Finite Group Databasebeta

← #639956

Subgroups — #639956

Order 512.

3674 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2280C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#17059C212yesmaximal, complemented
4#17058C212yesmaximal, complemented
5#17058C212yesmaximal, complemented
6#17057C212yesmaximal, complemented
7#15063C214yesmaximal, complemented
8#15061C214yesmaximal, complemented
9#14727C214yesmaximal
10#14681C211yesmaximal, characteristic
11#14342C212yesmaximal, complemented
12#14263C212yesmaximal, complemented
13#14128C212yesmaximal, complemented
14#13958C212yesmaximal, complemented
15C25×C4C2211yescharacteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2212yes—
21C2×#397C2218yes—
22C2×#385C2214yes—
23#2280—28no—
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#381C2212yes—
27C2×#380—28no—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×#380C2211yescharacteristic
31C2×#380C2211yescharacteristic
32C2×#380C2212yes—
33C2×#380C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#379C2212yes—
37C2×#379C2212yes—
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2212yes—
40C22×#95C2212yes—
41#1445C2214yescomplemented
42#1445C2214yescomplemented
43#1445C2214yescomplemented
44#1445C2214yescomplemented
45#1445C2214yescomplemented
46#1445C2214yescomplemented
47#1444C2214yescomplemented
48#1444C2214yescomplemented
49#1444C2214yescomplemented
50#1442C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1440C2218yes—
54#1440C2218yes—
55#1439C2218yes—
56#1436C2218yescomplemented
57#1434C2218yescomplemented
58C22⋊#95C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2214yes—
65C26C2311yescharacteristic
66C24×C4—28no—
67C24×C4—28no—
68C24×C4—44no—
69C24×C4D418yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2312yes—
77C24×C4C2312yes—
78C2×#117—24no—
79C2×#117—24no—
80C2×#117—28no—
81C2×#117—44no—
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2318yes—
95C22×C22⋊C4—28no—
96C22×C22⋊C4—48no—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C42C2312yes—
109C22×C42C2312yes—
110#397—416no—
111#385—48no—
112#381—28no—
113#381—28no—
114#381—28no—
115#380—28no—
116#380—28no—
117#380—28no—
118#380—28no—
119#380—24no—
120#380—24no—
121#380—24no—
122#380—24no—
123#379—28no—
124#379—44no—
125C4×C22⋊C4—28no—
126C4×C22⋊C4—28no—
127C2×#95—28no—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2312yes—
134C2×#95C2312yes—
135C2×#95C2314yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—42no—
151C25—42no—
152C25C2×D414yes—
153C25C2411yescharacteristic
154C25C2411yescharacteristic
155C25C2411yescharacteristic
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—232no—
161C23×C4—232no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—28no—
173C23×C4—28no—
174C23×C4—416no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—464no—
180C23×C4—44no—
181C23×C4—44no—
182C23×C4—44no—
183C23×C4—44no—
184C23×C4C2×D414yes—
185C23×C4C2×D414yes—
186C23×C4C2×D414yes—
187C23×C4C2×D414yes—
188C23×C4C2×D414yes—
189C23×C4C2×D414yes—
190C23×C4#4118yes—
191C23×C4#4118yes—
192C23×C4C2412yes—
193C23×C4C2418yes—
194C23×C4C2412yes—
195C23×C4C2412yes—
196C23×C4C2412yes—
197C23×C4C2414yes—
198C23×C4C2414yes—
199C23×C4C2411yescharacteristic
200C23×C4C2411yescharacteristic
201C23×C4C2411yescharacteristic
202C23×C4C2411yescharacteristic
203#117—416no—
204#117—416no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—28no—
214C2×C4⋊C4—28no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—44no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—416no—
222C2×C4⋊C4—416no—
223C2×C4⋊C4—48no—
224C2×C4⋊C4—48no—
225C2×C4⋊C4—48no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—216no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—432no—
248C2×C22⋊C4—44no—
249C2×C22⋊C4—44no—
250C2×C22⋊C4—44no—
251C2×C22⋊C4—44no—
252C2×C42—24no—
253C2×C42—24no—
254C2×C42—28no—
255C2×C42—44no—
256#95—28no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—48no—
267#95—48no—
268#95—48no—
269#95—48no—
270#95—48no—
271#95—48no—
272#95—48no—
273C24—22no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—232no—
278C24—28no—
279C24—28no—
280C24—28no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—28no—
294C24—28no—
295C24—44no—
296C24—44no—
297C24—416no—
298C24—416no—
299C24—416no—
300C24—416no—
301C24—416no—
302C24—416no—
303C24C22×D412yes—
304C24C22×D412yes—
305C24C22×D412yes—
306C24C2×#4114yes—
307C24C2×#4114yes—
308C24H(5,𝔽2)14yes—
309C24H(5,𝔽2)14yes—
310C242−1+414yes—
311C24C2511yescharacteristic
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—28no—
316C22×C4—232no—
317C22×C4—232no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—416no—
372C22×C4—416no—
373C22×C4—416no—
374C22×C4—416no—
375C22×C4—4128no—
376C22×C4—48no—
377C22×C4—48no—
378C22×C4—48no—
379C22×C4—48no—
380C22×C4—48no—
381C22×C4—48no—
382C22×C4—816no—
383C22×C4—816no—
384C22×C4#123116yes—
385C22×C4#123116yes—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C2×#4114yes—
391C22×C4C2×#4114yes—
392C22×C4C2×#4114yes—
393C22×C4C2×#4114yes—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—48no—
399C4⋊C4—48no—
400C4⋊C4—44no—
401C4⋊C4—44no—
402C4⋊C4—816no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—416no—
406C22⋊C4—48no—
407C22⋊C4—48no—
408C22⋊C4—48no—
409C22⋊C4—416no—
410C22⋊C4—48no—
411C22⋊C4—416no—
412C22⋊C4—88no—
413C22⋊C4—88no—
414C42—48no—
415C42—48no—
416C23—22no—
417C23—22no—
418C23—22no—
419C23—22no—
420C23—28no—
421C23—28no—
422C23—28no—
423C23—28no—
424C23—28no—
425C23—28no—
426C23—232no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—28no—
440C23—28no—
441C23—28no—
442C23—28no—
443C23—28no—
444C23—28no—
445C23—416no—
446C23—416no—
447C23—416no—
448C23—416no—
449C23—416no—
450C23—4128no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—416no—
458C23—44no—
459C23—44no—
460C23C2×#12314yes—
461C23C2×#12314yes—
462C23#53414yes—
463C23#53514yes—
464C23#53614yes—
465C23#551116yes—
466C23C23×D411yescharacteristic
467C23C22×#4112yes—
468C23C22×#4112yes—
469C23C2×H(5,𝔽2)12yes—
470C23C2×H(5,𝔽2)12yes—
471C23C2×2−1+412yes—
472C23#58512yes—
473C23#58512yes—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—28no—
483C2×C4—28no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—232no—
487C2×C4—232no—
488C2×C4—416no—
489C2×C4—416no—
490C2×C4—416no—
491C2×C4—416no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—48no—
518C2×C4—48no—
519C2×C4—48no—
520C2×C4—48no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—816no—
526C2×C4—816no—
527C2×C4C2×#12314yes—
528C2×C4C2×#12314yes—
529C2×C4C2×#12314yes—
530C2×C4C2×#12314yes—
531C22—24no—
532C22—24no—
533C22—24no—
534C22—24no—
535C22—24no—
536C22—232no—
537C22—24no—
538C22—24no—
539C22—24no—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—464no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—416no—
553C22—416no—
554C22—42no—
555C22#248318yes—
556C22C22×#12311yescharacteristic
557C22C22×#12311yescharacteristic
558C22C2×#53411yescharacteristic
559C22C2×#53511yescharacteristic
560C22C2×#53611yescharacteristic
561C22#344911yescharacteristic
562C22#344911yescharacteristic
563C22#345514yes—
564C22C2×#55114yes—
565C22#349214yes—
566C22#349414yes—
567C22#349514yes—
568C22#353414yes—
569C22#353514yes—
570C4—24no—
571C4—24no—
572C4—24no—
573C4—24no—
574C4—82no—
575C4—84no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—84no—
580C4—88no—
581C2—28no—
582C2—44no—
583C2—44no—
584C2#3463511yescharacteristic
585C2C2×#248311yescharacteristic
586C2#3522711yescharacteristic
587C2#3526211yescharacteristic
588C2#3531111yescharacteristic
589C2#3744611yescharacteristic
590C2#3744711yescharacteristic
591C2#3785418yes—
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
3. #17059C2semidirect
4. #17058C2semidirect
5. #17058C2semidirect
6. #17057C2semidirect
7. #15063C2semidirect
8. #15061C2semidirect
11. #14342C2semidirect
12. #14263C2semidirect
13. #14128C2semidirect
14. #13958C2semidirect
41. #1445C22semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1444C22semidirect
48. #1444C22semidirect
49. #1444C22semidirect
50. #1442C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
56. #1436C22semidirect
57. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22802
C22×#3802
#170592
#170582
#170572
#150632
#150612
#147272
#146812
#143422
#142632
#141282
#139582