Finite Group Databasebeta

← #552728

Subgroups — #552728

Order 512.

4046 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2308C211yesmaximal, characteristic
2#14666C212yesmaximal, complemented
3#14516C212yesmaximal, complemented
4#14415C214yesmaximal, complemented
5#14410C212yesmaximal, complemented
6#14335C211yesmaximal, characteristic
7#14267C214yesmaximal, complemented
8#14145C212yesmaximal, complemented
9#13991C214yesmaximal, complemented
10#13972C214yesmaximal, complemented
11#13877C212yesmaximal, complemented
12C2×#1438C211yesmaximal, characteristic
13C22⋊C42C212yesmaximal, complemented
14C24×D4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2212yes—
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19#2308C4116yescomplemented
20C2×#388C2214yescomplemented
21C2×#388C2214yescomplemented
22C2×#386C2212yes—
23C2×#386C2212yes—
24C2×#386C2212yes—
25C2×#386C2212yes—
26C2×#385C2214yes—
27C2×#381C2212yes—
28C2×#381C2212yes—
29C2×#380C2214yescomplemented
30C2×#379C2212yes—
31C2×#379C2212yes—
32C2×#379C2212yes—
33C2×C4×C22⋊C4C2214yes—
34C2×C4×C22⋊C4C2214yescomplemented
35C22×#95C2212yes—
36#1443C2214yescomplemented
37#1443C2214yescomplemented
38#1442C2214yescomplemented
39#1442C2214yescomplemented
40#1440C2214yescomplemented
41#1440C2214yescomplemented
42#1439C2214yescomplemented
43#1439C2214yescomplemented
44#1439C2214yescomplemented
45#1439C2214yescomplemented
46#1438C2211yescharacteristic
47#1438C2212yes—
48#1438C2211yescharacteristic
49#1438C2212yes—
50#1438C2212yes—
51#1438C2212yes—
52#1438C2211yescharacteristic
53#1438C2212yes—
54#1438C2211yescharacteristic
55#1438C2212yes—
56#1437C2214yescomplemented
57#1437C2214yescomplemented
58#1437C2214yescomplemented
59#1437C2214yescomplemented
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62#1434C2214yescomplemented
63#1434C2214yescomplemented
64C22⋊#95C2214yes—
65C22⋊#95C2214yes—
66C22⋊#95C2214yescomplemented
67C22⋊#95C2214yescomplemented
68C26C2311yescharacteristic
69C26C2311yescharacteristic
70C23×D4—28no—
71C23×D4—44no—
72C23×D4—44no—
73C23×D4C2×C418yes—
74C23×D4C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C23×D4C2311yescharacteristic
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2314yes—
80C24×C4C2312yes—
81C24×C4C2312yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—48no—
86C22×C22⋊C4—48no—
87C22×C22⋊C4—44no—
88C22×C22⋊C4—44no—
89C22×C22⋊C4—44no—
90C22×C22⋊C4C2×C418yes—
91C22×C22⋊C4C2×C418yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C42C2314yes—
112C22×C42C2314yes—
113#388—48no—
114#388—48no—
115#386C2×C4116yescomplemented
116#386C2×C4116yescomplemented
117#386C2×C4116yescomplemented
118#386C2×C4116yescomplemented
119#385—48no—
120#381—44no—
121#381C2×C4116yes—
122#380—48no—
123#379—28no—
124#379—44no—
125#379C2×C4116yes—
126C4×C22⋊C4—48no—
127C4×C22⋊C4—48no—
128C2×#95—44no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2312yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C25—24no—
150C25—24no—
151C25—24no—
152C25—24no—
153C25—24no—
154C25—44no—
155C25—44no—
156C25—44no—
157C25—44no—
158C25C22×C414yes—
159C25C22×C414yes—
160C25#4118yes—
161C25C2411yescharacteristic
162C25C2411yescharacteristic
163C25C2411yescharacteristic
164C25C2411yescharacteristic
165C25C2411yescharacteristic
166C22×D4—232no—
167C22×D4—24no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—24no—
171C22×D4—416no—
172C22×D4—416no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—42no—
181C22×D4—464no—
182C22×D4C22×C414yes—
183C22×D4C22×C414yes—
184C22×D4C22×C414yes—
185C22×D4C22×C414yes—
186C23×C4—24no—
187C23×C4—24no—
188C23×C4—24no—
189C23×C4—28no—
190C23×C4—28no—
191C23×C4—28no—
192C23×C4—28no—
193C23×C4—42no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—44no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4C22×C414yes—
205C23×C4C22×C418yes—
206C23×C4C22×C418yes—
207C23×C4C2414yes—
208C23×C4C2414yes—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C2×C4⋊C4—28no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—416no—
237C2×C22⋊C4—416no—
238C2×C22⋊C4—432no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—432no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4C22×C418yes—
265C2×C22⋊C4C22×C418yes—
266C2×C22⋊C4C22×C418yes—
267C2×C22⋊C4C22×C418yes—
268C2×C22⋊C4C22×C4116yes—
269C2×C22⋊C4C22×C4116yes—
270C2×C42—28no—
271C2×C42—44no—
272C2×C42—44no—
273C2×C42—44no—
274C2×C42—44no—
275C2×C42—44no—
276#95—28no—
277#95—28no—
278#95—42no—
279#95—44no—
280#95—42no—
281#95—44no—
282#95—44no—
283#95—44no—
284#95—44no—
285#95—44no—
286#95—44no—
287#95—42no—
288#95—44no—
289#95—42no—
290#95—44no—
291#95—44no—
292#95—44no—
293#95—44no—
294#95C22×C4116yes—
295#95C22×C4116yes—
296C24—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—22no—
301C24—22no—
302C24—22no—
303C24—22no—
304C24—24no—
305C24—24no—
306C24—28no—
307C24—28no—
308C24—232no—
309C24—28no—
310C24—22no—
311C24—22no—
312C24—24no—
313C24—24no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—24no—
320C24—28no—
321C24—28no—
322C24—48no—
323C24—416no—
324C24—416no—
325C24—48no—
326C24—416no—
327C24—48no—
328C24—48no—
329C24—416no—
330C24—416no—
331C24—84no—
332C24—84no—
333C24—88no—
334C24—84no—
335C24—84no—
336C24—88no—
337C24—88no—
338C24—88no—
339C24—84no—
340C24—84no—
341C24—84no—
342C24—84no—
343C24#117116yes—
344C24C23×C412yes—
345C24C23×C412yes—
346C24C23×C412yes—
347C24C23×C412yes—
348C24C23×C412yes—
349C24C2×#4114yes—
350C24C2×#4114yes—
351C24C2×#4114yes—
352C24H(5,𝔽2)14yes—
353C24H(5,𝔽2)14yes—
354C24C2511yescharacteristic
355C2×D4—28no—
356C2×D4—28no—
357C2×D4—28no—
358C2×D4—28no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—44no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—4128no—
368C2×D4—416no—
369C2×D4—416no—
370C2×D4—416no—
371C2×D4—416no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
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—48no—
421C22×C4—48no—
422C22×C4—416no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—416no—
426C22×C4—416no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—88no—
430C22×C4—88no—
431C22×C4—88no—
432C22×C4—84no—
433C22×C4—84no—
434C22×C4—88no—
435C22×C4—88no—
436C22×C4—84no—
437C22×C4—84no—
438C22×C4C23×C412yes—
439C22×C4C23×C412yes—
440C22×C4C23×C418yes—
441C22×C4C23×C418yes—
442C22×C4H(5,𝔽2)18yes—
443C4⋊C4—88no—
444C4⋊C4—88no—
445C22⋊C4—216no—
446C22⋊C4—216no—
447C22⋊C4—216no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—416no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—416no—
454C22⋊C4—416no—
455C22⋊C4—416no—
456C22⋊C4—48no—
457C22⋊C4—48no—
458C22⋊C4—416no—
459C22⋊C4—84no—
460C22⋊C4—84no—
461C22⋊C4—88no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C42—88no—
465C42—88no—
466C23—28no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—24no—
471C23—24no—
472C23—28no—
473C23—28no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—28no—
481C23—28no—
482C23—28no—
483C23—28no—
484C23—24no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—28no—
489C23—28no—
490C23—232no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—28no—
498C23—28no—
499C23—432no—
500C23—432no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—416no—
514C23—44no—
515C23—44no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—44no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—832no—
535C23—816no—
536C23—816no—
537C23—816no—
538C23—832no—
539C23—816no—
540C23—832no—
541C23—832no—
542C23C2×#11714yes—
543C23C2×#11714yes—
544C23C2×#11714yes—
545C23#51814yes—
546C23#51814yes—
547C23#538116yes—
548C23C24×C411yescharacteristic
549C23C22×#4112yes—
550C23C2×H(5,𝔽2)12yes—
551C23C2×H(5,𝔽2)12yes—
552C23C2×H(5,𝔽2)14yes—
553C23#58514yes—
554D4—416no—
555D4—416no—
556D4—416no—
557D4—416no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—28no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—28no—
572C2×C4—28no—
573C2×C4—416no—
574C2×C4—44no—
575C2×C4—44no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—816no—
610C2×C4—816no—
611C2×C4—832no—
612C22—216no—
613C22—28no—
614C22—28no—
615C22—216no—
616C22—28no—
617C22—28no—
618C22—28no—
619C22—24no—
620C22—24no—
621C22—48no—
622C22—48no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—44no—
627C22—48no—
628C22—48no—
629C22—44no—
630C22—44no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—42no—
636C22—42no—
637C22—832no—
638C22—832no—
639C22—84no—
640C22—84no—
641C22—84no—
642C22—84no—
643C22#231218yes—
644C22C22×#11711yescharacteristic
645C22C2×#51811yescharacteristic
646C22C2×#51811yescharacteristic
647C22C4×H(5,𝔽2)14yes—
648C22C2×#53814yes—
649C22#349012yes—
650C22#349112yes—
651C22#350514yes—
652C22#353212yes—
653C22#353312yes—
654C22#353414yes—
655C22#354214yes—
656C22#354312yes—
657C22#354512yes—
658C4—48no—
659C4—48no—
660C4—48no—
661C4—42no—
662C4—42no—
663C4—84no—
664C4—84no—
665C4—84no—
666C2—28no—
667C2—44no—
668C2—44no—
669C2—82no—
670C2—82no—
671C2C2×#231211yescharacteristic
672C2#3425112yes—
673C2#3440712yes—
674C2#3440812yes—
675C2#4053614yes—
676C2#4055614yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14666C2semidirect
3. #14516C2semidirect
4. #14415C2semidirect
5. #14410C2semidirect
7. #14267C2semidirect
8. #14145C2semidirect
9. #13991C2semidirect
10. #13972C2semidirect
11. #13877C2semidirect
13. C22⋊C42C2semidirect
19. #2308C4semidirect
20. C2×#388C22semidirect
21. C2×#388C22semidirect
29. C2×#380C22semidirect
34. C2×C4×C22⋊C4C22semidirect
36. #1443C22semidirect
37. #1443C22semidirect
38. #1442C22semidirect
39. #1442C22semidirect
40. #1440C22semidirect
41. #1440C22semidirect
42. #1439C22semidirect
43. #1439C22semidirect
44. #1439C22semidirect
45. #1439C22semidirect
56. #1437C22semidirect
57. #1437C22semidirect
58. #1437C22semidirect
59. #1437C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect
66. C22⋊#95C22semidirect
67. C22⋊#95C22semidirect
115. #386C2×C4semidirect
116. #386C2×C4semidirect
117. #386C2×C4semidirect
118. #386C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23082
#146662
#145162
#144152
#144102
#143352
#142672
#141452
#139912
#139722
#138772
C2×#14382
C22⋊C422