Finite Group Databasebeta

← #768858

Subgroups — #768858

Order 512.

4606 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2483C212yesmaximal, complemented
2C2×#2405C212yesmaximal
3C22×#394C211yesmaximal, characteristic
4C2×#2315C211yesmaximal, characteristic, complemented
5#15549C218yesmaximal, complemented
6#15384C218yesmaximal, complemented
7#14936C212yesmaximal, complemented
8#14734C212yesmaximal, complemented
9#13903C212yesmaximal, complemented
10#13880C212yesmaximal, complemented
11#13775C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C22×#123C2214yes—
14C23×C4⋊C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C23×C22⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20#2483—216no—
21#2405—216no—
22C2×#399C2214yescomplemented
23C2×#397C2218yescomplemented
24C2×#394—28no—
25C2×#394C2218yes—
26C2×#394C2218yes—
27#2315—28no—
28C2×#386C2214yes—
29C2×#385C2214yes—
30C2×#385C2214yes—
31C2×#385C2214yescomplemented
32C2×#385C2214yescomplemented
33C2×#385C2214yescomplemented
34C2×#384C2214yescomplemented
35C2×#381C2214yes—
36C2×#381C2214yescomplemented
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#379C2214yes—
40C2×#379C2212yes—
41C2×C4×C22⋊C4C2212yes—
42C22×#95C2211yescharacteristic
43#1445C2218yescomplemented
44#1445C2214yescomplemented
45#1444C2214yescomplemented
46#1444C2218yescomplemented
47#1439C2214yescomplemented
48#1439C2218yescomplemented
49#1439C2214yescomplemented
50#1439C2214yescomplemented
51#1437C2218yescomplemented
52#1437C2218yescomplemented
53C4⋊#95C2214yescomplemented
54C22⋊#95C2212yes—
55C26C2311yescharacteristic
56C26C2311yescharacteristic
57C23×D4—24no—
58C23×D4—28no—
59C23×D4—44no—
60C23×D4—44no—
61C23×D4C2314yes—
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C24×C4C2311yescharacteristic
65C24×C4C2312yes—
66C24×C4C2312yes—
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C2×#123—216no—
70C2×#123—48no—
71C2×#123—48no—
72C22×C4⋊C4—24no—
73C22×C4⋊C4—44no—
74C22×C4⋊C4C2314yes—
75C22×C4⋊C4C2318yes—
76C22×C4⋊C4C2318yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C22⋊C4—28no—
82C22×C22⋊C4—28no—
83C22×C22⋊C4—24no—
84C22×C22⋊C4—28no—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—28no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—44no—
91C22×C22⋊C4D418yes—
92C22×C22⋊C4C2314yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2318yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2314yes—
100C22×C22⋊C4C2318yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C42C2314yes—
104#399—48no—
105#397—232no—
106#394—232no—
107#394—232no—
108#386—216no—
109#385—216no—
110#385—216no—
111#385—216no—
112#385—216no—
113#385—216no—
114#384—48no—
115#381—216no—
116#381—216no—
117#380—28no—
118#380—28no—
119#379—216no—
120#379D4116yes—
121C4×C22⋊C4—28no—
122C2×#95—24no—
123C2×#95C2314yes—
124C2×#95C2318yes—
125C2×#95C2318yes—
126C2×#95C2318yes—
127C2×#95C2318yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C25—24no—
134C25—24no—
135C25—22no—
136C25—24no—
137C25—24no—
138C25—24no—
139C25—24no—
140C25—42no—
141C25—42no—
142C25—42no—
143C25—42no—
144C25—42no—
145C25—42no—
146C25—42no—
147C25C2×D414yes—
148C25C2×D418yes—
149C25C2412yes—
150C25C2412yes—
151C25C2411yescharacteristic
152C22×D4—28no—
153C22×D4—28no—
154C22×D4—28no—
155C22×D4—28no—
156C22×D4—28no—
157C22×D4—44no—
158C22×D4—44no—
159C22×D4—416no—
160C22×D4—464no—
161C22×D4—416no—
162C22×D4—416no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—22no—
167C23×C4—22no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—24no—
176C23×C4—24no—
177C23×C4—24no—
178C23×C4—24no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—44no—
186C23×C4—44no—
187C23×C4—42no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4C2×D414yes—
191C23×C4C2×D414yes—
192C23×C4#4118yes—
193C23×C4C2412yes—
194C23×C4C2414yes—
195C23×C4C2414yes—
196C23×C4C2412yes—
197C23×C4C2418yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201#123—464no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—216no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—216no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—216no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—416no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—48no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—44no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—232no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—28no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—28no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—216no—
236C2×C22⋊C4—28no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—216no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—216no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—44no—
247C2×C22⋊C4—48no—
248C2×C22⋊C4—416no—
249C2×C22⋊C4—432no—
250C2×C22⋊C4—416no—
251C2×C22⋊C4—416no—
252C2×C22⋊C4—48no—
253C2×C22⋊C4—48no—
254C2×C22⋊C4C2×D4116yes—
255C2×C22⋊C4C2×D4116yes—
256C2×C22⋊C4C2×D4116yes—
257C2×C22⋊C4C2×D4116yes—
258C2×C42—28no—
259C2×C42—44no—
260C2×C42—44no—
261#95—28no—
262#95—28no—
263#95—28no—
264#95—216no—
265#95—216no—
266#95—216no—
267#95—216no—
268#95—28no—
269#95—44no—
270#95—44no—
271C24—22no—
272C24—22no—
273C24—22no—
274C24—22no—
275C24—24no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—28no—
282C24—28no—
283C24—232no—
284C24—28no—
285C24—24no—
286C24—22no—
287C24—24no—
288C24—24no—
289C24—22no—
290C24—24no—
291C24—24no—
292C24—24no—
293C24—44no—
294C24—44no—
295C24—416no—
296C24—44no—
297C24—44no—
298C24—416no—
299C24—416no—
300C24—416no—
301C24—416no—
302C24—44no—
303C24—44no—
304C24—44no—
305C24—44no—
306C24—44no—
307C24—44no—
308C24—416no—
309C24—88no—
310C24—88no—
311C24—88no—
312C24—88no—
313C24—88no—
314C24—88no—
315C24—88no—
316C24#127116yes—
317C24C22×D414yes—
318C24C22×D414yes—
319C24C22×D414yes—
320C24C22×D412yes—
321C24C22×D414yes—
322C24C2×#4114yes—
323C24H(5,𝔽2)14yes—
324C242−1+414yes—
325C24C2511yescharacteristic
326C2×D4—232no—
327C2×D4—416no—
328C2×D4—416no—
329C2×D4—416no—
330C2×D4—416no—
331C2×D4—416no—
332C2×D4—4128no—
333C2×D4—416no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—28no—
337C22×C4—22no—
338C22×C4—24no—
339C22×C4—28no—
340C22×C4—24no—
341C22×C4—28no—
342C22×C4—24no—
343C22×C4—28no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—416no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—416no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—416no—
380C22×C4—416no—
381C22×C4—44no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—416no—
388C22×C4—88no—
389C22×C4—88no—
390C22×C4—84no—
391C22×C4—88no—
392C22×C4—84no—
393C22×C4—88no—
394C22×C4—84no—
395C22×C4—84no—
396C22×C4#123116yescomplemented
397C22×C4C22×D4116yes—
398C22×C4C22×D414yes—
399C22×C4C22×D414yes—
400C22×C4C2×#4114yes—
401C22×C4C2×#4114yes—
402C22×C4H(5,𝔽2)14yes—
403C4⋊C4—416no—
404C4⋊C4—432no—
405C4⋊C4—84no—
406C4⋊C4—84no—
407C4⋊C4—816no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C22⋊C4—232no—
411C22⋊C4—232no—
412C22⋊C4—416no—
413C22⋊C4—416no—
414C22⋊C4—416no—
415C22⋊C4—432no—
416C22⋊C4—416no—
417C22⋊C4—416no—
418C22⋊C4—416no—
419C22⋊C4—416no—
420C22⋊C4—816no—
421C42—416no—
422C23—22no—
423C23—22no—
424C23—28no—
425C23—22no—
426C23—22no—
427C23—28no—
428C23—28no—
429C23—28no—
430C23—28no—
431C23—28no—
432C23—24no—
433C23—24no—
434C23—28no—
435C23—28no—
436C23—28no—
437C23—24no—
438C23—28no—
439C23—24no—
440C23—28no—
441C23—24no—
442C23—28no—
443C23—24no—
444C23—232no—
445C23—28no—
446C23—24no—
447C23—28no—
448C23—28no—
449C23—416no—
450C23—416no—
451C23—416no—
452C23—416no—
453C23—416no—
454C23—48no—
455C23—48no—
456C23—48no—
457C23—48no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—48no—
462C23—42no—
463C23—44no—
464C23—48no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23—48no—
469C23—44no—
470C23—48no—
471C23—88no—
472C23—864no—
473C23—88no—
474C23—88no—
475C23—88no—
476C23—864no—
477C23—88no—
478C23—88no—
479C23—88no—
480C23—88no—
481C23C2×#12314yes—
482C23C2×#12718yes—
483C23C2×#12714yes—
484C23#53614yes—
485C23#547116yes—
486C23C23×D411yescharacteristic
487C23C23×D412yes—
488C23C22×#4112yes—
489C23C2×H(5,𝔽2)12yes—
490C23C2×H(5,𝔽2)12yes—
491C23C2×2−1+412yes—
492C23#58512yes—
493C23#58512yes—
494D4—464no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—28no—
498C2×C4—28no—
499C2×C4—24no—
500C2×C4—216no—
501C2×C4—216no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—28no—
505C2×C4—216no—
506C2×C4—24no—
507C2×C4—416no—
508C2×C4—416no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—48no—
514C2×C4—48no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—48no—
524C2×C4—42no—
525C2×C4—48no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—48no—
529C2×C4—48no—
530C2×C4—44no—
531C2×C4—48no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—48no—
535C2×C4—88no—
536C2×C4—88no—
537C2×C4—816no—
538C2×C4—816no—
539C2×C4—88no—
540C2×C4C2×#12318yes—
541C22—24no—
542C22—232no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—28no—
549C22—28no—
550C22—28no—
551C22—44no—
552C22—416no—
553C22—416no—
554C22—44no—
555C22—44no—
556C22—44no—
557C22—416no—
558C22—44no—
559C22—44no—
560C22—44no—
561C22—832no—
562C22—832no—
563C22—82no—
564C22—88no—
565C22—82no—
566C22—82no—
567C22—82no—
568C22#250118yes—
569C22C22×#12311yescharacteristic
570C22C22×#12711yescharacteristic
571C22C2×#53611yescharacteristic
572C22C2×#54714yes—
573C22#348411yescharacteristic
574C22#348414yes—
575C22#348411yescharacteristic
576C22#348511yescharacteristic
577C22#348514yes—
578C22#348511yescharacteristic
579C22#349414yes—
580C22#350514yes—
581C22#353514yes—
582C22#356114yes—
583C4—28no—
584C4—416no—
585C4—44no—
586C4—44no—
587C4—82no—
588C4—84no—
589C4—84no—
590C4—82no—
591C2—28no—
592C2—48no—
593C2—84no—
594C2C2×#250111yescharacteristic
595C2#3535111yescharacteristic
596C2#3539011yescharacteristic
597C2#3546311yescharacteristic
598C2#3554011yescharacteristic
599C2#3672011yescharacteristic
600C2#3696611yescharacteristic
601C2#4305318yes—
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
4. C2×#2315C2semidirect
5. #15549C2semidirect
6. #15384C2semidirect
7. #14936C2semidirect
8. #14734C2semidirect
9. #13903C2semidirect
10. #13880C2semidirect
11. #13775C2semidirect
22. C2×#399C22semidirect
23. C2×#397C22semidirect
31. C2×#385C22semidirect
32. C2×#385C22semidirect
33. C2×#385C22semidirect
34. C2×#384C22semidirect
36. C2×#381C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1439C22semidirect
48. #1439C22semidirect
49. #1439C22semidirect
50. #1439C22semidirect
51. #1437C22semidirect
52. #1437C22semidirect
53. C4⋊#95C22semidirect
396. C22×C4#123semidirect

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×#3942
C2×#23152
#155492
#153842
#149362
#147342
#139032
#138802
#137752