Finite Group Databasebeta

← #639955

Subgroups — #639955

Order 512.

3798 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#381C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#16699C214yesmaximal, complemented
4#16698C214yesmaximal, complemented
5#15019C212yesmaximal, complemented
6#15015C212yesmaximal, complemented
7#15014C212yesmaximal, complemented
8#14936C212yesmaximal, complemented
9#14919C214yesmaximal, complemented
10#14681C211yesmaximal, characteristic
11#14342C212yesmaximal, complemented
12#14263C212yesmaximal, complemented
13#14127C212yesmaximal
14#13869C212yesmaximal
15C25×C4C2211yescharacteristic
16C23×C4⋊C4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C2×#385C2214yes—
22C2×#385C2214yes—
23C2×#381—28no—
24C2×#381C2212yes—
25C2×#381C2212yes—
26C2×#381C2212yes—
27C2×#381C2212yes—
28C2×#381C2212yes—
29C2×#381C2214yes—
30C2×#381C2212yes—
31C2×#380—28no—
32C2×#380C2212yes—
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×#379C2212yes—
37C22×#95C2212yes—
38C22×#95C2211yescharacteristic
39C22×#95C2211yescharacteristic
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42#1445C2214yescomplemented
43#1445C2214yescomplemented
44#1445C2214yescomplemented
45#1445C2214yescomplemented
46#1445C2218yescomplemented
47#1444C2214yescomplemented
48#1444C2214yescomplemented
49#1443C2218yescomplemented
50#1443C2218yescomplemented
51#1442C2214yescomplemented
52#1442C2218yescomplemented
53#1442C2214yescomplemented
54#1440C2218yes—
55#1439C2218yes—
56C22⋊#95C2214yes—
57C22⋊#95C2214yes—
58C22⋊#95C2214yes—
59C22⋊#95C2214yes—
60C22⋊#95C2214yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C26C2311yescharacteristic
66C24×C4—28no—
67C24×C4—28no—
68C24×C4—44no—
69C24×C4D418yes—
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2311yescharacteristic
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C22×C4⋊C4—24no—
83C22×C4⋊C4—24no—
84C22×C4⋊C4—28no—
85C22×C4⋊C4—44no—
86C22×C4⋊C4C2312yes—
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2312yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2318yes—
91C22×C22⋊C4—28no—
92C22×C22⋊C4—24no—
93C22×C22⋊C4—24no—
94C22×C22⋊C4—28no—
95C22×C22⋊C4—48no—
96C22×C22⋊C4—44no—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2314yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2312yes—
101C22×C22⋊C4C2314yes—
102C22×C22⋊C4C2314yes—
103C22×C22⋊C4C2312yes—
104C22×C22⋊C4C2314yes—
105C22×C22⋊C4C2314yes—
106#385—48no—
107#385—48no—
108#381—28no—
109#381—28no—
110#381—28no—
111#381—28no—
112#381—28no—
113#381—48no—
114#381—44no—
115#380—28no—
116#380—28no—
117#380—28no—
118#380—28no—
119#379—28no—
120C2×#95—28no—
121C2×#95—24no—
122C2×#95—24no—
123C2×#95—24no—
124C2×#95—24no—
125C2×#95C2314yes—
126C2×#95C2314yes—
127C2×#95C2312yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2312yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2318yes—
140C2×#95C2314yes—
141C2×#95C2314yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C2×#95C2318yes—
145C2×#95C2318yes—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—24no—
150C25—42no—
151C25—42no—
152C25C2×D414yes—
153C25#4118yes—
154C25C2411yescharacteristic
155C25C2411yescharacteristic
156C25C2411yescharacteristic
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—232no—
160C23×C4—24no—
161C23×C4—24no—
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—22no—
175C23×C4—22no—
176C23×C4—22no—
177C23×C4—22no—
178C23×C4—24no—
179C23×C4—24no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—28no—
183C23×C4—28no—
184C23×C4—24no—
185C23×C4—24no—
186C23×C4—416no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—42no—
190C23×C4—42no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4—44no—
196C23×C4—42no—
197C23×C4—42no—
198C23×C4—42no—
199C23×C4—44no—
200C23×C4—42no—
201C23×C4—464no—
202C23×C4C2×D414yes—
203C23×C4C2×D414yes—
204C23×C4C2×D414yes—
205C23×C4C2×D414yes—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4#4118yes—
209C23×C4C2418yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—416no—
229C2×C4⋊C4—416no—
230C2×C4⋊C4—48no—
231C2×C4⋊C4—48no—
232C2×C4⋊C4—48no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—28no—
235C2×C22⋊C4—24no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—28no—
240C2×C22⋊C4—28no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—28no—
243C2×C22⋊C4—28no—
244C2×C22⋊C4—28no—
245C2×C22⋊C4—28no—
246C2×C22⋊C4—28no—
247C2×C22⋊C4—28no—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—44no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4—44no—
254C2×C22⋊C4—416no—
255C2×C22⋊C4—432no—
256C2×C22⋊C4—416no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259#95—28no—
260#95—28no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—24no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—28no—
270#95—28no—
271#95—48no—
272#95—48no—
273#95—48no—
274#95—44no—
275#95—44no—
276#95—48no—
277#95—48no—
278#95—48no—
279#95—48no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—28no—
289C24—28no—
290C24—28no—
291C24—28no—
292C24—232no—
293C24—232no—
294C24—22no—
295C24—22no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—44no—
301C24—44no—
302C24—416no—
303C24—416no—
304C24—416no—
305C24—416no—
306C24—416no—
307C24#123116yes—
308C24C22×D412yes—
309C24C22×D412yes—
310C24C22×D412yes—
311C24C2×#4114yes—
312C24C2×#4114yes—
313C24C2×#4114yes—
314C24C2×#4114yes—
315C24H(5,𝔽2)14yes—
316C24H(5,𝔽2)14yes—
317C242−1+414yes—
318C24C2511yescharacteristic
319C22×C4—28no—
320C22×C4—28no—
321C22×C4—232no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—28no—
340C22×C4—28no—
341C22×C4—28no—
342C22×C4—28no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—24no—
355C22×C4—28no—
356C22×C4—28no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—28no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—28no—
372C22×C4—24no—
373C22×C4—28no—
374C22×C4—28no—
375C22×C4—24no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—416no—
381C22×C4—416no—
382C22×C4—416no—
383C22×C4—416no—
384C22×C4—4128no—
385C22×C4—416no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—48no—
389C22×C4—48no—
390C22×C4—48no—
391C22×C4—48no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—44no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—44no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—416no—
405C22×C4—88no—
406C22×C4—816no—
407C22×C4—88no—
408C22×C4—816no—
409C22×C4—88no—
410C22×C4#123116yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C2×#4114yes—
416C22×C4C2×#4114yes—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—48no—
420C4⋊C4—48no—
421C4⋊C4—816no—
422C22⋊C4—48no—
423C22⋊C4—416no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—416no—
428C22⋊C4—416no—
429C22⋊C4—416no—
430C22⋊C4—88no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—28no—
440C23—28no—
441C23—28no—
442C23—28no—
443C23—28no—
444C23—28no—
445C23—28no—
446C23—28no—
447C23—28no—
448C23—28no—
449C23—28no—
450C23—28no—
451C23—232no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—28no—
459C23—28no—
460C23—28no—
461C23—232no—
462C23—416no—
463C23—416no—
464C23—416no—
465C23—416no—
466C23—4128no—
467C23—416no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23C2×#12314yes—
473C23C2×#12314yes—
474C23C2×#12314yes—
475C23C2×#12314yes—
476C23#53414yes—
477C23#53514yes—
478C23#53614yes—
479C23#539116yes—
480C23#551116yes—
481C23C23×D411yescharacteristic
482C23C22×#4112yes—
483C23C22×#4112yes—
484C23C2×H(5,𝔽2)12yes—
485C23C2×H(5,𝔽2)12yes—
486C23C2×2−1+412yes—
487C23#58512yes—
488C23#58512yes—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—232no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—416no—
503C2×C4—416no—
504C2×C4—416no—
505C2×C4—416no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—416no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—48no—
529C2×C4—48no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—48no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—48no—
537C2×C4—48no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—48no—
541C2×C4—48no—
542C2×C4—88no—
543C2×C4—816no—
544C2×C4—88no—
545C2×C4—88no—
546C2×C4—816no—
547C2×C4—88no—
548C2×C4C2×#12314yes—
549C2×C4C2×#12314yes—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—24no—
557C22—24no—
558C22—24no—
559C22—24no—
560C22—232no—
561C22—232no—
562C22—24no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—464no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—416no—
572C22—42no—
573C22#248318yes—
574C22#262318yes—
575C22C22×#12311yescharacteristic
576C22C22×#12311yescharacteristic
577C22C2×#53411yescharacteristic
578C22C2×#53511yescharacteristic
579C22C2×#53611yescharacteristic
580C22#344911yescharacteristic
581C22#344911yescharacteristic
582C22C2×#53914yes—
583C22C2×#55114yes—
584C22#349114yes—
585C22#349414yes—
586C22#349514yes—
587C22#353414yes—
588C22#353514yes—
589C4—24no—
590C4—24no—
591C4—42no—
592C4—42no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—84no—
597C4—82no—
598C4—84no—
599C4—88no—
600C2—28no—
601C2—28no—
602C2—44no—
603C2C2×#262311yescharacteristic
604C2C2×#248311yescharacteristic
605C2#3509711yescharacteristic
606C2#3526211yescharacteristic
607C2#3531111yescharacteristic
608C2#3744011yescharacteristic
609C2#3744111yescharacteristic
610C2#3784918yes—
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. #16699C2semidirect
4. #16698C2semidirect
5. #15019C2semidirect
6. #15015C2semidirect
7. #15014C2semidirect
8. #14936C2semidirect
9. #14919C2semidirect
11. #14342C2semidirect
12. #14263C2semidirect
42. #1445C22semidirect
43. #1445C22semidirect
44. #1445C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1444C22semidirect
48. #1444C22semidirect
49. #1443C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3812
C22×#3802
#166992
#166982
#150192
#150152
#150142
#149362
#149192
#146812
#143422
#142632
#141272
#138692