Finite Group Databasebeta

← #613322

Subgroups — #613322

Order 512.

5394 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#117C211yesmaximal, characteristic
2C2×#2484C211yesmaximal, characteristic, complemented
3C2×#2482C211yesmaximal, characteristic
4C2×#2296C211yesmaximal, characteristic, complemented
5#15092C212yesmaximal, complemented
6#14918C212yesmaximal
7#14907C214yesmaximal, complemented
8#14893C214yesmaximal, complemented
9#14705C212yesmaximal, complemented
10#14693C212yesmaximal
11#14684C211yesmaximal, characteristic, complemented
12#14063C218yesmaximal, complemented
13#13870C211yesmaximal, characteristic, complemented
14#13868C211yesmaximal, characteristic
15C25×C4C2211yescharacteristic
16C22×#117—28no—
17C22×#117—28no—
18C22×#117—28no—
19C22×#117C2211yescharacteristic
20C22×#117C2214yes—
21C22×#117C2214yes—
22C22×#117C2211yescharacteristic
23C23×C4⋊C4C2211yescharacteristic
24C23×C4⋊C4C2211yescharacteristic
25C23×C22⋊C4C2211yescharacteristic
26C23×C22⋊C4C2211yescharacteristic
27C23×C42C2211yescharacteristic
28C23×C42C2211yescharacteristic
29#2484—28no—
30#2482—28no—
31C2×#400C2212yescomplemented
32C2×#399C2212yes—
33C2×#398C2212yescomplemented
34C2×#397C2212yes—
35#2296—28no—
36C2×#389C2212yescomplemented
37C2×#389C2212yescomplemented
38C2×#388C2214yes—
39C2×#387C2214yescomplemented
40C2×#385C2218yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#385C2212yes—
44C2×#385C2212yes—
45C2×#381C2211yescharacteristic
46C2×#381C2212yes—
47C2×#381C2211yescharacteristic
48C2×#381C2212yes—
49C2×#379C2212yes—
50C2×#379C2212yes—
51C2×C4×C4⋊C4C2214yescomplemented
52#1444C2218yescomplemented
53#1441C2214yescomplemented
54#1441C2218yescomplemented
55#1440C2218yes—
56#1439C2214yes—
57#1439C2218yes—
58#1437C2214yes—
59#1436C2218yescomplemented
60#1436C2214yescomplemented
61#1436C2218yescomplemented
62#1434C2214yescomplemented
63#1434C2214yescomplemented
64C22⋊#95C2212yes—
65C22⋊#95C2218yes—
66C22⋊#95C2212yes—
67C22⋊#95C2212yes—
68C22⋊#95C2212yes—
69C26C2311yescharacteristic
70C24×C4—24no—
71C24×C4—24no—
72C24×C4—24no—
73C24×C4—216no—
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2314yes—
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C2×#117—2128no—
82C2×#117—24no—
83C2×#117—24no—
84C2×#117—24no—
85C2×#117—24no—
86C2×#117—216no—
87C2×#117—216no—
88C2×#117—216no—
89C2×#117—216no—
90C2×#117—28no—
91C2×#117—216no—
92C2×#117—216no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4D418yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2318yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C42—24no—
126C22×C42—24no—
127C22×C42—24no—
128C22×C42—24no—
129C22×C42—24no—
130C22×C42D418yes—
131C22×C42C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136#400—28no—
137#399—28no—
138#398—28no—
139#397—28no—
140#389—28no—
141#389—28no—
142#388—216no—
143#387—216no—
144#385—28no—
145#385—28no—
146#385—28no—
147#385—28no—
148#385—232no—
149#381—24no—
150#381—24no—
151#381—28no—
152#381—28no—
153#379—28no—
154#379—28no—
155C4×C4⋊C4—216no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2318yes—
163C2×#95C2314yes—
164C2×#95C2318yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2318yes—
168C2×#95C2314yes—
169C2×#95C2318yes—
170C2×#95C2318yes—
171C25—22no—
172C25—22no—
173C25—28no—
174C25—24no—
175C25—24no—
176C25—44no—
177C25C2×D414yes—
178C25C2412yes—
179C25C2411yescharacteristic
180C23×C4—232no—
181C23×C4—264no—
182C23×C4—264no—
183C23×C4—264no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—28no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—28no—
191C23×C4—24no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—28no—
195C23×C4—28no—
196C23×C4—28no—
197C23×C4—28no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4#41116yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229#117—232no—
230#117—2128no—
231#117—2128no—
232#117—416no—
233C2×C4⋊C4—232no—
234C2×C4⋊C4—232no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—216no—
254C2×C4⋊C4—216no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—48no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4C2×D418yes—
259C2×C4⋊C4C2×D418yes—
260C2×C22⋊C4—232no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—232no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—216no—
280C2×C22⋊C4—216no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—216no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4—48no—
287C2×C22⋊C4C2×D418yes—
288C2×C22⋊C4C2×D418yes—
289C2×C22⋊C4#4118yes—
290C2×C42—232no—
291C2×C42—232no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—28no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42C2×D418yes—
305C2×C42C2×D418yes—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—216no—
317#95—216no—
318#95—216no—
319#95—48no—
320#95—48no—
321C24—24no—
322C24—24no—
323C24—216no—
324C24—216no—
325C24—28no—
326C24—216no—
327C24—22no—
328C24—22no—
329C24—24no—
330C24—24no—
331C24—24no—
332C24—28no—
333C24—216no—
334C24—48no—
335C24—432no—
336C24—432no—
337C24—48no—
338C24—48no—
339C24—44no—
340C24—44no—
341C24—48no—
342C24C22×D414yes—
343C24C22×D412yes—
344C24C22×D412yes—
345C24C22×D412yes—
346C24C2×#4118yes—
347C24C2×#4114yes—
348C242−1+414yes—
349C24C2511yescharacteristic
350C22×C4—232no—
351C22×C4—2256no—
352C22×C4—232no—
353C22×C4—232no—
354C22×C4—216no—
355C22×C4—216no—
356C22×C4—216no—
357C22×C4—216no—
358C22×C4—232no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—232no—
363C22×C4—28no—
364C22×C4—216no—
365C22×C4—216no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—232no—
379C22×C4—232no—
380C22×C4—28no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—216no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—28no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—28no—
408C22×C4—24no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—28no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—28no—
415C22×C4—28no—
416C22×C4—28no—
417C22×C4—48no—
418C22×C4—48no—
419C22×C4—48no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4#120116yes—
427C22×C4C22×D414yes—
428C22×C4C22×D414yes—
429C22×C4C22×D414yes—
430C22×C4C22×D412yes—
431C22×C4C22×D412yes—
432C22×C4C22×D412yes—
433C22×C4C22×D412yes—
434C22×C4C22×D412yes—
435C22×C4C22×D412yes—
436C22×C4C22×D414yes—
437C22×C4C22×D414yes—
438C22×C4C2×#4118yes—
439C22×C4C2×#4118yes—
440C22×C4C2×#4118yes—
441C4⋊C4—232no—
442C4⋊C4—216no—
443C4⋊C4—216no—
444C4⋊C4—216no—
445C4⋊C4—416no—
446C4⋊C4—44no—
447C4⋊C4—44no—
448C4⋊C4—416no—
449C4⋊C4—416no—
450C4⋊C4—88no—
451C22⋊C4—232no—
452C22⋊C4—216no—
453C22⋊C4—216no—
454C22⋊C4—216no—
455C22⋊C4—432no—
456C22⋊C4—48no—
457C22⋊C4—416no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—88no—
461C42—232no—
462C42—216no—
463C42—216no—
464C42—216no—
465C42—48no—
466C23—24no—
467C23—24no—
468C23—24no—
469C23—24no—
470C23—216no—
471C23—24no—
472C23—24no—
473C23—216no—
474C23—232no—
475C23—24no—
476C23—24no—
477C23—28no—
478C23—28no—
479C23—28no—
480C23—216no—
481C23—28no—
482C23—216no—
483C23—464no—
484C23—432no—
485C23—464no—
486C23—416no—
487C23—432no—
488C23—416no—
489C23—42no—
490C23—42no—
491C23—44no—
492C23—44no—
493C23—48no—
494C23—44no—
495C23—48no—
496C23C2×#12014yes—
497C23#52518yes—
498C23#52514yes—
499C23#53614yes—
500C23#54718yes—
501C23#54718yes—
502C23C23×D411yescharacteristic
503C23C23×D411yescharacteristic
504C23C23×D411yescharacteristic
505C23C22×#4114yes—
506C23C22×#4112yes—
507C23C2×2−1+412yes—
508C23#58514yes—
509C2×C4—232no—
510C2×C4—232no—
511C2×C4—2128no—
512C2×C4—232no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—28no—
536C2×C4—216no—
537C2×C4—28no—
538C2×C4—416no—
539C2×C4—416no—
540C2×C4—416no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—48no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—48no—
562C2×C4—48no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—48no—
568C2×C4—48no—
569C2×C4C2×#12014yes—
570C2×C4C2×#12014yes—
571C2×C4#52518yes—
572C2×C4#52518yes—
573C22—24no—
574C22—28no—
575C22—216no—
576C22—24no—
577C22—216no—
578C22—28no—
579C22—28no—
580C22—216no—
581C22—464no—
582C22—48no—
583C22—416no—
584C22—44no—
585C22—44no—
586C22—48no—
587C22—48no—
588C22#240618yes—
589C22C22×#12011yescharacteristic
590C22C2×#52512yes—
591C22C2×#52512yes—
592C22C2×#52512yes—
593C22C2×#52511yescharacteristic
594C22C2×#53611yescharacteristic
595C22#344912yes—
596C22C2×#54712yes—
597C22C2×#54712yes—
598C22#347518yes—
599C22#348512yes—
600C22#348712yes—
601C22#349518yes—
602C4—28no—
603C4—28no—
604C4—24no—
605C4—24no—
606C4—48no—
607C4—44no—
608C4—44no—
609C4—44no—
610C4—82no—
611C4—84no—
612C4—82no—
613C4—84no—
614C2—28no—
615C2—48no—
616C2C2×#240611yescharacteristic
617C2#3487512yes—
618C2#3514911yescharacteristic
619C2#3532512yes—
620C2#3634114yes—
621C2#3642714yes—
622C2#3661411yescharacteristic
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×#2484C2semidirect
4. C2×#2296C2semidirect
5. #15092C2semidirect
7. #14907C2semidirect
8. #14893C2semidirect
9. #14705C2semidirect
11. #14684C2semidirect
12. #14063C2semidirect
13. #13870C2semidirect
31. C2×#400C22semidirect
33. C2×#398C22semidirect
36. C2×#389C22semidirect
37. C2×#389C22semidirect
39. C2×#387C22semidirect
51. C2×C4×C4⋊C4C22semidirect
52. #1444C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
59. #1436C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. #1434C22semidirect
63. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C23×#1172
C2×#24842
C2×#24822
C2×#22962
#150922
#149182
#149072
#148932
#147052
#146932
#146842
#140632
#138702
#138682