Finite Group Databasebeta

← #2811778

Subgroups — #2811778

Order 512.

2274 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic, complemented
2C2×#2500C211yesmaximal, characteristic
3#16684C214yesmaximal, complemented
4#16599C212yesmaximal, complemented
5#15946C212yesmaximal, complemented
6#15696C214yesmaximal, complemented
7#15674C212yesmaximal
8#15351C214yesmaximal, complemented
9#15084C212yesmaximal, complemented
10#14893C212yesmaximal, complemented
11#14454C211yesmaximal, characteristic, complemented
12#14399C214yesmaximal, complemented
13#14225C212yesmaximal, complemented
14C22×#121C2211yescharacteristic
15C22×#117C2211yescharacteristic
16#2571—28no—
17#2500—28no—
18C2×#400C2212yes—
19C2×#400C2212yes—
20C2×#399C2212yes—
21C2×#399C2214yes—
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#394C2214yes—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#390C2212yes—
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#388C2214yes—
31C2×#388C2214yes—
32C2×#388C2214yes—
33C2×#388C2214yes—
34C2×#384C2211yescharacteristic
35C2×#382C2214yes—
36C2×#382C2211yescharacteristic
37C2×C4×C4⋊C4C2212yes—
38C2×C4×C22⋊C4C2211yescharacteristic
39#1445C2212yes—
40#1444C2214yes—
41#1444C2212yes—
42#1444C2214yes—
43#1442C2214yes—
44#1442C2214yes—
45#1441C2214yes—
46#1441C2212yes—
47#1440C2214yes—
48#1439C2212yes—
49#1439C2212yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2214yes—
53#1439C2214yes—
54#1439C2214yes—
55#1439C2214yes—
56#1438C2212yes—
57#1438C2214yes—
58#1438C2214yes—
59#1437C2212yes—
60#1436C2214yes—
61#1436C2212yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65#1434C2212yes—
66#1434C2214yes—
67#1434C2212yes—
68C22⋊#95C2214yes—
69C23×D4C2311yescharacteristic
70C23×D4C2312yes—
71C24×C4C2311yescharacteristic
72C2×#121—24no—
73C2×#121—42no—
74C2×#121—42no—
75C2×#117—24no—
76C2×#117—42no—
77C2×#117—42no—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2311yescharacteristic
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2314yes—
85C22×C4⋊C4C2314yes—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2314yes—
88C22×C4⋊C4C2311yescharacteristic
89C22×C4⋊C4C2311yescharacteristic
90C22×C22⋊C4C2312yes—
91C22×C22⋊C4C2314yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2314yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2314yes—
101C22×C42C2314yes—
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2312yes—
106#400—28no—
107#400—28no—
108#399—48no—
109#399—44no—
110#397—28no—
111#397—44no—
112#394—216no—
113#390—24no—
114#390—24no—
115#390—44no—
116#389—216no—
117#389—44no—
118#388—216no—
119#388—216no—
120#388—48no—
121#388—48no—
122#384D418yes—
123#382—216no—
124#382—24no—
125C4×C4⋊C4—28no—
126C4×C22⋊C4—24no—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2314yes—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2314yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2312yes—
149C25C2412yes—
150C25C2411yescharacteristic
151C22×D4—24no—
152C22×D4—22no—
153C22×D4—24no—
154C22×D4—24no—
155C22×D4—41no—
156C22×D4—41no—
157C22×D4—42no—
158C22×D4—42no—
159C22×D4—44no—
160C22×D4—44no—
161C23×C4—22no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—42no—
166C23×C4C2414yes—
167C23×C4C2412yes—
168C23×C4C2414yes—
169C23×C4C2414yes—
170C23×C4C2412yes—
171C23×C4C2412yes—
172C23×C4C2412yes—
173C23×C4C2414yes—
174C23×C4C2411yescharacteristic
175C23×C4C2412yes—
176C23×C4C2411yescharacteristic
177#121—416no—
178#117—416no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4C2×D414yes—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D4116yes—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—28no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—44no—
229C2×C22⋊C4—44no—
230C2×C22⋊C4—44no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—42no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—44no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—44no—
241C2×C22⋊C4—44no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C42—24no—
247C2×C42—24no—
248C2×C42—22no—
249C2×C42—28no—
250C2×C42—28no—
251C2×C42—24no—
252C2×C42—28no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42—44no—
257C2×C42C2×D414yes—
258#95—24no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—28no—
263#95—28no—
264#95—28no—
265#95—28no—
266#95—28no—
267#95—24no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—44no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277#95—42no—
278#95—42no—
279#95#4118yes—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—21no—
284C24—22no—
285C24—22no—
286C24—24no—
287C24—22no—
288C24—24no—
289C24—24no—
290C24—22no—
291C24—22no—
292C24—24no—
293C24—24no—
294C24—24no—
295C24—24no—
296C24H(5,𝔽2)12yes—
297C24C2511yescharacteristic
298C2×D4—48no—
299C2×D4—44no—
300C2×D4—44no—
301C2×D4—48no—
302C2×D4—44no—
303C2×D4—84no—
304C2×D4—84no—
305C2×D4—88no—
306C2×D4—84no—
307C2×D4—88no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—28no—
314C22×C4—21no—
315C22×C4—24no—
316C22×C4—24no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
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—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—22no—
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—22no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—22no—
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—48no—
364C22×C4—48no—
365C22×C4—44no—
366C22×C4—44no—
367C22×C4C22×D414yes—
368C22×C4C22×D414yes—
369C22×C4C22×D412yes—
370C22×C4C22×D414yes—
371C22×C4C2×#4118yes—
372C22×C4C2×#4114yes—
373C22×C4H(5,𝔽2)12yes—
374C22×C4H(5,𝔽2)14yes—
375C22×C42−1+414yes—
376C4⋊C4—416no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—48no—
380C4⋊C4—44no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—84no—
384C4⋊C4—84no—
385C4⋊C4—88no—
386C4⋊C4—88no—
387C4⋊C4—88no—
388C22⋊C4—48no—
389C22⋊C4—416no—
390C22⋊C4—48no—
391C22⋊C4—416no—
392C22⋊C4—416no—
393C22⋊C4—48no—
394C22⋊C4—416no—
395C22⋊C4—88no—
396C22⋊C4—88no—
397C22⋊C4—88no—
398C22⋊C4—88no—
399C42—28no—
400C42—48no—
401C42—44no—
402C42—416no—
403C42—84no—
404C23—22no—
405C23—24no—
406C23—24no—
407C23—24no—
408C23—41no—
409C23—42no—
410C23—41no—
411C23—42no—
412C23—41no—
413C23—44no—
414C23—44no—
415C23—42no—
416C23—44no—
417C23—42no—
418C23—41no—
419C23—44no—
420C23—42no—
421C23—44no—
422C23—42no—
423C23—42no—
424C23—44no—
425C23—42no—
426C23—44no—
427C23—44no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—44no—
434C23—44no—
435C23—44no—
436C23—44no—
437C23—44no—
438C23—42no—
439C23—44no—
440C23—44no—
441C23C23×D411yescharacteristic
442C23C22×#4112yes—
443C23C2×H(5,𝔽2)12yes—
444C23C2×H(5,𝔽2)11yescharacteristic
445C23C2×H(5,𝔽2)11yescharacteristic
446C23C2×2−1+412yes—
447C23#58514yes—
448C23#58512yes—
449D4—88no—
450D4—816no—
451C2×C4—24no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—28no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—28no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—28no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—416no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—44no—
493C2×C4—42no—
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—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4#53614yes—
514C2×C4#55018yes—
515C22—42no—
516C22—42no—
517C22—42no—
518C22—48no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—84no—
524C22—84no—
525C22—84no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22—84no—
530C22—84no—
531C22—84no—
532C22C2×#53611yescharacteristic
533C22C2×#55012yes—
534C22#348311yescharacteristic
535C22#348411yescharacteristic
536C22#348512yes—
537C22#348711yescharacteristic
538C22#349314yes—
539C22#351514yes—
540C22#351912yes—
541C22#353312yes—
542C22#353512yes—
543C22#353512yes—
544C22#354012yes—
545C22#354014yes—
546C22#354911yescharacteristic
547C22#355914yes—
548C4—24no—
549C4—28no—
550C4—44no—
551C4—44no—
552C4—42no—
553C4—48no—
554C4—84no—
555C4—82no—
556C4—84no—
557C4—84no—
558C4—82no—
559C2—82no—
560C2—162no—
561C2#3671111yescharacteristic
562C2#3913412yes—
563C2#4382312yes—
564C2#4538812yes—
565C2#4808714yes—
566C2#5205411yescharacteristic
567C2#5538112yes—
568C2#5905611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
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×#2571C2semidirect
3. #16684C2semidirect
4. #16599C2semidirect
5. #15946C2semidirect
6. #15696C2semidirect
8. #15351C2semidirect
9. #15084C2semidirect
10. #14893C2semidirect
11. #14454C2semidirect
12. #14399C2semidirect
13. #14225C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
C2×#25002
#166842
#165992
#159462
#156962
#156742
#153512
#150842
#148932
#144542
#143992
#142252