Finite Group Databasebeta

← #4456853

Subgroups — #4456853

Order 512.

2414 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic
3#16428C214yesmaximal, complemented
4#16427C214yesmaximal, complemented
5#16393C214yesmaximal, complemented
6#16391C214yesmaximal
7#15006C214yesmaximal, complemented
8#14998C214yesmaximal, complemented
9#14297C211yesmaximal, characteristic, complemented
10#13953C211yesmaximal, characteristic, complemented
11#13939C211yesmaximal, characteristic, complemented
12#13903C211yesmaximal, characteristic
13#13853C211yesmaximal, characteristic, complemented
14C22×#117C2211yescharacteristic
15C23×C22⋊C4C2211yescharacteristic
16#2482—28no—
17C2×#399C2212yes—
18C2×#398C2214yescomplemented
19C2×#397C2212yes—
20C2×#397C2214yes—
21#2312—28no—
22C2×#388C2214yes—
23C2×#388C2214yes—
24C2×#388C2214yes—
25C2×#385C2212yes—
26C2×#381C2211yescharacteristic
27C2×#381C2212yes—
28C2×#380C2212yes—
29C2×#380C2211yescharacteristic
30C2×#379C2212yes—
31C2×#379C2212yes—
32C2×#379C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2211yescharacteristic
35C2×C4×C22⋊C4C2211yescharacteristic
36#1445C2214yescomplemented
37#1445C2214yescomplemented
38#1445C2214yescomplemented
39#1445C2212yescomplemented
40#1445C2212yescomplemented
41#1442C2214yescomplemented
42#1442C2214yescomplemented
43#1442C2214yescomplemented
44#1441C2214yescomplemented
45#1441C2214yescomplemented
46#1440C2214yes—
47#1440C2214yes—
48#1440C2214yes—
49#1440C2214yes—
50#1440C2214yes—
51#1440C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1437C2212yes—
59#1437C2212yes—
60#1436C2214yescomplemented
61#1436C2214yescomplemented
62C4⋊#95C2212yescomplemented
63C4⋊#95C2212yescomplemented
64C4⋊#95C2212yescomplemented
65#1434C2214yescomplemented
66#1434C2212yescomplemented
67C4×#95C2212yescomplemented
68C4×#95C2212yescomplemented
69C26C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C2×#117—24no—
73C2×#117—24no—
74C2×#117—24no—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2314yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2314yes—
82C22×C22⋊C4—24no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4D418yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2312yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C42C2312yes—
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2312yes—
104#399—28no—
105#398—48no—
106#397—28no—
107#397—48no—
108#388—216no—
109#388—216no—
110#388—216no—
111#385—28no—
112#381—24no—
113#381—44no—
114#380—28no—
115#380—24no—
116#379—28no—
117#379—28no—
118#379—44no—
119C4×C22⋊C4—24no—
120C4×C22⋊C4—24no—
121C4×C22⋊C4—24no—
122C2×#95C2314yes—
123C2×#95C2312yes—
124C2×#95C2314yes—
125C2×#95C2314yes—
126C2×#95C2312yes—
127C2×#95C2312yes—
128C2×#95C2312yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C25—22no—
147C25—22no—
148C25—44no—
149C25—42no—
150C25—42no—
151C25—42no—
152C25—42no—
153C25C2×D414yes—
154C25C2412yes—
155C25C2411yescharacteristic
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—42no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4C2×D414yes—
163C23×C4C2×D414yes—
164C23×C4#4114yes—
165C23×C4#4114yes—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2412yes—
174C23×C4C2412yes—
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177#117—416no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C22⋊C4—232no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—28no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—44no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—42no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4C2×D418yes—
239C2×C22⋊C4C2×D418yes—
240C2×C42—24no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—42no—
248C2×C42—42no—
249C2×C42—42no—
250C2×C42—42no—
251C2×C42—42no—
252C2×C42—42no—
253#95—24no—
254#95—24no—
255#95—28no—
256#95—24no—
257#95—24no—
258#95—28no—
259#95—24no—
260#95—24no—
261#95—28no—
262#95—28no—
263#95—44no—
264#95—44no—
265#95—44no—
266#95—44no—
267#95—44no—
268#95—44no—
269#95—44no—
270#95—44no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277C24—22no—
278C24—24no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—24no—
285C24—24no—
286C24—24no—
287C24—24no—
288C24—24no—
289C24—24no—
290C24—28no—
291C24—48no—
292C24—48no—
293C24—48no—
294C24—48no—
295C24—48no—
296C24—84no—
297C24—82no—
298C24—82no—
299C24—84no—
300C24—82no—
301C24—82no—
302C24—88no—
303C24—88no—
304C24—88no—
305C24—88no—
306C24C22×D414yes—
307C24C22×D412yes—
308C24C2×#4112yes—
309C24C2×#4112yes—
310C24C2511yescharacteristic
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—28no—
324C22×C4—24no—
325C22×C4—22no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—24no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—48no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—84no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4#123116yes—
376C22×C4C22×D414yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C2×#4112yes—
380C22×C4C2×#4112yes—
381C22×C4C2×#4112yes—
382C22×C4C2×#4112yes—
383C22×C4H(5,𝔽2)14yes—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—84no—
387C4⋊C4—88no—
388C4⋊C4—88no—
389C4⋊C4—88no—
390C4⋊C4—88no—
391C22⋊C4—216no—
392C22⋊C4—216no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—416no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—88no—
400C22⋊C4—88no—
401C22⋊C4—88no—
402C22⋊C4—88no—
403C22⋊C4—84no—
404C22⋊C4—88no—
405C42—48no—
406C42—84no—
407C42—84no—
408C42—82no—
409C42—82no—
410C23—24no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—22no—
416C23—22no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—28no—
425C23—28no—
426C23—416no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—44no—
437C23—42no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—88no—
448C23—88no—
449C23—832no—
450C23—816no—
451C23—816no—
452C23—816no—
453C23—816no—
454C23C2×#12314yes—
455C23C23×D411yescharacteristic
456C23C22×#4111yescharacteristic
457C23C22×#4111yescharacteristic
458C23C2×H(5,𝔽2)12yes—
459C23#58514yes—
460C23#58512yes—
461C23#58512yes—
462C23#58512yes—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—24no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—416no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—88no—
526C2×C4C2×#12314yes—
527C2×C4C2×#12314yes—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—44no—
532C22—44no—
533C22—48no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—44no—
538C22—48no—
539C22—48no—
540C22—44no—
541C22—832no—
542C22—84no—
543C22—84no—
544C22—84no—
545C22—84no—
546C22C22×#12311yescharacteristic
547C22#344912yes—
548C22#345512yes—
549C22#345512yes—
550C22#348512yes—
551C22#349112yes—
552C22#349111yescharacteristic
553C22#349111yescharacteristic
554C22#349412yes—
555C22#349612yes—
556C22#350811yescharacteristic
557C22#351011yescharacteristic
558C22#354514yes—
559C22#356214yes—
560C22#356614yes—
561C22#356614yes—
562C4—24no—
563C4—24no—
564C4—44no—
565C4—44no—
566C4—82no—
567C4—84no—
568C4—84no—
569C4—84no—
570C4—84no—
571C4—82no—
572C4—82no—
573C2—44no—
574C2—84no—
575C2#3463511yescharacteristic
576C2#3544811yescharacteristic
577C2#3554111yescharacteristic
578C2#3701012yes—
579C2#3701412yes—
580C2#4137814yes—
581C2#4144212yes—
582C2#4144312yes—
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. #16428C2semidirect
4. #16427C2semidirect
5. #16393C2semidirect
7. #15006C2semidirect
8. #14998C2semidirect
9. #14297C2semidirect
10. #13953C2semidirect
11. #13939C2semidirect
13. #13853C2semidirect
18. C2×#398C22semidirect
36. #1445C22semidirect
37. #1445C22semidirect
38. #1445C22semidirect
39. #1445C22semidirect
40. #1445C22semidirect
41. #1442C22semidirect
42. #1442C22semidirect
43. #1442C22semidirect
44. #1441C22semidirect
45. #1441C22semidirect
60. #1436C22semidirect
61. #1436C22semidirect
62. C4⋊#95C22semidirect
63. C4⋊#95C22semidirect
64. C4⋊#95C22semidirect
65. #1434C22semidirect
66. #1434C22semidirect
67. C4×#95C22semidirect
68. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C2×#23122
#164282
#164272
#163932
#163912
#150062
#149982
#142972
#139532
#139392
#139032
#138532