Finite Group Databasebeta

← #765935

Subgroups — #765935

Order 512.

4406 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C212yesmaximal, complemented
2C2×#2405C212yesmaximal
3C22×#397C211yesmaximal, characteristic, complemented
4C2×#2308C211yesmaximal, characteristic
5#15549C218yesmaximal, complemented
6#15539C214yesmaximal, complemented
7#15480C214yesmaximal, complemented
8#15069C212yesmaximal, complemented
9#14946C212yesmaximal, complemented
10#13996C214yesmaximal, complemented
11#13775C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C22×#123C2214yes—
14C23×C4⋊C4C2211yescharacteristic, complemented
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18#2483—216no—
19#2405—216no—
20C2×#397—28no—
21C2×#397C2218yes—
22C2×#397C2214yes—
23C2×#397C2214yescomplemented
24C2×#394C2218yes—
25#2308—28no—
26C2×#386C2214yes—
27C2×#386C2212yes—
28C2×#386C2212yes—
29C2×#386C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yescomplemented
32C2×#385C2214yescomplemented
33C2×#381C2212yes—
34C2×#381C2214yescomplemented
35C2×#381C2214yescomplemented
36C2×#379C2214yes—
37C2×#379C2212yes—
38C2×#379C2212yes—
39C2×C4×C22⋊C4C2212yescomplemented
40C22×#95C2211yescharacteristic, complemented
41C22×#95C2212yescomplemented
42#1445C2218yescomplemented
43#1445C2218yescomplemented
44#1444C2218yescomplemented
45#1439C2218yescomplemented
46#1439C2218yescomplemented
47#1439C2218yescomplemented
48#1437C2214yes—
49#1437C2218yescomplemented
50#1437C2214yescomplemented
51C4⋊#95C2218yescomplemented
52C22⋊#95C2212yescomplemented
53C26C2311yescharacteristic
54C26C2311yescharacteristic
55C23×D4—24no—
56C23×D4—28no—
57C23×D4—28no—
58C23×D4—44no—
59C23×D4C2312yes—
60C23×D4C2312yes—
61C24×C4C2312yes—
62C24×C4C2311yescharacteristic
63C24×C4C2312yes—
64C24×C4C2312yes—
65C24×C4C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C2×#123—216no—
68C2×#123—216no—
69C2×#123—48no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4—44no—
72C22×C4⋊C4C2318yes—
73C22×C4⋊C4C2314yes—
74C22×C4⋊C4C2318yes—
75C22×C22⋊C4—24no—
76C22×C22⋊C4—28no—
77C22×C22⋊C4—28no—
78C22×C22⋊C4—28no—
79C22×C22⋊C4—28no—
80C22×C22⋊C4—44no—
81C22×C22⋊C4D418yes—
82C22×C22⋊C4C2314yes—
83C22×C22⋊C4C2312yes—
84C22×C22⋊C4C2314yes—
85C22×C22⋊C4C2312yes—
86C22×C22⋊C4C2318yes—
87C22×C22⋊C4C2314yes—
88C22×C22⋊C4C2318yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C42C2314yes—
96#397—216no—
97#397—216no—
98#397—232no—
99#394—232no—
100#386—28no—
101#386—28no—
102#386—216no—
103#386—216no—
104#385—216no—
105#385—216no—
106#385—216no—
107#381—28no—
108#381—216no—
109#381—216no—
110#379—28no—
111#379—216no—
112#379D4116yes—
113C4×C22⋊C4—44no—
114C2×#95—28no—
115C2×#95—24no—
116C2×#95C2318yes—
117C2×#95C2318yes—
118C2×#95C2314yes—
119C2×#95C2318yes—
120C2×#95C2318yes—
121C2×#95C2318yes—
122C2×#95C2314yes—
123C2×#95C2314yes—
124C2×#95C2318yes—
125C2×#95C2314yes—
126C25—24no—
127C25—24no—
128C25—22no—
129C25—24no—
130C25—24no—
131C25—42no—
132C25—44no—
133C25—44no—
134C25—44no—
135C25—44no—
136C25C2×D414yes—
137C25C2×D418yes—
138C25C2412yes—
139C25C2411yescharacteristic
140C25C2411yescharacteristic
141C25C2411yescharacteristic
142C22×D4—232no—
143C22×D4—28no—
144C22×D4—24no—
145C22×D4—24no—
146C22×D4—24no—
147C22×D4—24no—
148C22×D4—28no—
149C22×D4—28no—
150C22×D4—24no—
151C22×D4—44no—
152C22×D4—464no—
153C22×D4—416no—
154C22×D4—416no—
155C22×D4#4118yes—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—28no—
167C23×C4—28no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—44no—
173C23×C4—42no—
174C23×C4—44no—
175C23×C4—44no—
176C23×C4—44no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4C2×D414yes—
180C23×C4C2×D414yes—
181C23×C4C2414yes—
182C23×C4C2414yes—
183C23×C4C2414yes—
184C23×C4C2418yes—
185C23×C4C2412yes—
186C23×C4C2412yes—
187C23×C4C2412yes—
188#123—464no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—216no—
191C2×C4⋊C4—216no—
192C2×C4⋊C4—216no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—48no—
196C2×C4⋊C4—416no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—48no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—28no—
204C2×C22⋊C4—28no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—28no—
207C2×C22⋊C4—28no—
208C2×C22⋊C4—28no—
209C2×C22⋊C4—28no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—216no—
212C2×C22⋊C4—216no—
213C2×C22⋊C4—28no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—216no—
216C2×C22⋊C4—416no—
217C2×C22⋊C4—42no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—44no—
222C2×C22⋊C4—44no—
223C2×C22⋊C4—44no—
224C2×C22⋊C4—44no—
225C2×C22⋊C4—48no—
226C2×C22⋊C4—416no—
227C2×C22⋊C4—432no—
228C2×C22⋊C4—48no—
229C2×C22⋊C4—48no—
230C2×C22⋊C4—48no—
231C2×C22⋊C4—48no—
232C2×C22⋊C4C2×D4116yes—
233C2×C22⋊C4C2×D418yes—
234C2×C22⋊C4C2×D418yes—
235C2×C22⋊C4C2×D4116yes—
236C2×C22⋊C4C2×D4116yes—
237C2×C42—44no—
238C2×C42—48no—
239#95—28no—
240#95—216no—
241#95—216no—
242#95—216no—
243#95—216no—
244#95—28no—
245#95—28no—
246#95—216no—
247#95—48no—
248#95—44no—
249C24—24no—
250C24—24no—
251C24—22no—
252C24—22no—
253C24—22no—
254C24—22no—
255C24—24no—
256C24—24no—
257C24—24no—
258C24—24no—
259C24—28no—
260C24—28no—
261C24—232no—
262C24—28no—
263C24—22no—
264C24—24no—
265C24—22no—
266C24—24no—
267C24—28no—
268C24—28no—
269C24—48no—
270C24—48no—
271C24—416no—
272C24—416no—
273C24—416no—
274C24—44no—
275C24—416no—
276C24—44no—
277C24—816no—
278C24—88no—
279C24—88no—
280C24—84no—
281C24—84no—
282C24—88no—
283C24—816no—
284C24—88no—
285C24—88no—
286C24#127116yes—
287C24C22×D412yes—
288C24C22×D414yes—
289C24C22×D412yes—
290C24C22×D414yes—
291C24C22×D412yes—
292C24C22×D414yes—
293C24C2×#4114yes—
294C24C2×#4114yes—
295C24H(5,𝔽2)14yes—
296C24C2511yescharacteristic
297C2×D4—216no—
298C2×D4—216no—
299C2×D4—232no—
300C2×D4—416no—
301C2×D4—416no—
302C2×D4—4128no—
303C2×D4—48no—
304C2×D4—48no—
305C2×D4—48no—
306C2×D4—48no—
307C2×D4—416no—
308C22×C4—28no—
309C22×C4—28no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—28no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—28no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—28no—
321C22×C4—28no—
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
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—28no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—48no—
343C22×C4—48no—
344C22×C4—416no—
345C22×C4—416no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—48no—
349C22×C4—48no—
350C22×C4—416no—
351C22×C4—48no—
352C22×C4—48no—
353C22×C4—48no—
354C22×C4—48no—
355C22×C4—48no—
356C22×C4—416no—
357C22×C4—416no—
358C22×C4—88no—
359C22×C4—88no—
360C22×C4—84no—
361C22×C4—84no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4C22×D414yes—
365C22×C4C22×D4116yes—
366C22×C4C22×D414yes—
367C22×C4C2×#4114yes—
368C4⋊C4—432no—
369C4⋊C4—416no—
370C4⋊C4—816no—
371C22⋊C4—232no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—416no—
375C22⋊C4—416no—
376C22⋊C4—432no—
377C22⋊C4—48no—
378C22⋊C4—416no—
379C22⋊C4—416no—
380C22⋊C4—48no—
381C22⋊C4—816no—
382C22⋊C4—88no—
383C22⋊C4—88no—
384C42—88no—
385C23—24no—
386C23—24no—
387C23—28no—
388C23—28no—
389C23—216no—
390C23—28no—
391C23—28no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—28no—
397C23—28no—
398C23—28no—
399C23—28no—
400C23—28no—
401C23—24no—
402C23—24no—
403C23—28no—
404C23—24no—
405C23—24no—
406C23—232no—
407C23—24no—
408C23—28no—
409C23—216no—
410C23—432no—
411C23—416no—
412C23—48no—
413C23—48no—
414C23—44no—
415C23—44no—
416C23—44no—
417C23—44no—
418C23—42no—
419C23—42no—
420C23—44no—
421C23—48no—
422C23—44no—
423C23—44no—
424C23—48no—
425C23—48no—
426C23—48no—
427C23—48no—
428C23—864no—
429C23—88no—
430C23—816no—
431C23—88no—
432C23—816no—
433C23—816no—
434C23—864no—
435C23—88no—
436C23—88no—
437C23#52514yes—
438C23C2×#12714yes—
439C23C2×#12714yes—
440C23C2×#12714yes—
441C23#53414yes—
442C23#546116yes—
443C23C23×D411yescharacteristic
444C23C23×D412yes—
445C23C22×#4112yes—
446C23C2×H(5,𝔽2)12yes—
447C23#58514yes—
448C23#58514yes—
449D4—432no—
450D4—432no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—216no—
455C2×C4—28no—
456C2×C4—216no—
457C2×C4—216no—
458C2×C4—28no—
459C2×C4—28no—
460C2×C4—416no—
461C2×C4—416no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—48no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—48no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—48no—
480C2×C4—48no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—48no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—88no—
489C2×C4—816no—
490C2×C4—816no—
491C2×C4—816no—
492C22—232no—
493C22—24no—
494C22—28no—
495C22—28no—
496C22—24no—
497C22—24no—
498C22—24no—
499C22—28no—
500C22—28no—
501C22—416no—
502C22—416no—
503C22—42no—
504C22—42no—
505C22—44no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—44no—
510C22—416no—
511C22—48no—
512C22—832no—
513C22—832no—
514C22—88no—
515C22—82no—
516C22—84no—
517C22—82no—
518C22#253918yes—
519C22C2×#52511yescharacteristic
520C22C22×#12711yescharacteristic
521C22C2×#53411yescharacteristic
522C22C2×#54614yes—
523C22#348514yes—
524C22#348514yes—
525C22#348512yes—
526C22#348512yes—
527C22#352118yes—
528C22#353314yes—
529C22#354514yes—
530C4—44no—
531C4—416no—
532C4—44no—
533C4—44no—
534C4—84no—
535C4—84no—
536C4—84no—
537C2—28no—
538C2—44no—
539C2—44no—
540C2—84no—
541C2C2×#253911yescharacteristic
542C2#3545212yes—
543C2#3546312yes—
544C2#3696511yescharacteristic
545C2#3711911yescharacteristic
546C2#4352518yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2483C2semidirect
3. C22×#397C2semidirect
5. #15549C2semidirect
6. #15539C2semidirect
7. #15480C2semidirect
8. #15069C2semidirect
9. #14946C2semidirect
10. #13996C2semidirect
11. #13775C2semidirect
14. C23×C4⋊C4C22semidirect
23. C2×#397C22semidirect
31. C2×#385C22semidirect
32. C2×#385C22semidirect
34. C2×#381C22semidirect
35. C2×#381C22semidirect
39. C2×C4×C22⋊C4C22semidirect
40. C22×#95C22semidirect
41. C22×#95C22semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1444C22semidirect
45. #1439C22semidirect
46. #1439C22semidirect
47. #1439C22semidirect
49. #1437C22semidirect
50. #1437C22semidirect
51. C4⋊#95C22semidirect
52. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24832
C2×#24052
C22×#3972
C2×#23082
#155492
#155392
#154802
#150692
#149462
#139962
#137752