Finite Group Databasebeta

← #4414562

Subgroups — #4414562

Order 512.

3214 conjugacy classes of subgroups.

The subgroup lattice has 631 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#16902C212yesmaximal, complemented
2#16902C212yesmaximal, complemented
3#16902C212yesmaximal, complemented
4#16825C212yesmaximal, complemented
5#16825C212yesmaximal, complemented
6#16825C212yesmaximal, complemented
7#14936C214yesmaximal, complemented
8#14935C214yesmaximal, complemented
9#14726C214yesmaximal
10#14681C211yesmaximal, characteristic
11#14681C211yesmaximal, characteristic
12#14637C214yesmaximal, complemented
13C2×C22⋊#95C211yesmaximal, characteristic
14C25×C4C2211yescharacteristic
15C23×C22⋊C4C2212yes—
16C2×#396C2214yes—
17C2×#396C2214yes—
18C2×#385C2214yes—
19C2×#385C2214yes—
20C2×#381C2212yes—
21C2×#381C2212yes—
22C2×#381C2212yes—
23C2×#381C2212yes—
24C2×#380C2214yes—
25C2×#379C2212yes—
26C2×#379C2212yes—
27C22×#95C2212yes—
28C22×#95C2212yes—
29#1445C2214yescomplemented
30#1445C2214yescomplemented
31#1445C2214yescomplemented
32#1445C2214yescomplemented
33#1445C2214yescomplemented
34#1445C2214yescomplemented
35#1444C2214yescomplemented
36#1444C2214yescomplemented
37#1443C2214yescomplemented
38#1443C2214yescomplemented
39#1441C2214yescomplemented
40#1441C2214yescomplemented
41#1441C2214yescomplemented
42#1441C2214yescomplemented
43#1441C2214yescomplemented
44#1441C2214yescomplemented
45#1439C2214yes—
46#1439C2214yes—
47#1439C2214yes—
48#1439C2214yes—
49#1439C2214yes—
50#1439C2214yes—
51C22⋊#95C2211yescharacteristic
52C22⋊#95C2211yescharacteristic
53C22⋊#95C2211yescharacteristic
54C22⋊#95C2212yes—
55C22⋊#95C2212yes—
56C22⋊#95C2212yes—
57C22⋊#95C2212yes—
58C22⋊#95C2212yes—
59C22⋊#95C2212yes—
60C22⋊#95C2212yes—
61C22⋊#95C2212yes—
62C22⋊#95C2212yes—
63C22⋊#95C2214yes—
64C22⋊#95C2211yescharacteristic
65C22⋊#95C2211yescharacteristic
66C22⋊#95C2211yescharacteristic
67C26C2311yescharacteristic
68C24×C4—44no—
69C24×C4—44no—
70C24×C4—44no—
71C24×C4Q818yes—
72C24×C4C2311yescharacteristic
73C24×C4C2311yescharacteristic
74C24×C4C2314yes—
75C24×C4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81C22×C4⋊C4C2314yes—
82C22×C4⋊C4C2314yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2314yes—
85C22×C22⋊C4—28no—
86C22×C22⋊C4—44no—
87C22×C22⋊C4—44no—
88C22×C22⋊C4C2314yes—
89C22×C22⋊C4C2314yes—
90C22×C22⋊C4C2314yes—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2314yes—
94C22×C22⋊C4C2314yes—
95C22×C22⋊C4C2312yes—
96C22×C22⋊C4C2312yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2312yes—
99#396—216no—
100#396—48no—
101#385—48no—
102#385—48no—
103#381—44no—
104#381—44no—
105#381—44no—
106#381—44no—
107#380—48no—
108#379—44no—
109#379—44no—
110C2×#95—44no—
111C2×#95—44no—
112C2×#95C2314yes—
113C2×#95C2314yes—
114C2×#95C2312yes—
115C2×#95C2312yes—
116C2×#95C2312yes—
117C2×#95C2314yes—
118C2×#95C2314yes—
119C2×#95C2314yes—
120C2×#95C2312yes—
121C2×#95C2312yes—
122C2×#95C2312yes—
123C2×#95C2312yes—
124C2×#95C2312yes—
125C2×#95C2312yes—
126C2×#95C2314yes—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2312yes—
130C2×#95C2312yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C25—24no—
141C25—22no—
142C25—24no—
143C25—22no—
144C25—24no—
145C25—24no—
146C25—42no—
147C25—42no—
148C25C2×Q814yes—
149C25C2411yescharacteristic
150C25C2411yescharacteristic
151C25C2411yescharacteristic
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—24no—
157C23×C4—24no—
158C23×C4—28no—
159C23×C4—28no—
160C23×C4—28no—
161C23×C4—28no—
162C23×C4—416no—
163C23×C4—416no—
164C23×C4—416no—
165C23×C4—42no—
166C23×C4—42no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—44no—
178C23×C4—44no—
179C23×C4—44no—
180C23×C4—464no—
181C23×C4C2×Q814yes—
182C23×C4C2×Q814yes—
183C23×C4C2×Q814yes—
184C23×C4C2×Q814yes—
185C23×C4C2×Q814yes—
186C23×C4C2×Q814yes—
187C23×C4C2414yes—
188C23×C4C2414yes—
189C23×C4C2414yes—
190C23×C4C2414yes—
191C23×C4C2414yes—
192C23×C4C2411yescharacteristic
193C23×C4C2411yescharacteristic
194C23×C4C2411yescharacteristic
195C23×C4C2411yescharacteristic
196C23×C4C2414yes—
197C2×C4⋊C4—28no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—28no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—28no—
222C2×C22⋊C4—28no—
223C2×C22⋊C4—28no—
224C2×C22⋊C4—28no—
225C2×C22⋊C4—28no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—24no—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—24no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4—42no—
232C2×C22⋊C4—44no—
233C2×C22⋊C4—42no—
234C2×C22⋊C4—42no—
235C2×C22⋊C4—432no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—44no—
239C2×C22⋊C4—44no—
240C2×C22⋊C4—42no—
241C2×C22⋊C4—42no—
242C2×C22⋊C4—42no—
243C2×C22⋊C4—44no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—44no—
246C2×C22⋊C4—42no—
247C2×C22⋊C4—42no—
248C2×C22⋊C4#4118yes—
249#95—28no—
250#95—42no—
251#95—44no—
252#95—42no—
253#95—44no—
254#95—42no—
255#95—42no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260#95—42no—
261#95—44no—
262#95—42no—
263#95—44no—
264#95—42no—
265#95—42no—
266#95—42no—
267#95—44no—
268#95—44no—
269#95—42no—
270#95—42no—
271#95—42no—
272#95—44no—
273#95—44no—
274#95—44no—
275#95—44no—
276#95—44no—
277C24—22no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—216no—
283C24—216no—
284C24—28no—
285C24—28no—
286C24—28no—
287C24—22no—
288C24—21no—
289C24—22no—
290C24—22no—
291C24—22no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—21no—
296C24—22no—
297C24—22no—
298C24—22no—
299C24—22no—
300C24—28no—
301C24—28no—
302C24—44no—
303C24—44no—
304C24—416no—
305C24—416no—
306C24—48no—
307C24—48no—
308C24—48no—
309C24—48no—
310C24—416no—
311C24—416no—
312C24C22×Q812yes—
313C24C22×Q812yes—
314C24C22×Q812yes—
315C24C2×#4114yes—
316C24H(5,𝔽2)12yes—
317C24H(5,𝔽2)14yes—
318C24H(5,𝔽2)14yes—
319C24H(5,𝔽2)14yes—
320C242−1+412yes—
321C24C2511yescharacteristic
322C22×C4—28no—
323C22×C4—28no—
324C22×C4—28no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—28no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—24no—
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—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—44no—
393C22×C4—416no—
394C22×C4—416no—
395C22×C4—416no—
396C22×C4—4128no—
397C22×C4—416no—
398C22×C4—416no—
399C22×C4—416no—
400C22×C4—416no—
401C22×C4—416no—
402C22×C4—416no—
403C22×C4—416no—
404C22×C4—88no—
405C22×C4—88no—
406C22×C4—88no—
407C22×C4C22×Q812yes—
408C22×C4C22×Q812yes—
409C22×C4C22×Q812yes—
410C22×C4C22×Q812yes—
411C22×C4C2×#4118yes—
412C4⋊C4—88no—
413C4⋊C4—88no—
414C4⋊C4—88no—
415C4⋊C4—88no—
416C4⋊C4—88no—
417C4⋊C4—88no—
418C22⋊C4—48no—
419C22⋊C4—416no—
420C22⋊C4—48no—
421C22⋊C4—416no—
422C22⋊C4—416no—
423C22⋊C4—88no—
424C22⋊C4—88no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C22⋊C4—84no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—24no—
434C23—24no—
435C23—28no—
436C23—28no—
437C23—28no—
438C23—24no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—232no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—24no—
456C23—24no—
457C23—24no—
458C23—24no—
459C23—28no—
460C23—28no—
461C23—24no—
462C23—24no—
463C23—24no—
464C23—24no—
465C23—416no—
466C23—416no—
467C23—416no—
468C23—416no—
469C23—416no—
470C23—464no—
471C23—464no—
472C23—416no—
473C23—44no—
474C23—44no—
475C23—44no—
476C23—44no—
477C23—44no—
478C23—44no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23#53314yes—
484C23#54314yes—
485C23#54314yes—
486C23#54314yes—
487C23#54414yes—
488C23#560116yes—
489C23C23×Q811yescharacteristic
490C23C22×#4112yes—
491C23C2×H(5,𝔽2)12yes—
492C23C2×H(5,𝔽2)12yes—
493C23C2×H(5,𝔽2)12yes—
494C23C2×H(5,𝔽2)11yescharacteristic
495C23C2×2−1+411yescharacteristic
496C23#58512yes—
497C23#58512yes—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—28no—
511C2×C4—28no—
512C2×C4—28no—
513C2×C4—416no—
514C2×C4—416no—
515C2×C4—416no—
516C2×C4—416no—
517C2×C4—416no—
518C2×C4—416no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—832no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—216no—
567C22—216no—
568C22—24no—
569C22—24no—
570C22—24no—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—24no—
575C22—24no—
576C22—24no—
577C22—464no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—48no—
585C22—48no—
586C22—48no—
587C22—48no—
588C22—42no—
589C22—42no—
590C22#284218yes—
591C22C2×#53311yescharacteristic
592C22C2×#54311yescharacteristic
593C22C2×#54311yescharacteristic
594C22C2×#54311yescharacteristic
595C22C2×#54411yescharacteristic
596C22#345811yescharacteristic
597C22#345811yescharacteristic
598C22#349012yes—
599C22#349212yes—
600C22#351314yes—
601C22#351314yes—
602C22C2×#56014yes—
603C22#353314yes—
604C22#353412yes—
605C22#353412yes—
606C22#353512yes—
607C22#353512yes—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—42no—
612C4—48no—
613C4—84no—
614C4—84no—
615C4—84no—
616C4—84no—
617C4—84no—
618C2—28no—
619C2—44no—
620C2—44no—
621C2#3505211yescharacteristic
622C2#3506511yescharacteristic
623C2#3506511yescharacteristic
624C2C2×#284211yescharacteristic
625C2#3747811yescharacteristic
626C2#3747811yescharacteristic
627C2#3747911yescharacteristic
628C2#4553614yes—
629C2#4553814yes—
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
1. #16902C2semidirect
2. #16902C2semidirect
3. #16902C2semidirect
4. #16825C2semidirect
5. #16825C2semidirect
6. #16825C2semidirect
7. #14936C2semidirect
8. #14935C2semidirect
12. #14637C2semidirect
29. #1445C22semidirect
30. #1445C22semidirect
31. #1445C22semidirect
32. #1445C22semidirect
33. #1445C22semidirect
34. #1445C22semidirect
35. #1444C22semidirect
36. #1444C22semidirect
37. #1443C22semidirect
38. #1443C22semidirect
39. #1441C22semidirect
40. #1441C22semidirect
41. #1441C22semidirect
42. #1441C22semidirect
43. #1441C22semidirect
44. #1441C22semidirect

Maximal subgroups

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

subgroupindex
#169022
#168252
#149362
#149352
#147262
#146812
#146372
C2×C22⋊#952