Finite Group Databasebeta

← #576159

Subgroups — #576159

Order 512.

3950 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2308C211yesmaximal, characteristic
2#14508C214yesmaximal, complemented
3#14427C214yesmaximal, complemented
4#14391C214yesmaximal, complemented
5#14262C212yesmaximal, complemented
6#14068C214yesmaximal, complemented
7#13988C214yesmaximal, complemented
8#13873C214yesmaximal, complemented
9#13867C212yesmaximal
10#13827C211yesmaximal, characteristic, complemented
11C22⋊C42C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C23×C22⋊C4C2212yes—
14C23×C22⋊C4C2212yes—
15#2308C4116yescomplemented
16C2×#388C2214yescomplemented
17C2×#388C2214yescomplemented
18C2×#386C2214yes—
19C2×#386C2214yes—
20C2×#386C2214yes—
21C2×#386C2214yes—
22C2×#385C2214yescomplemented
23C2×#381C2212yes—
24C2×#380C2214yescomplemented
25C2×#379C2212yes—
26C2×#379C2214yes—
27C2×#379C2211yescharacteristic
28C2×#379C2211yescharacteristic
29C2×C4×C22⋊C4C2214yescomplemented
30C2×C4×C22⋊C4C2214yescomplemented
31#1443C2214yescomplemented
32#1443C2214yescomplemented
33#1442C2214yescomplemented
34#1442C2214yescomplemented
35#1440C2214yescomplemented
36#1440C2214yescomplemented
37#1439C2214yes—
38#1439C2214yes—
39#1439C2214yescomplemented
40#1439C2214yescomplemented
41#1438C2212yes—
42#1438C2212yes—
43#1438C2212yes—
44#1438C2212yes—
45#1437C2214yes—
46#1437C2214yes—
47#1437C2214yescomplemented
48#1437C2214yescomplemented
49#1436C2214yescomplemented
50#1436C2214yescomplemented
51#1434C2214yescomplemented
52#1434C2214yescomplemented
53C22⋊#95C2214yes—
54C22⋊#95C2214yescomplemented
55C22⋊#95C2214yescomplemented
56C22⋊#95C2214yescomplemented
57C22⋊#95C2214yescomplemented
58C26C2311yescharacteristic
59C26C2311yescharacteristic
60C23×D4—28no—
61C23×D4—48no—
62C23×D4C2×C418yes—
63C23×D4C2312yes—
64C23×D4C2312yes—
65C24×C4C2314yes—
66C24×C4C2311yescharacteristic
67C24×C4C2314yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C22⋊C4—28no—
71C22×C22⋊C4—48no—
72C22×C22⋊C4—48no—
73C22×C22⋊C4C2×C4116yescomplemented
74C22×C22⋊C4C2312yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C22⋊C4C2314yes—
78C22×C22⋊C4C2314yes—
79C22×C22⋊C4C2312yes—
80C22×C22⋊C4C2314yes—
81C22×C22⋊C4C2312yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2314yes—
84C22×C22⋊C4C2312yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2312yes—
87C22×C22⋊C4C2312yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C42C2314yes—
91C22×C42C2314yes—
92#388—48no—
93#388—48no—
94#386—48no—
95#386—48no—
96#386C2×C4132yescomplemented
97#386C2×C4132yescomplemented
98#385—48no—
99#381C2×C4116yescomplemented
100#380—48no—
101#379—28no—
102#379—48no—
103#379C2×C418yes—
104#379C2×C418yes—
105C4×C22⋊C4—48no—
106C4×C22⋊C4—48no—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2314yes—
115C2×#95C2314yes—
116C2×#95C2314yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2314yes—
121C2×#95C2314yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2314yes—
125C25—24no—
126C25—28no—
127C25—44no—
128C25—44no—
129C25—44no—
130C25—44no—
131C25—44no—
132C25C22×C414yes—
133C25C22×C414yes—
134C25#4118yes—
135C25C2411yescharacteristic
136C25C2412yes—
137C25C2412yes—
138C22×D4—232no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—24no—
142C22×D4—24no—
143C22×D4—28no—
144C22×D4—28no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—432no—
148C22×D4—464no—
149C22×D4C22×C418yes—
150C22×D4C22×C418yes—
151C23×C4—24no—
152C23×C4—28no—
153C23×C4—28no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—44no—
163C23×C4—44no—
164C23×C4—44no—
165C23×C4—44no—
166C23×C4C22×C414yes—
167C23×C4C22×C4116yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2414yes—
171C23×C4C2414yes—
172C23×C4C2414yes—
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C23×C4C2414yes—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—28no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—28no—
190C2×C22⋊C4—28no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—432no—
200C2×C22⋊C4—432no—
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—44no—
211C2×C22⋊C4—44no—
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⋊C4C22×C4116yes—
222C2×C22⋊C4C22×C418yes—
223C2×C22⋊C4C22×C418yes—
224C2×C22⋊C4C22×C4116yes—
225C2×C22⋊C4C22×C4116yes—
226C2×C42—28no—
227C2×C42—44no—
228C2×C42—44no—
229C2×C42—44no—
230C2×C42—44no—
231C2×C42—44no—
232#95—28no—
233#95—28no—
234#95—44no—
235#95—44no—
236#95—44no—
237#95—44no—
238#95—44no—
239#95—44no—
240#95—44no—
241#95—44no—
242#95—44no—
243#95—44no—
244#95—44no—
245#95—44no—
246#95—44no—
247#95—44no—
248#95C22×C4116yes—
249#95C22×C4116yes—
250C24—232no—
251C24—22no—
252C24—22no—
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—28no—
268C24—216no—
269C24—48no—
270C24—48no—
271C24—416no—
272C24—416no—
273C24—48no—
274C24—416no—
275C24—416no—
276C24—84no—
277C24—84no—
278C24—88no—
279C24—88no—
280C24—88no—
281C24—88no—
282C24—88no—
283C24—88no—
284C24—88no—
285C24—88no—
286C24—88no—
287C24#117116yes—
288C24C23×C412yes—
289C24C23×C414yes—
290C24C23×C414yes—
291C24C2×#4114yes—
292C24C2×#4114yes—
293C24C2×#4114yes—
294C24H(5,𝔽2)18yes—
295C24C2511yescharacteristic
296C2×D4—216no—
297C2×D4—216no—
298C2×D4—216no—
299C2×D4—216no—
300C2×D4—48no—
301C2×D4—48no—
302C2×D4—416no—
303C2×D4—416no—
304C2×D4—416no—
305C2×D4—416no—
306C2×D4—4128no—
307C22×C4—28no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—24no—
318C22×C4—24no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—24no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—24no—
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—48no—
352C22×C4—416no—
353C22×C4—416no—
354C22×C4—416no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—416no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—88no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4C23×C412yes—
368C22×C4C23×C412yes—
369C22×C4C23×C418yes—
370C22×C4C23×C418yes—
371C22×C4H(5,𝔽2)18yes—
372C4⋊C4—88no—
373C4⋊C4—88no—
374C22⋊C4—216no—
375C22⋊C4—232no—
376C22⋊C4—48no—
377C22⋊C4—48no—
378C22⋊C4—416no—
379C22⋊C4—416no—
380C22⋊C4—416no—
381C22⋊C4—416no—
382C22⋊C4—416no—
383C22⋊C4—48no—
384C22⋊C4—84no—
385C22⋊C4—88no—
386C22⋊C4—84no—
387C22⋊C4—88no—
388C22⋊C4—88no—
389C22⋊C4—88no—
390C42—88no—
391C42—88no—
392C23—232no—
393C23—24no—
394C23—24no—
395C23—24no—
396C23—24no—
397C23—24no—
398C23—28no—
399C23—28no—
400C23—28no—
401C23—28no—
402C23—28no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—28no—
407C23—28no—
408C23—28no—
409C23—28no—
410C23—28no—
411C23—24no—
412C23—24no—
413C23—24no—
414C23—28no—
415C23—28no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—28no—
421C23—416no—
422C23—432no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—42no—
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—44no—
444C23—44no—
445C23—832no—
446C23—816no—
447C23—832no—
448C23—832no—
449C23—832no—
450C23—832no—
451C23—832no—
452C23C2×#11714yes—
453C23C2×#11718yes—
454C23#51818yes—
455C23C24×C411yescharacteristic
456C23C22×#4112yes—
457C23C2×H(5,𝔽2)14yes—
458C23C2×H(5,𝔽2)14yes—
459C23#58514yes—
460D4—432no—
461D4—432no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—28no—
472C2×C4—28no—
473C2×C4—28no—
474C2×C4—28no—
475C2×C4—416no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
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—
510C2×C4—44no—
511C2×C4—832no—
512C2×C4—832no—
513C22—24no—
514C22—28no—
515C22—28no—
516C22—216no—
517C22—28no—
518C22—28no—
519C22—28no—
520C22—48no—
521C22—48no—
522C22—48no—
523C22—48no—
524C22—48no—
525C22—48no—
526C22—48no—
527C22—48no—
528C22—48no—
529C22—48no—
530C22—48no—
531C22—48no—
532C22—48no—
533C22—48no—
534C22—44no—
535C22—44no—
536C22—832no—
537C22—832no—
538C22C22×#11711yescharacteristic
539C22C2×#51812yes—
540C22C4×H(5,𝔽2)14yes—
541C22#349012yes—
542C22#349014yes—
543C22#349112yes—
544C22#350514yes—
545C22#353414yes—
546C22#353714yes—
547C22#355814yes—
548C22#356014yes—
549C4—48no—
550C4—42no—
551C4—42no—
552C4—48no—
553C4—48no—
554C4—84no—
555C4—84no—
556C4—84no—
557C2—48no—
558C2—48no—
559C2—44no—
560C2#3417611yescharacteristic
561C2#3425112yes—
562C2#3443714yes—
563C2#4787714yes—
564C2#4789714yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14508C2semidirect
3. #14427C2semidirect
4. #14391C2semidirect
5. #14262C2semidirect
6. #14068C2semidirect
7. #13988C2semidirect
8. #13873C2semidirect
10. #13827C2semidirect
11. C22⋊C42C2semidirect
15. #2308C4semidirect
16. C2×#388C22semidirect
17. C2×#388C22semidirect
22. C2×#385C22semidirect
24. C2×#380C22semidirect
29. C2×C4×C22⋊C4C22semidirect
30. C2×C4×C22⋊C4C22semidirect
31. #1443C22semidirect
32. #1443C22semidirect
33. #1442C22semidirect
34. #1442C22semidirect
35. #1440C22semidirect
36. #1440C22semidirect
39. #1439C22semidirect
40. #1439C22semidirect
47. #1437C22semidirect
48. #1437C22semidirect
49. #1436C22semidirect
50. #1436C22semidirect
51. #1434C22semidirect
52. #1434C22semidirect
54. C22⋊#95C22semidirect
55. C22⋊#95C22semidirect
56. C22⋊#95C22semidirect
57. C22⋊#95C22semidirect
73. C22×C22⋊C4C2×C4semidirect
96. #386C2×C4semidirect
97. #386C2×C4semidirect
99. #381C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23082
#145082
#144272
#143912
#142622
#140682
#139882
#138732
#138672
#138272
C22⋊C422