Finite Group Databasebeta

← #4416355

Subgroups — #4416355

Order 512.

3638 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#17059C212yesmaximal, complemented
4#16815C214yesmaximal, complemented
5#16394C214yesmaximal, complemented
6#16366C212yesmaximal, complemented
7#15092C212yesmaximal, complemented
8#15015C212yesmaximal, complemented
9#14914C212yesmaximal, complemented
10#14843C212yesmaximal, complemented
11#14727C214yesmaximal
12#14681C211yesmaximal, characteristic
13#14674C212yesmaximal, complemented
14#14660C212yesmaximal, complemented
15C25×C4C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22#2848—28no—
23C2×#400C2214yescomplemented
24C2×#397C2218yes—
25C2×#385C2212yes—
26C2×#385C2212yes—
27C2×#385C2212yes—
28C2×#385C2214yes—
29C2×#381C2212yes—
30C2×#381C2212yes—
31C2×#380—28no—
32C2×#380C2211yescharacteristic
33C2×#380C2211yescharacteristic
34C2×#380C2211yescharacteristic
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#380C2212yes—
38C2×#380C2212yes—
39C2×#379C2212yes—
40C2×#379C2212yes—
41C22×#95C2212yes—
42#1446C2214yescomplemented
43#1445C2214yescomplemented
44#1445C2214yescomplemented
45#1444C2214yescomplemented
46#1444C2214yescomplemented
47#1444C2214yescomplemented
48#1443C2218yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2218yescomplemented
52#1441C2214yescomplemented
53#1441C2214yescomplemented
54#1441C2214yescomplemented
55#1440C2218yes—
56#1440C2218yes—
57#1439C2218yes—
58C22⋊#95C2212yes—
59C22⋊#95C2214yes—
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2214yes—
64C22⋊#95C2214yes—
65C26C2311yescharacteristic
66C23×Q8C2311yescharacteristic
67C24×C4—28no—
68C24×C4—44no—
69C24×C4—44no—
70C24×C4D418yes—
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2312yes—
76C24×C4C2312yes—
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C2×C22⋊Q8—24no—
80C2×C22⋊Q8—24no—
81C2×C22⋊Q8—24no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8—44no—
84C2×C22⋊Q8—44no—
85C22×C4⋊C4—24no—
86C22×C4⋊C4—24no—
87C22×C4⋊C4—44no—
88C22×C4⋊C4—44no—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C4⋊C4C2312yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2318yes—
100C22×C22⋊C4—28no—
101C22×C22⋊C4—48no—
102C22×C22⋊C4C2312yes—
103C22×C22⋊C4C2314yes—
104C22×C22⋊C4C2312yes—
105C22×C22⋊C4C2312yes—
106C22×C22⋊C4C2314yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2314yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113#400—48no—
114#397—416no—
115#385—28no—
116#385—28no—
117#385—28no—
118#385—48no—
119#381—28no—
120#381—44no—
121#380—24no—
122#380—24no—
123#380—24no—
124#380—28no—
125#380—28no—
126#380—28no—
127#380—28no—
128#379—28no—
129#379—44no—
130C2×#95—28no—
131C2×#95C2314yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2312yes—
136C2×#95C2314yes—
137C2×#95C2312yes—
138C2×#95C2312yes—
139C2×#95C2318yes—
140C2×#95C2318yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2314yes—
145C2×#95C2318yes—
146C2×#95C2318yes—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—24no—
151C25—24no—
152C25—42no—
153C25—42no—
154C25C2×D414yes—
155C25C2411yescharacteristic
156C25C2411yescharacteristic
157C25C2411yescharacteristic
158C22×Q8—22no—
159C22×Q8—24no—
160C22×Q8—42no—
161C22×Q8—42no—
162C23×C4—24no—
163C23×C4—24no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—232no—
168C23×C4—24no—
169C23×C4—24no—
170C23×C4—24no—
171C23×C4—24no—
172C23×C4—24no—
173C23×C4—24no—
174C23×C4—24no—
175C23×C4—28no—
176C23×C4—28no—
177C23×C4—416no—
178C23×C4—416no—
179C23×C4—42no—
180C23×C4—42no—
181C23×C4—42no—
182C23×C4—42no—
183C23×C4—42no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—44no—
189C23×C4—44no—
190C23×C4—464no—
191C23×C4C2×D414yes—
192C23×C4C2×D414yes—
193C23×C4C2×D414yes—
194C23×C4C2×D414yes—
195C23×C4C2×D414yes—
196C23×C4C2×D414yes—
197C23×C4#4118yes—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2418yes—
203C23×C4C2414yes—
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C22⋊Q8—416no—
210C22⋊Q8—416no—
211C22⋊Q8—416no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—44no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—44no—
232C2×C4⋊C4—416no—
233C2×C4⋊C4—416no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—48no—
236C2×C4⋊C4—48no—
237C2×C22⋊C4—28no—
238C2×C22⋊C4—28no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—24no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—216no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—28no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—48no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—432no—
259C2×C22⋊C4—44no—
260C2×C22⋊C4—44no—
261C2×C22⋊C4—44no—
262C2×C22⋊C4—44no—
263#95—28no—
264#95—28no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—28no—
269#95—28no—
270#95—28no—
271#95—48no—
272#95—44no—
273#95—48no—
274#95—48no—
275#95—48no—
276#95—48no—
277#95—48no—
278#95—48no—
279C24—22no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—232no—
284C24—28no—
285C24—28no—
286C24—28no—
287C24—22no—
288C24—22no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—28no—
300C24—28no—
301C24—44no—
302C24—44no—
303C24—416no—
304C24—416no—
305C24—416no—
306C24—416no—
307C24—416no—
308C24—416no—
309C24C22×D412yes—
310C24C22×D412yes—
311C24C22×D412yes—
312C24C2×#4114yes—
313C24H(5,𝔽2)14yes—
314C24H(5,𝔽2)14yes—
315C242−1+414yes—
316C242−1+414yes—
317C24C2511yescharacteristic
318C2×Q8—44no—
319C2×Q8—44no—
320C2×Q8—44no—
321C2×Q8—88no—
322C22×C4—232no—
323C22×C4—28no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—28no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—28no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—28no—
355C22×C4—24no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—28no—
360C22×C4—28no—
361C22×C4—28no—
362C22×C4—28no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—28no—
367C22×C4—28no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—416no—
382C22×C4—416no—
383C22×C4—416no—
384C22×C4—416no—
385C22×C4—4128no—
386C22×C4—416no—
387C22×C4—48no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—48no—
393C22×C4—816no—
394C22×C4—816no—
395C22×C4#123116yes—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C2×#4114yes—
401C22×C4C2×#4114yes—
402C22×C4H(5,𝔽2)14yes—
403C4⋊C4—48no—
404C4⋊C4—48no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—44no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—48no—
413C4⋊C4—816no—
414C22⋊C4—48no—
415C22⋊C4—416no—
416C22⋊C4—48no—
417C22⋊C4—48no—
418C22⋊C4—48no—
419C22⋊C4—48no—
420C22⋊C4—416no—
421C22⋊C4—48no—
422C22⋊C4—416no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—28no—
430C23—28no—
431C23—28no—
432C23—28no—
433C23—28no—
434C23—28no—
435C23—232no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—28no—
452C23—28no—
453C23—28no—
454C23—416no—
455C23—416no—
456C23—416no—
457C23—416no—
458C23—416no—
459C23—4128no—
460C23—416no—
461C23—44no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23C2×#12314yes—
470C23#53414yes—
471C23#53514yes—
472C23#53614yes—
473C23#53614yes—
474C23#563116yes—
475C23C23×D411yescharacteristic
476C23C22×#4112yes—
477C23C2×H(5,𝔽2)12yes—
478C23C2×H(5,𝔽2)12yes—
479C23C2×H(5,𝔽2)12yes—
480C23C2×2−1+412yes—
481C23C2×2−1+412yes—
482C23#58512yes—
483Q8—88no—
484C2×C4—232no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—28no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—416no—
498C2×C4—416no—
499C2×C4—416no—
500C2×C4—416no—
501C2×C4—416no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—48no—
523C2×C4—48no—
524C2×C4—48no—
525C2×C4—48no—
526C2×C4—48no—
527C2×C4—48no—
528C2×C4—48no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—416no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—816no—
536C2×C4—816no—
537C2×C4C2×#12314yes—
538C2×C4C2×#12314yes—
539C2×C4#53414yes—
540C22—24no—
541C22—24no—
542C22—24no—
543C22—232no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—464no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—42no—
559C22—42no—
560C22—416no—
561C22—416no—
562C22—42no—
563C22—42no—
564C22#281718yes—
565C22C22×#12311yescharacteristic
566C22C2×#53411yescharacteristic
567C22C2×#53411yescharacteristic
568C22C2×#53511yescharacteristic
569C22C2×#53611yescharacteristic
570C22C2×#53611yescharacteristic
571C22#344911yescharacteristic
572C22#349014yes—
573C22#349214yes—
574C22#349514yes—
575C22C2×#56314yes—
576C22#353414yes—
577C22#353614yes—
578C22#354114yes—
579C4—24no—
580C4—24no—
581C4—24no—
582C4—42no—
583C4—82no—
584C4—82no—
585C4—84no—
586C4—84no—
587C4—88no—
588C4—82no—
589C4—82no—
590C2—28no—
591C2—44no—
592C2—44no—
593C2#3499211yescharacteristic
594C2#3522711yescharacteristic
595C2#3531111yescharacteristic
596C2C2×#281711yescharacteristic
597C2#3743811yescharacteristic
598C2#3744611yescharacteristic
599C2#3745111yescharacteristic
600C2#4546318yes—
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. #17059C2semidirect
4. #16815C2semidirect
5. #16394C2semidirect
6. #16366C2semidirect
7. #15092C2semidirect
8. #15015C2semidirect
9. #14914C2semidirect
10. #14843C2semidirect
13. #14674C2semidirect
14. #14660C2semidirect
23. C2×#400C22semidirect
42. #1446C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1444C22semidirect
46. #1444C22semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C22×#3802
#170592
#168152
#163942
#163662
#150922
#150152
#149142
#148432
#147272
#146812
#146742
#146602