Finite Group Databasebeta

← #5236484

Subgroups — #5236484

Order 512.

3086 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17271C211yesmaximal, characteristic, complemented
2#17024C214yesmaximal, complemented
3#16544C214yesmaximal, complemented
4#16255C214yesmaximal, complemented
5#15947C211yesmaximal, characteristic
6#15709C214yesmaximal, complemented
7#15407C214yesmaximal, complemented
8#15281C214yesmaximal, complemented
9#15259C212yesmaximal, complemented
10#14662C211yesmaximal, characteristic, complemented
11#14358C212yesmaximal, complemented
12C22×#124C2211yescharacteristic
13C22×#120C2212yes—
14C22×#120C2214yes—
15C2×#397C2214yescomplemented
16C2×#394C2214yes—
17C2×#394C2214yes—
18C2×#393C2214yescomplemented
19C2×#390C2214yes—
20C2×#388C2214yescomplemented
21C2×#388C2214yescomplemented
22C2×#388C2214yescomplemented
23C2×#382C2212yescomplemented
24C2×#379C2214yescomplemented
25C2×#379C2212yescomplemented
26C2×#379C2212yescomplemented
27#1446C2212yescomplemented
28#1444C2214yescomplemented
29#1443C2214yescomplemented
30#1442C2214yescomplemented
31#1440C2212yescomplemented
32#1440C2214yescomplemented
33#1440C2214yescomplemented
34#1440C2212yescomplemented
35#1439C2214yescomplemented
36#1439C2214yescomplemented
37#1439C2214yescomplemented
38#1439C2214yescomplemented
39#1439C2214yescomplemented
40#1439C2214yescomplemented
41#1439C2214yescomplemented
42#1439C2214yescomplemented
43#1438C2211yescharacteristic
44#1438C2212yes—
45#1438C2214yes—
46#1438C2214yes—
47#1438C2214yes—
48#1438C2211yescharacteristic
49#1438C2214yes—
50#1438C2214yes—
51#1438C2214yes—
52#1438C2214yes—
53#1437C2214yescomplemented
54#1437C2214yescomplemented
55#1436C2214yescomplemented
56#1434C2214yescomplemented
57C26C2312yes—
58C23×Q8C2311yescharacteristic
59C23×D4C2314yes—
60C23×D4C2314yes—
61C23×D4C2311yescharacteristic
62C2×#124—24no—
63C2×#124—44no—
64C2×#120—28no—
65C2×#120—48no—
66C2×#120—48no—
67C2×#120—48no—
68C2×#120—48no—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2314yes—
72C22×C22⋊C4C2312yes—
73C22×C22⋊C4C2314yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2312yes—
77C22×C22⋊C4C2312yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2314yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2314yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2312yes—
91C22×C42C2311yescharacteristic
92C22×C42C2314yes—
93#397—48no—
94#394—48no—
95#394—48no—
96#393—48no—
97#390—48no—
98#388—48no—
99#388—48no—
100#388—48no—
101#382—44no—
102#379—44no—
103#379—44no—
104#379—48no—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2312yes—
117C2×#95C2314yes—
118C2×#95C2312yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C25—24no—
122C25—28no—
123C25—44no—
124C25—44no—
125C25—44no—
126C25—44no—
127C25—44no—
128C25—42no—
129C25—42no—
130C25C2412yes—
131C25C2414yes—
132C22×Q8—22no—
133C22×Q8—42no—
134C22×Q8—44no—
135C22×D4—28no—
136C22×D4—28no—
137C22×D4—28no—
138C22×D4—28no—
139C22×D4—28no—
140C22×D4—44no—
141C22×D4—44no—
142C22×D4—44no—
143C22×D4—44no—
144C22×D4—44no—
145C22×D4—44no—
146C22×D4—42no—
147C22×D4—44no—
148C22×D4—44no—
149C22×D4—44no—
150C22×D4—44no—
151C22×D4#4114yes—
152C23×C4C2414yes—
153C23×C4C2414yes—
154C23×C4C2414yes—
155C23×C4C2411yescharacteristic
156C23×C4C2411yescharacteristic
157C23×C4C2414yes—
158C23×C4C2411yescharacteristic
159C23×C4C2412yes—
160C23×C4C2414yes—
161#124—416no—
162#120—432no—
163#120—464no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4—44no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—28no—
230C2×C42—44no—
231C2×C42—44no—
232#95—28no—
233#95—28no—
234#95—28no—
235#95—28no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—42no—
240#95—42no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248C24—24no—
249C24—24no—
250C24—22no—
251C24—22no—
252C24—24no—
253C24—24no—
254C24—24no—
255C24—24no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—24no—
260C24—24no—
261C24—24no—
262C24—24no—
263C24—24no—
264C24—24no—
265C24—24no—
266C24—24no—
267C24—24no—
268C24—24no—
269C24—24no—
270C24—416no—
271C24—48no—
272C24—48no—
273C24—416no—
274C24—48no—
275C24—84no—
276C24—84no—
277C24—88no—
278C24—88no—
279C24—88no—
280C24—88no—
281C24—88no—
282C24—88no—
283C24—84no—
284C24—84no—
285C24—88no—
286C24—88no—
287C24—88no—
288C24—88no—
289C24—88no—
290C24—88no—
291C24C2×#4114yes—
292C24H(5,𝔽2)18yes—
293C24C2511yescharacteristic
294C2×Q8—44no—
295C2×Q8—84no—
296C2×Q8—88no—
297C2×D4—48no—
298C2×D4—44no—
299C2×D4—416no—
300C2×D4—416no—
301C2×D4—416no—
302C2×D4—416no—
303C2×D4—416no—
304C2×D4—88no—
305C2×D4—88no—
306C2×D4—88no—
307C2×D4—88no—
308C2×D4—88no—
309C2×D4—88no—
310C2×D4—88no—
311C2×D4—88no—
312C2×D4—88no—
313C2×D4—88no—
314C22×C4—22no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—24no—
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—24no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4C2×#4112yes—
361C22×C4H(5,𝔽2)18yes—
362C22×C4H(5,𝔽2)14yes—
363C4⋊C4—88no—
364C4⋊C4—88no—
365C4⋊C4—88no—
366C22⋊C4—216no—
367C22⋊C4—416no—
368C22⋊C4—416no—
369C22⋊C4—416no—
370C22⋊C4—48no—
371C22⋊C4—48no—
372C22⋊C4—416no—
373C22⋊C4—416no—
374C22⋊C4—416no—
375C22⋊C4—88no—
376C22⋊C4—88no—
377C22⋊C4—88no—
378C22⋊C4—88no—
379C22⋊C4—88no—
380C22⋊C4—84no—
381C22⋊C4—84no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C22⋊C4—88no—
385C42—28no—
386C42—88no—
387C23—24no—
388C23—24no—
389C23—24no—
390C23—28no—
391C23—28no—
392C23—28no—
393C23—28no—
394C23—28no—
395C23—28no—
396C23—28no—
397C23—28no—
398C23—28no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—44no—
410C23—44no—
411C23—44no—
412C23—44no—
413C23—44no—
414C23—42no—
415C23—44no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—42no—
420C23—44no—
421C23—44no—
422C23—44no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—44no—
439C23—44no—
440C23—44no—
441C23—44no—
442C23—44no—
443C23—816no—
444C23—832no—
445C23—832no—
446C23—832no—
447C23—832no—
448C23—832no—
449C23—816no—
450C23—816no—
451C23—832no—
452C23C22×#4111yescharacteristic
453C23C2×H(5,𝔽2)14yes—
454C23C2×H(5,𝔽2)12yes—
455C23C2×H(5,𝔽2)14yes—
456C23#58514yes—
457Q8—88no—
458D4—88no—
459D4—832no—
460D4—832no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—28no—
464C2×C4—24no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—28no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—44no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C22—48no—
511C22—48no—
512C22—48no—
513C22—48no—
514C22—44no—
515C22—44no—
516C22—48no—
517C22—44no—
518C22—48no—
519C22—48no—
520C22—48no—
521C22—48no—
522C22—864no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—84no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22—84no—
533C22—84no—
534C22—84no—
535C22#349011yescharacteristic
536C22#349012yes—
537C22#351314yes—
538C22#353214yes—
539C22#353414yes—
540C22#353712yes—
541C22#354012yes—
542C22#354214yes—
543C22#354914yes—
544C22#355814yes—
545C22#356014yes—
546C4—44no—
547C4—42no—
548C4—48no—
549C4—84no—
550C4—84no—
551C4—84no—
552C4—84no—
553C4—81no—
554C4—81no—
555C2—88no—
556C2—84no—
557C2#4756311yescharacteristic
558C2#4835914yes—
559C2#5181512yes—
560C2#5481514yes—
561C2#5739814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15947C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #17271C2semidirect
2. #17024C2semidirect
3. #16544C2semidirect
4. #16255C2semidirect
6. #15709C2semidirect
7. #15407C2semidirect
8. #15281C2semidirect
9. #15259C2semidirect
10. #14662C2semidirect
11. #14358C2semidirect
15. C2×#397C22semidirect
18. C2×#393C22semidirect
20. C2×#388C22semidirect
21. C2×#388C22semidirect
22. C2×#388C22semidirect
23. C2×#382C22semidirect
24. C2×#379C22semidirect
25. C2×#379C22semidirect
26. C2×#379C22semidirect
27. #1446C22semidirect
28. #1444C22semidirect
29. #1443C22semidirect
30. #1442C22semidirect
31. #1440C22semidirect
32. #1440C22semidirect
33. #1440C22semidirect
34. #1440C22semidirect
35. #1439C22semidirect
36. #1439C22semidirect
37. #1439C22semidirect
38. #1439C22semidirect
39. #1439C22semidirect
40. #1439C22semidirect
41. #1439C22semidirect
42. #1439C22semidirect
53. #1437C22semidirect
54. #1437C22semidirect
55. #1436C22semidirect
56. #1434C22semidirect

Maximal subgroups

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

subgroupindex
#172712
#170242
#165442
#162552
#159472
#157092
#154072
#152812
#152592
#146622
#143582