Finite Group Databasebeta

← #4415856

Subgroups — #4415856

Order 512.

3042 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic, complemented
2C22×#388C211yesmaximal, characteristic
3#17060C212yesmaximal, complemented
4#17023C214yesmaximal, complemented
5#16676C214yesmaximal, complemented
6#16511C212yesmaximal, complemented
7#15015C212yesmaximal, complemented
8#15014C212yesmaximal, complemented
9#14820C212yesmaximal
10#14734C212yesmaximal
11#14720C212yesmaximal, complemented
12#14717C212yesmaximal
13#14683C211yesmaximal, characteristic, complemented
14#14630C214yesmaximal, complemented
15C22×C4⋊Q8C2211yescharacteristic, complemented
16C22×C22⋊Q8C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2857—28no—
23C2×#399C2214yes—
24C2×#397C2214yes—
25C2×#395C2214yescomplemented
26C2×#393C2214yes—
27C2×#391C2212yescomplemented
28C2×#389C2212yescomplemented
29C2×#389C2212yescomplemented
30C2×#388—28no—
31C2×#388C2212yes—
32C2×#388C2212yes—
33C2×#388C2212yes—
34C2×#388C2212yes—
35C2×#385C2214yes—
36C2×#384C2211yescharacteristic, complemented
37C2×#383C2212yescomplemented
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×#379C2214yes—
41C22×#95C2211yescharacteristic
42C22×#95C2211yescharacteristic
43#1445C2214yescomplemented
44#1445C2214yescomplemented
45#1444C2218yescomplemented
46#1444C2214yescomplemented
47#1443C2218yescomplemented
48#1442C2218yescomplemented
49#1442C2214yescomplemented
50#1441C2214yescomplemented
51#1440C2218yes—
52#1439C2218yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2218yes—
56#1439C2214yes—
57#1434C2212yescomplemented
58#1434C2212yescomplemented
59#1434C2212yescomplemented
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C26C2311yescharacteristic
66C23×Q8C2312yes—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C2×C4⋊Q8—24no—
74C2×C4⋊Q8—44no—
75C2×C22⋊Q8—28no—
76C2×C22⋊Q8—48no—
77C22×C4⋊C4—24no—
78C22×C4⋊C4—44no—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2318yes—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—24no—
91C22×C22⋊C4—24no—
92C22×C22⋊C4—44no—
93C22×C22⋊C4—44no—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2318yes—
101C22×C22⋊C4C2312yes—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2318yes—
104C22×C42—44no—
105C22×C42D418yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#399—48no—
111#397—216no—
112#395—48no—
113#393—48no—
114#391—28no—
115#389—28no—
116#389—28no—
117#388—28no—
118#388—28no—
119#388—28no—
120#388—28no—
121#385—48no—
122#384—24no—
123#383—44no—
124#380—28no—
125#380—28no—
126#379—48no—
127C2×#95—24no—
128C2×#95—24no—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2318yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C25—22no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—24no—
151C25—42no—
152C25—42no—
153C25—42no—
154C25C2412yes—
155C25C2411yescharacteristic
156C22×Q8—24no—
157C22×Q8—28no—
158C22×Q8—44no—
159C22×Q8—44no—
160C23×C4—22no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—22no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—42no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4C2×D414yes—
182C23×C4C2×D414yes—
183C23×C4C2×D414yes—
184C23×C4C2414yes—
185C23×C4C2414yes—
186C23×C4C2418yes—
187C23×C4C2412yes—
188C23×C4C2412yes—
189C23×C4C2411yescharacteristic
190C23×C4C2411yescharacteristic
191C23×C4C2411yescharacteristic
192C23×C4C2412yes—
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C4⋊Q8—416no—
197C22⋊Q8—432no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—28no—
208C2×C4⋊C4—28no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—416no—
216C2×C4⋊C4—48no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—48no—
219C2×C22⋊C4—232no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—216no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—416no—
241C2×C22⋊C4—416no—
242C2×C22⋊C4—48no—
243C2×C22⋊C4—48no—
244C2×C22⋊C4—48no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4#4118yes—
248C2×C42—24no—
249C2×C42—24no—
250C2×C42—416no—
251C2×C42—42no—
252C2×C42—42no—
253C2×C42C2×D414yes—
254C2×C42C2×D414yes—
255C2×C42C2×D414yes—
256C2×C42C2×D414yes—
257#95—24no—
258#95—24no—
259#95—24no—
260#95—24no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—216no—
265#95—28no—
266#95—28no—
267#95—44no—
268#95—48no—
269#95—44no—
270#95—44no—
271#95—48no—
272#95—48no—
273#95—48no—
274C24—22no—
275C24—22no—
276C24—22no—
277C24—22no—
278C24—28no—
279C24—22no—
280C24—22no—
281C24—24no—
282C24—24no—
283C24—24no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—416no—
288C24—416no—
289C24—416no—
290C24—44no—
291C24—44no—
292C24—44no—
293C24—44no—
294C24—44no—
295C24—44no—
296C24—416no—
297C24—88no—
298C24—88no—
299C24—88no—
300C24C22×D412yes—
301C24C2×#4114yes—
302C24H(5,𝔽2)14yes—
303C242−1+414yes—
304C242−1+414yes—
305C24C2511yescharacteristic
306C2×Q8—48no—
307C2×Q8—48no—
308C2×Q8—48no—
309C2×Q8—816no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—28no—
317C22×C4—28no—
318C22×C4—28no—
319C22×C4—28no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—28no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—28no—
359C22×C4—24no—
360C22×C4—28no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—416no—
365C22×C4—416no—
366C22×C4—416no—
367C22×C4—416no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—88no—
377C22×C4—88no—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C2×#4114yes—
385C22×C4C2×#4114yes—
386C22×C4H(5,𝔽2)14yes—
387C22×C4H(5,𝔽2)14yes—
388C4⋊C4—48no—
389C4⋊C4—48no—
390C4⋊C4—48no—
391C4⋊C4—416no—
392C4⋊C4—48no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—816no—
397C22⋊C4—216no—
398C22⋊C4—216no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—416no—
402C22⋊C4—416no—
403C22⋊C4—48no—
404C22⋊C4—48no—
405C22⋊C4—816no—
406C22⋊C4—816no—
407C42—28no—
408C42—28no—
409C42—44no—
410C42—44no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—28no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—24no—
422C23—28no—
423C23—28no—
424C23—28no—
425C23—416no—
426C23—416no—
427C23—416no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—48no—
438C23—48no—
439C23—48no—
440C23—48no—
441C23—864no—
442C23—88no—
443C23—88no—
444C23—88no—
445C23—88no—
446C23#53514yes—
447C23#53614yes—
448C23#53614yes—
449C23#563116yes—
450C23C23×D411yescharacteristic
451C23C22×#4112yes—
452C23C2×H(5,𝔽2)12yes—
453C23C2×H(5,𝔽2)12yes—
454C23C2×H(5,𝔽2)12yes—
455C23C2×2−1+412yes—
456C23C2×2−1+412yes—
457C23#58512yes—
458Q8—816no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—22no—
469C2×C4—24no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—28no—
479C2×C4—416no—
480C2×C4—416no—
481C2×C4—416no—
482C2×C4—416no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—48no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—88no—
515C2×C4—88no—
516C2×C4#52514yes—
517C2×C4#53414yes—
518C2×C4#53514yes—
519C22—24no—
520C22—232no—
521C22—24no—
522C22—24no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—44no—
527C22—416no—
528C22—44no—
529C22—44no—
530C22—832no—
531C22—88no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—82no—
536C22#282118yes—
537C22C2×#52511yescharacteristic
538C22C2×#53411yescharacteristic
539C22C2×#53511yescharacteristic
540C22C2×#53511yescharacteristic
541C22C2×#53611yescharacteristic
542C22C2×#53611yescharacteristic
543C22#344911yescharacteristic
544C22#349414yes—
545C22#352014yes—
546C22#352014yes—
547C22C2×#56314yes—
548C22#353514yes—
549C22#353714yes—
550C22#356114yes—
551C4—24no—
552C4—24no—
553C4—24no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—44no—
558C4—88no—
559C4—82no—
560C4—84no—
561C4—82no—
562C4—84no—
563C2—28no—
564C2—84no—
565C2#3508411yescharacteristic
566C2#3524011yescharacteristic
567C2#3530111yescharacteristic
568C2C2×#282111yescharacteristic
569C2#3744111yescharacteristic
570C2#3745011yescharacteristic
571C2#3753711yescharacteristic
572C2#4612418yes—
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
1. C2×#2857C2semidirect
3. #17060C2semidirect
4. #17023C2semidirect
5. #16676C2semidirect
6. #16511C2semidirect
7. #15015C2semidirect
8. #15014C2semidirect
11. #14720C2semidirect
13. #14683C2semidirect
14. #14630C2semidirect
15. C22×C4⋊Q8C22semidirect
25. C2×#395C22semidirect
27. C2×#391C22semidirect
28. C2×#389C22semidirect
29. C2×#389C22semidirect
36. C2×#384C22semidirect
37. C2×#383C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1443C22semidirect
48. #1442C22semidirect
49. #1442C22semidirect
50. #1441C22semidirect
57. #1434C22semidirect
58. #1434C22semidirect
59. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
C22×#3882
#170602
#170232
#166762
#165112
#150152
#150142
#148202
#147342
#147202
#147172
#146832
#146302