Finite Group Databasebeta

← #768961

Subgroups — #768961

Order 512.

3146 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#396C211yesmaximal, characteristic
2C2×#2315C211yesmaximal, characteristic
3#15470C218yesmaximal, complemented
4#15385C218yesmaximal, complemented
5#15015C212yesmaximal, complemented
6#14822C212yesmaximal, complemented
7#13923C212yesmaximal, complemented
8#13881C212yesmaximal, complemented
9#13869C212yesmaximal
10#13869C212yesmaximal
11#13775C211yesmaximal, characteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C2×#400C2214yescomplemented
20C2×#396—28no—
21C2×#396C2218yes—
22#2315—28no—
23C2×#385C2214yes—
24C2×#385C2214yes—
25C2×#382C2214yescomplemented
26C2×#381C2214yes—
27C2×#381C2214yes—
28C2×#380C2212yes—
29C2×#380C2212yes—
30C2×#379C2212yes—
31C2×C4×C22⋊C4C2212yes—
32C22×#95C2211yescharacteristic
33#1445C2214yescomplemented
34#1443C2218yescomplemented
35#1442C2218yescomplemented
36#1442C2218yescomplemented
37#1442C2214yescomplemented
38#1441C2214yescomplemented
39#1441C2214yescomplemented
40#1441C2214yescomplemented
41#1440C2218yes—
42#1440C2218yes—
43#1437C2218yes—
44#1436C2218yescomplemented
45C4⋊#95C2214yescomplemented
46C22⋊#95C2214yes—
47C22⋊#95C2214yes—
48C22⋊#95C2214yes—
49C22⋊#95C2214yes—
50C22⋊#95C2214yes—
51C22⋊#95C2214yes—
52C22⋊#95C2212yes—
53C26C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C24×C4C2311yescharacteristic
57C24×C4C2311yescharacteristic
58C24×C4C2312yes—
59C24×C4C2312yes—
60C24×C4C2311yescharacteristic
61C24×C4C2311yescharacteristic
62C22×C4⋊C4—24no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—28no—
66C22×C4⋊C4—44no—
67C22×C4⋊C4—44no—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2314yes—
70C22×C4⋊C4C2314yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C22⋊C4—28no—
76C22×C22⋊C4—28no—
77C22×C22⋊C4—24no—
78C22×C22⋊C4—24no—
79C22×C22⋊C4—28no—
80C22×C22⋊C4—28no—
81C22×C22⋊C4—44no—
82C22×C22⋊C4—44no—
83C22×C22⋊C4D418yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2314yes—
86C22×C22⋊C4C2314yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C42C2314yes—
93#400—48no—
94#396—232no—
95#385—216no—
96#385—216no—
97#382—48no—
98#381—48no—
99#381—48no—
100#380—28no—
101#380—28no—
102#379—28no—
103C4×C22⋊C4—28no—
104C2×#95—24no—
105C2×#95C2314yes—
106C2×#95C2318yes—
107C2×#95C2318yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2318yes—
112C2×#95C2314yes—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2318yes—
117C2×#95C2318yes—
118C2×#95C2314yes—
119C25—24no—
120C25—24no—
121C25—42no—
122C25—42no—
123C25—42no—
124C25—42no—
125C25C2×D414yes—
126C25#4118yes—
127C25C2412yes—
128C25C2411yescharacteristic
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—24no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—42no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4—44no—
160C23×C4C2×D414yes—
161C23×C4C2×D414yes—
162C23×C4#4118yes—
163C23×C4C2418yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2412yes—
167C23×C4C2412yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—416no—
186C2×C4⋊C4—416no—
187C2×C4⋊C4—416no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C22⋊C4—232no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—28no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—416no—
213C2×C22⋊C4—416no—
214C2×C22⋊C4—432no—
215C2×C22⋊C4—44no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4C2×D4116yes—
220C2×C42—28no—
221C2×C42—44no—
222C2×C42—44no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—28no—
227#95—216no—
228#95—216no—
229#95—48no—
230#95—44no—
231#95—48no—
232#95—48no—
233#95—48no—
234#95—48no—
235#95—48no—
236#95—44no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24—22no—
241C24—24no—
242C24—24no—
243C24—24no—
244C24—24no—
245C24—24no—
246C24—24no—
247C24—28no—
248C24—28no—
249C24—232no—
250C24—44no—
251C24—44no—
252C24—44no—
253C24—44no—
254C24—416no—
255C24—416no—
256C24—88no—
257C24—88no—
258C24—88no—
259C24—88no—
260C24#124116yes—
261C24C22×D414yes—
262C24C22×D412yes—
263C24C2×#4114yes—
264C24C2×#4114yes—
265C24C2×#4114yes—
266C24C2×#4114yes—
267C242−1+414yes—
268C24C2511yescharacteristic
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—28no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—28no—
298C22×C4—28no—
299C22×C4—28no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—28no—
304C22×C4—24no—
305C22×C4—28no—
306C22×C4—24no—
307C22×C4—28no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—416no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—416no—
316C22×C4—416no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4—44no—
325C22×C4—416no—
326C22×C4—416no—
327C22×C4—44no—
328C22×C4—48no—
329C22×C4—48no—
330C22×C4—44no—
331C22×C4—88no—
332C22×C4—84no—
333C22×C4—84no—
334C22×C4—84no—
335C22×C4—84no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—88no—
339C22×C4—88no—
340C22×C4#123116yes—
341C22×C4C22×D414yes—
342C22×C4C22×D414yes—
343C22×C4C2×#4114yes—
344C22×C4C2×#4114yes—
345C22×C4H(5,𝔽2)14yes—
346C22×C4H(5,𝔽2)14yes—
347C4⋊C4—416no—
348C4⋊C4—416no—
349C4⋊C4—416no—
350C4⋊C4—84no—
351C4⋊C4—84no—
352C4⋊C4—84no—
353C4⋊C4—84no—
354C22⋊C4—232no—
355C22⋊C4—416no—
356C22⋊C4—416no—
357C22⋊C4—416no—
358C22⋊C4—416no—
359C22⋊C4—416no—
360C22⋊C4—88no—
361C22⋊C4—88no—
362C42—416no—
363C23—22no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—28no—
368C23—28no—
369C23—28no—
370C23—28no—
371C23—28no—
372C23—28no—
373C23—24no—
374C23—24no—
375C23—28no—
376C23—28no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—24no—
381C23—28no—
382C23—28no—
383C23—28no—
384C23—28no—
385C23—232no—
386C23—416no—
387C23—416no—
388C23—48no—
389C23—48no—
390C23—48no—
391C23—48no—
392C23—88no—
393C23—864no—
394C23—88no—
395C23—88no—
396C23—88no—
397C23C2×#12314yes—
398C23C2×#12418yes—
399C23C2×#12414yes—
400C23#53614yes—
401C23#552116yes—
402C23C23×D411yescharacteristic
403C23C22×#4112yes—
404C23C22×#4112yes—
405C23C2×H(5,𝔽2)12yes—
406C23C2×H(5,𝔽2)12yes—
407C23C2×2−1+412yes—
408C23#58512yes—
409C23#58512yes—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—416no—
422C2×C4—416no—
423C2×C4—416no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—48no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—48no—
445C2×C4—48no—
446C2×C4—48no—
447C2×C4—48no—
448C2×C4—48no—
449C2×C4—48no—
450C2×C4—48no—
451C2×C4—48no—
452C2×C4—48no—
453C2×C4—48no—
454C2×C4—48no—
455C2×C4—48no—
456C2×C4—88no—
457C2×C4—88no—
458C2×C4—816no—
459C2×C4—816no—
460C2×C4—88no—
461C2×C4C2×#12318yes—
462C22—24no—
463C22—232no—
464C22—24no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—28no—
470C22—28no—
471C22—28no—
472C22—44no—
473C22—416no—
474C22—416no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22—832no—
479C22#250318yes—
480C22C22×#12311yescharacteristic
481C22C22×#12411yescharacteristic
482C22C2×#53611yescharacteristic
483C22C2×#55214yes—
484C22#348411yescharacteristic
485C22#348411yescharacteristic
486C22#348511yescharacteristic
487C22#348511yescharacteristic
488C22#349414yes—
489C22#350514yes—
490C22#350514yes—
491C22#350514yes—
492C22#353614yes—
493C22#356414yes—
494C4—28no—
495C4—44no—
496C4—44no—
497C4—44no—
498C4—82no—
499C4—88no—
500C4—84no—
501C4—84no—
502C4—82no—
503C2—28no—
504C2—48no—
505C2C2×#250311yescharacteristic
506C2#3535111yescharacteristic
507C2#3537711yescharacteristic
508C2#3537711yescharacteristic
509C2#3554011yescharacteristic
510C2#3671811yescharacteristic
511C2#3697011yescharacteristic
512C2#4305218yes—
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. #15470C2semidirect
4. #15385C2semidirect
5. #15015C2semidirect
6. #14822C2semidirect
7. #13923C2semidirect
8. #13881C2semidirect
19. C2×#400C22semidirect
25. C2×#382C22semidirect
33. #1445C22semidirect
34. #1443C22semidirect
35. #1442C22semidirect
36. #1442C22semidirect
37. #1442C22semidirect
38. #1441C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
44. #1436C22semidirect
45. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3962
C2×#23152
#154702
#153852
#150152
#148222
#139232
#138812
#138692
#137752