D411yescharacteristic
516C23C23×D411yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4112yes—
519C23C22×#4112yes—
520C23C2×H(5,𝔽2)12yes—
521C23C2×H(5,𝔽2)14yes—
522C23C2×2−1+412yes—
523Q8—88no—
524D4—432no—
525D4—432no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—232no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—28no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—28no—
546C2×C4—28no—
547C2×C4—28no—
548C2×C4—432no—
549C2×C4—42no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—48no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—816no—
582C2×C4C2×#12114yes—
583C2×C4C2×#12114yes—
584C2×C4#52514yes—
585C2×C4#53514yes—
586C2×C4#54618yes—
587C2×C4#55118yes—
588C22—28no—
589C22—216no—
590C22—24no—
591C22—28no—
592C22—28no—
593C22—24no—
594C22—28no—
595C22—48no—
596C22—416no—
597C22—48no—
598C22—416no—
599C22—48no—
600C22—864no—
601C22—84no—
602C22—88no—
603C22—84no—
604C22—84no—
605C22—84no—
606C22—88no—
607C22#254518yes—
608C22C22×#12111yescharacteristic
609C22C2×#52511yescharacteristic
610C22C2×#53211yescharacteristic
611C22C2×#53511yescharacteristic
612C22C2×#53611yescharacteristic
613C22C2×#54612yes—
614C22C2×#54812yes—
615C22C2×#55112yes—
616C22C2×#55612yes—
617C22#348114yes—
618C22#348412yes—
619C22#349812yes—
620C22#350812yes—
621C22#351914yes—
622C22#352014yes—
623C22#353614yes—
624C4—24no—
625C4—44no—
626C4—44no—
627C4—44no—
628C4—44no—
629C4—44no—
630C4—44no—
631C4—48no—
632C4—84no—
633C4—82no—
634C4—82no—
635C4#241118yes—
636C2—28no—
637C2—88no—
638C2C2×#241111yescharacteristic
639C2C2×#254511yescharacteristic
640C2#3552811yescharacteristic
641C2#3636212yes—
642C2#3652812yes—
643C2#3807812yes—
644C2#3912012yes—
645C2#5942414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2502C2semidirect
3. C2×#2345C2semidirect
5. #15661C2semidirect
6. #15226C2semidirect
7. #15134C2semidirect
8. #14726C2semidirect
9. #14702C2semidirect
10. #14153C2semidirect
11. #14001C2semidirect
12. #13971C2semidirect
14. C4×#385C2semidirect
23. C2×#398C22semidirect
24. C2×#396C22semidirect
27. C2×#391C22semidirect
28. C2×#391C22semidirect
29. C2×#391C22semidirect
37. C2×#389C22semidirect
38. C2×#388C22semidirect
39. C2×#387C22semidirect
42. C2×#384C22semidirect
43. C2×#384C22semidirect
49. C2×C4×C4⋊C4C22semidirect
50. C2×C4×C4⋊C4C22semidirect
54. #1440C22semidirect
55. #1439C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
60. #1437C22semidirect
61. #1436C22semidirect
62. #1436C22semidirect
63. C4⋊#95C22semidirect
66. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25022
C2×#24352
C2×#23452
C22×#3902
#156612
#152262
#151342
#147262
#147022
#141532
#140012
#139712
#139022
C4×#3852