Finite Group Databasebeta

← #4865297

Subgroups — #4865297

Order 512.

2482 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic
2C2×#2621C211yesmaximal, characteristic, complemented
3#17176C214yesmaximal, complemented
4#17029C212yesmaximal, complemented
5#16779C214yesmaximal, complemented
6#16665C212yesmaximal, complemented
7#16602C212yesmaximal, complemented
8#16470C212yesmaximal, complemented
9#16391C212yesmaximal
10#16364C212yesmaximal, complemented
11#15722C212yesmaximal, complemented
12#15480C212yesmaximal
13#14866C212yesmaximal, complemented
14#14717C212yesmaximal
15#13938C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C22×C22⋊Q8C2212yes—
18C22×#117C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20#2621—28no—
21#2482—28no—
22C2×#399C2212yes—
23C2×#398C2212yescomplemented
24C2×#398C2212yescomplemented
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#397C2214yes—
29C2×#393C2212yes—
30C2×#393C2212yes—
31C2×#391C2212yescomplemented
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#385C2211yescharacteristic
35C2×#385C2211yescharacteristic
36C2×#383C2211yescharacteristic, complemented
37C2×#382C2211yescharacteristic, complemented
38C2×#381C2211yescharacteristic
39C2×#379C2211yescharacteristic
40C2×#379C2211yescharacteristic
41C2×#379C2214yes—
42C2×C4×C4⋊C4C2211yescharacteristic, complemented
43#1446C2212yescomplemented
44#1446C2212yescomplemented
45#1444C2212yescomplemented
46#1443C2214yescomplemented
47#1442C2212yescomplemented
48#1442C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2212yescomplemented
54#1441C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1440C2214yes—
58#1440C2214yes—
59#1440C2214yes—
60#1439C2214yes—
61#1439C2212yes—
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2212yes—
70#1437C2214yes—
71#1437C2212yes—
72#1436C2212yescomplemented
73#1434C2214yescomplemented
74#1434C2212yescomplemented
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×C4⋊Q8—24no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8—42no—
84C2×C22⋊Q8—28no—
85C2×C22⋊Q8—48no—
86C2×#117—42no—
87C2×#117—42no—
88C2×#117D418yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2314yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C22⋊C4—24no—
100C22×C22⋊C4—42no—
101C22×C22⋊C4—42no—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2311yescharacteristic
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C42C2311yescharacteristic
119C22×C42C2311yescharacteristic
120C22×C42C2312yes—
121#399—28no—
122#398—28no—
123#398—28no—
124#397—28no—
125#397—28no—
126#397—28no—
127#397—48no—
128#393—44no—
129#393—44no—
130#391—28no—
131#388—28no—
132#388—28no—
133#385—24no—
134#385—24no—
135#383—42no—
136#382—24no—
137#381—24no—
138#379—24no—
139#379—24no—
140#379—48no—
141C4×C4⋊C4—42no—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2312yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25—22no—
169C25—24no—
170C25—24no—
171C25—41no—
172C25—41no—
173C25—42no—
174C25—44no—
175C25—42no—
176C25C2412yes—
177C25C2411yescharacteristic
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—44no—
182C22×Q8—44no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—42no—
188C23×C4—41no—
189C23×C4—41no—
190C23×C4—44no—
191C23×C4C2×D414yes—
192C23×C4C2414yes—
193C23×C4C2414yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2414yes—
197C23×C4C2414yes—
198C23×C4C2412yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203C23×C4C2411yescharacteristic
204C23×C4C2411yescharacteristic
205C4⋊Q8—416no—
206C22⋊Q8—432no—
207#117—416no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—28no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4—44no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4—44no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4C2×D414yes—
233C2×C4⋊C4C2×D414yes—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—41no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—416no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—44no—
267C2×C22⋊C4—44no—
268C2×C22⋊C4—44no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D414yes—
278C2×C22⋊C4C2×D414yes—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—42no—
284C2×C42—44no—
285C2×C42C2×D414yes—
286C2×C42C2×D414yes—
287#95—24no—
288#95—24no—
289#95—24no—
290#95—24no—
291#95—24no—
292#95—24no—
293#95—28no—
294#95—28no—
295#95—24no—
296#95—24no—
297#95—44no—
298#95—44no—
299#95—44no—
300#95—44no—
301#95—44no—
302#95—44no—
303#95—44no—
304#95—44no—
305#95—44no—
306#95—42no—
307#95—42no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313C24—21no—
314C24—21no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—24no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—24no—
323C24—22no—
324C24—22no—
325C24—24no—
326C24—22no—
327C24—24no—
328C24—24no—
329C24—48no—
330C24—44no—
331C24—416no—
332C24—44no—
333C24—48no—
334C24—44no—
335C24—44no—
336C24—44no—
337C24—84no—
338C24—84no—
339C24—84no—
340C24—88no—
341C24—88no—
342C24—84no—
343C24—84no—
344C24—88no—
345C24C22×D412yes—
346C24H(5,𝔽2)14yes—
347C242−1+414yes—
348C24C2511yescharacteristic
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—48no—
352C2×Q8—88no—
353C2×Q8—88no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—28no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—28no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
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—22no—
418C22×C4—24no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—44no—
423C22×C4—84no—
424C22×C4—84no—
425C22×C4—84no—
426C22×C4C22×D412yes—
427C22×C4C22×D412yes—
428C22×C4C22×D412yes—
429C22×C4C22×D412yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4H(5,𝔽2)12yes—
433C22×C4H(5,𝔽2)12yes—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—416no—
438C4⋊C4—48no—
439C4⋊C4—48no—
440C4⋊C4—48no—
441C4⋊C4—84no—
442C4⋊C4—84no—
443C4⋊C4—88no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C22⋊C4—48no—
447C22⋊C4—48no—
448C22⋊C4—48no—
449C22⋊C4—44no—
450C22⋊C4—44no—
451C22⋊C4—416no—
452C22⋊C4—48no—
453C22⋊C4—84no—
454C22⋊C4—84no—
455C22⋊C4—88no—
456C22⋊C4—88no—
457C22⋊C4—88no—
458C22⋊C4—88no—
459C22⋊C4—88no—
460C42—28no—
461C42—28no—
462C42—84no—
463C23—22no—
464C23—22no—
465C23—24no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—28no—
470C23—28no—
471C23—24no—
472C23—28no—
473C23—24no—
474C23—24no—
475C23—28no—
476C23—41no—
477C23—42no—
478C23—41no—
479C23—42no—
480C23—42no—
481C23—44no—
482C23—44no—
483C23—416no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23—42no—
488C23—42no—
489C23—42no—
490C23—44no—
491C23—44no—
492C23—44no—
493C23—42no—
494C23—44no—
495C23—42no—
496C23—44no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—816no—
503C23—88no—
504C23—816no—
505C23—816no—
506C23—832no—
507C23—816no—
508C23—88no—
509C23#53514yes—
510C23C23×D411yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×H(5,𝔽2)11yescharacteristic
513C23C2×H(5,𝔽2)12yes—
514C23C2×2−1+412yes—
515C23#58512yes—
516C23#58512yes—
517C23#58514yes—
518Q8—816no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—416no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—41no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
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—88no—
591C2×C4#53414yes—
592C22—28no—
593C22—24no—
594C22—44no—
595C22—44no—
596C22—42no—
597C22—44no—
598C22—48no—
599C22—42no—
600C22—48no—
601C22—44no—
602C22—44no—
603C22—48no—
604C22—44no—
605C22—832no—
606C22—84no—
607C22—82no—
608C22—82no—
609C22—82no—
610C22—82no—
611C22—84no—
612C22C2×#53411yescharacteristic
613C22C2×#53511yescharacteristic
614C22#344911yescharacteristic
615C22#348912yes—
616C22#350012yes—
617C22#353412yes—
618C22#353412yes—
619C22#353514yes—
620C22#353811yescharacteristic
621C22#354012yes—
622C22#354011yescharacteristic
623C22#354112yes—
624C22#354514yes—
625C22#355112yes—
626C22#356414yes—
627C22#356614yes—
628C4—24no—
629C4—42no—
630C4—42no—
631C4—42no—
632C4—42no—
633C4—42no—
634C4—84no—
635C4—82no—
636C4—84no—
637C4—82no—
638C4—84no—
639C4—82no—
640C4—84no—
641C2—44no—
642C2—84no—
643C2#3737411yescharacteristic
644C2#3741911yescharacteristic
645C2#3744011yescharacteristic
646C2#4229612yes—
647C2#4230412yes—
648C2#5420414yes—
649C2#5480512yes—
650C2#5659312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2621C2semidirect
3. #17176C2semidirect
4. #17029C2semidirect
5. #16779C2semidirect
6. #16665C2semidirect
7. #16602C2semidirect
8. #16470C2semidirect
10. #16364C2semidirect
11. #15722C2semidirect
13. #14866C2semidirect
15. #13938C2semidirect
16. C22×C4⋊Q8C22semidirect
23. C2×#398C22semidirect
24. C2×#398C22semidirect
31. C2×#391C22semidirect
36. C2×#383C22semidirect
37. C2×#382C22semidirect
42. C2×C4×C4⋊C4C22semidirect
43. #1446C22semidirect
44. #1446C22semidirect
45. #1444C22semidirect
46. #1443C22semidirect
47. #1442C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
72. #1436C22semidirect
73. #1434C22semidirect
74. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C2×#26212
#171762
#170292
#167792
#166652
#166022
#164702
#163912
#163642
#157222
#154802
#148662
#147172
#139382