Finite Group Databasebeta

← #640909

Subgroups — #640909

Order 512.

3062 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#17066C212yesmaximal, complemented
4#17062C212yesmaximal, complemented
5#17046C214yesmaximal, complemented
6#15174C214yesmaximal, complemented
7#15015C212yesmaximal, complemented
8#14821C212yesmaximal, complemented
9#14820C212yesmaximal
10#14734C212yesmaximal
11#14684C211yesmaximal, characteristic, complemented
12#14305C214yesmaximal, complemented
13#14128C212yesmaximal, complemented
14#13868C212yesmaximal
15C22×#117C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17C23×C22⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C42C2211yescharacteristic
21C2×#399C2214yes—
22C2×#398C2214yescomplemented
23C2×#397C2214yes—
24C2×#388—28no—
25C2×#388C2212yes—
26C2×#388C2212yes—
27C2×#388C2212yes—
28C2×#388C2212yes—
29C2×#385C2214yes—
30#2285—28no—
31C2×#384C2211yescharacteristic, complemented
32C2×#384C2211yescharacteristic, complemented
33C2×#382C2212yescomplemented
34C2×#381C2212yes—
35C2×#380C2212yes—
36C2×#380C2212yes—
37C2×#379C2214yes—
38C2×C4×C4⋊C4C2212yescomplemented
39C2×C4×C4⋊C4C2212yescomplemented
40C22×#95C2211yescharacteristic
41C22×#95C2211yescharacteristic
42#1445C2214yescomplemented
43#1444C2214yescomplemented
44#1443C2218yescomplemented
45#1442C2214yescomplemented
46#1442C2214yescomplemented
47#1441C2214yescomplemented
48#1441C2218yescomplemented
49#1440C2218yes—
50#1439C2214yes—
51#1439C2214yes—
52#1439C2218yes—
53#1439C2214yes—
54#1439C2214yes—
55#1437C2218yes—
56#1436C2218yescomplemented
57#1434C2212yescomplemented
58#1434C2212yescomplemented
59#1434C2212yescomplemented
60#1434C2212yescomplemented
61C22⋊#95C2214yes—
62C22⋊#95C2214yes—
63C22⋊#95C2212yes—
64C22⋊#95C2212yes—
65C26C2311yescharacteristic
66C24×C4C2311yescharacteristic
67C24×C4C2311yescharacteristic
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C24×C4C2311yescharacteristic
71C24×C4C2311yescharacteristic
72C2×#117—28no—
73C2×#117—48no—
74C22×C4⋊C4—24no—
75C22×C4⋊C4—44no—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2312yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2318yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2312yes—
84C22×C4⋊C4C2312yes—
85C22×C22⋊C4—24no—
86C22×C22⋊C4—28no—
87C22×C22⋊C4—24no—
88C22×C22⋊C4—24no—
89C22×C22⋊C4—28no—
90C22×C22⋊C4—44no—
91C22×C22⋊C4C2312yes—
92C22×C22⋊C4C2312yes—
93C22×C22⋊C4C2312yes—
94C22×C22⋊C4C2312yes—
95C22×C22⋊C4C2314yes—
96C22×C22⋊C4C2314yes—
97C22×C22⋊C4C2312yes—
98C22×C22⋊C4C2318yes—
99C22×C22⋊C4C2312yes—
100C22×C22⋊C4C2318yes—
101C22×C42—28no—
102C22×C42D418yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2311yescharacteristic
106C22×C42C2311yescharacteristic
107C22×C42C2314yes—
108#399—216no—
109#398—48no—
110#397—48no—
111#388—28no—
112#388—28no—
113#388—28no—
114#388—28no—
115#385—48no—
116#384—24no—
117#384—24no—
118#382—28no—
119#381—28no—
120#380—28no—
121#380—28no—
122#379—48no—
123C4×C4⋊C4—28no—
124C4×C4⋊C4—28no—
125C2×#95—24no—
126C2×#95—24no—
127C2×#95C2314yes—
128C2×#95C2314yes—
129C2×#95C2314yes—
130C2×#95C2314yes—
131C2×#95C2312yes—
132C2×#95C2314yes—
133C2×#95C2314yes—
134C2×#95C2312yes—
135C2×#95C2312yes—
136C2×#95C2312yes—
137C2×#95C2318yes—
138C2×#95C2318yes—
139C2×#95C2314yes—
140C2×#95C2314yes—
141C2×#95C2318yes—
142C2×#95C2318yes—
143C2×#95C2318yes—
144C25—22no—
145C25—24no—
146C25—24no—
147C25—24no—
148C25—24no—
149C25—42no—
150C25—42no—
151C25—42no—
152C25C2412yes—
153C25C2411yescharacteristic
154C23×C4—24no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—24no—
158C23×C4—24no—
159C23×C4—24no—
160C23×C4—24no—
161C23×C4—24no—
162C23×C4—22no—
163C23×C4—22no—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4—24no—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4—42no—
170C23×C4—42no—
171C23×C4—42no—
172C23×C4—42no—
173C23×C4—42no—
174C23×C4C2×D414yes—
175C23×C4C2×D414yes—
176C23×C4C2×D414yes—
177C23×C4#4118yes—
178C23×C4C2414yes—
179C23×C4C2414yes—
180C23×C4C2418yes—
181C23×C4C2412yes—
182C23×C4C2412yes—
183C23×C4C2411yescharacteristic
184C23×C4C2411yescharacteristic
185C23×C4C2412yes—
186C23×C4C2411yescharacteristic
187C23×C4C2411yescharacteristic
188C23×C4C2411yescharacteristic
189C23×C4C2411yescharacteristic
190#117—432no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—216no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—416no—
209C2×C4⋊C4—48no—
210C2×C4⋊C4—48no—
211C2×C22⋊C4—232no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
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—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—24no—
227C2×C22⋊C4—28no—
228C2×C22⋊C4—28no—
229C2×C22⋊C4—28no—
230C2×C22⋊C4—216no—
231C2×C22⋊C4—44no—
232C2×C22⋊C4—416no—
233C2×C22⋊C4—416no—
234C2×C22⋊C4—48no—
235C2×C22⋊C4—48no—
236C2×C22⋊C4—48no—
237C2×C22⋊C4—48no—
238C2×C22⋊C4—48no—
239C2×C22⋊C4#4118yes—
240C2×C42—232no—
241C2×C42—24no—
242C2×C42—24no—
243C2×C42—24no—
244C2×C42—24no—
245C2×C42—28no—
246C2×C42—44no—
247C2×C42—44no—
248C2×C42C2×D414yes—
249C2×C42C2×D414yes—
250C2×C42C2×D414yes—
251C2×C42C2×D414yes—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—24no—
256#95—28no—
257#95—28no—
258#95—28no—
259#95—28no—
260#95—28no—
261#95—28no—
262#95—44no—
263#95—44no—
264#95—48no—
265#95—48no—
266#95—48no—
267#95—48no—
268#95—48no—
269C24—22no—
270C24—22no—
271C24—22no—
272C24—22no—
273C24—28no—
274C24—22no—
275C24—22no—
276C24—24no—
277C24—24no—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—24no—
282C24—416no—
283C24—416no—
284C24—416no—
285C24—44no—
286C24—44no—
287C24—44no—
288C24—44no—
289C24—44no—
290C24—44no—
291C24—416no—
292C24—88no—
293C24—88no—
294C24—88no—
295C24C22×D412yes—
296C24C2×#4114yes—
297C24C2×#4114yes—
298C24H(5,𝔽2)14yes—
299C242−1+414yes—
300C24C2511yescharacteristic
301C22×C4—28no—
302C22×C4—28no—
303C22×C4—232no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—28no—
313C22×C4—28no—
314C22×C4—28no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—28no—
326C22×C4—28no—
327C22×C4—22no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—22no—
331C22×C4—28no—
332C22×C4—28no—
333C22×C4—24no—
334C22×C4—28no—
335C22×C4—28no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—28no—
339C22×C4—24no—
340C22×C4—28no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—28no—
345C22×C4—24no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—416no—
355C22×C4—416no—
356C22×C4—416no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—88no—
366C22×C4—88no—
367C22×C4#123116yes—
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D412yes—
372C22×C4C22×D412yes—
373C22×C4C22×D412yes—
374C22×C4C2×#4114yes—
375C22×C4C2×#4114yes—
376C22×C4C2×#4114yes—
377C22×C4C2×#4114yes—
378C22×C4H(5,𝔽2)14yes—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—48no—
382C4⋊C4—48no—
383C4⋊C4—48no—
384C4⋊C4—48no—
385C4⋊C4—48no—
386C4⋊C4—48no—
387C4⋊C4—816no—
388C22⋊C4—216no—
389C22⋊C4—216no—
390C22⋊C4—48no—
391C22⋊C4—416no—
392C22⋊C4—416no—
393C22⋊C4—48no—
394C22⋊C4—48no—
395C22⋊C4—48no—
396C22⋊C4—816no—
397C22⋊C4—816no—
398C42—28no—
399C42—28no—
400C42—28no—
401C42—28no—
402C42—416no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—22no—
407C23—28no—
408C23—28no—
409C23—28no—
410C23—28no—
411C23—28no—
412C23—28no—
413C23—24no—
414C23—28no—
415C23—28no—
416C23—28no—
417C23—416no—
418C23—416no—
419C23—416no—
420C23—42no—
421C23—42no—
422C23—42no—
423C23—42no—
424C23—42no—
425C23—42no—
426C23—48no—
427C23—44no—
428C23—48no—
429C23—44no—
430C23—44no—
431C23—48no—
432C23—48no—
433C23—864no—
434C23—88no—
435C23—88no—
436C23—88no—
437C23—88no—
438C23C2×#12314yes—
439C23#53514yes—
440C23#53614yes—
441C23#539116yes—
442C23C23×D411yescharacteristic
443C23C22×#4112yes—
444C23C22×#4112yes—
445C23C2×H(5,𝔽2)12yes—
446C23C2×H(5,𝔽2)12yes—
447C23C2×2−1+412yes—
448C23#58512yes—
449C23#58512yes—
450C2×C4—232no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—28no—
460C2×C4—28no—
461C2×C4—28no—
462C2×C4—28no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—24no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—28no—
471C2×C4—416no—
472C2×C4—416no—
473C2×C4—416no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—48no—
492C2×C4—48no—
493C2×C4—48no—
494C2×C4—48no—
495C2×C4—48no—
496C2×C4—48no—
497C2×C4—48no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—88no—
505C2×C4—88no—
506C2×C4C2×#12314yes—
507C2×C4C2×#12314yes—
508C2×C4#52514yes—
509C2×C4#53514yes—
510C22—232no—
511C22—24no—
512C22—24no—
513C22—24no—
514C22—24no—
515C22—24no—
516C22—24no—
517C22—416no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—832no—
522C22—88no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22—82no—
527C22#262318yes—
528C22C22×#12311yescharacteristic
529C22C2×#52511yescharacteristic
530C22C2×#53511yescharacteristic
531C22C2×#53511yescharacteristic
532C22C2×#53611yescharacteristic
533C22#344911yescharacteristic
534C22#344911yescharacteristic
535C22C2×#53914yes—
536C22#347514yes—
537C22#349414yes—
538C22#349514yes—
539C22#351914yes—
540C22#356014yes—
541C22#356114yes—
542C4—24no—
543C4—24no—
544C4—24no—
545C4—24no—
546C4—44no—
547C4—42no—
548C4—42no—
549C4—88no—
550C4—82no—
551C4—84no—
552C4—84no—
553C4—82no—
554C2—28no—
555C2—84no—
556C2C2×#262311yescharacteristic
557C2#3480911yescharacteristic
558C2#3523711yescharacteristic
559C2#3526211yescharacteristic
560C2#3532511yescharacteristic
561C2#3753711yescharacteristic
562C2#3754711yescharacteristic
563C2#3840218yes—
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
2. C2×#2285C2semidirect
3. #17066C2semidirect
4. #17062C2semidirect
5. #17046C2semidirect
6. #15174C2semidirect
7. #15015C2semidirect
8. #14821C2semidirect
11. #14684C2semidirect
12. #14305C2semidirect
13. #14128C2semidirect
22. C2×#398C22semidirect
31. C2×#384C22semidirect
32. C2×#384C22semidirect
33. C2×#382C22semidirect
38. C2×C4×C4⋊C4C22semidirect
39. C2×C4×C4⋊C4C22semidirect
42. #1445C22semidirect
43. #1444C22semidirect
44. #1443C22semidirect
45. #1442C22semidirect
46. #1442C22semidirect
47. #1441C22semidirect
48. #1441C22semidirect
56. #1436C22semidirect
57. #1434C22semidirect
58. #1434C22semidirect
59. #1434C22semidirect
60. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
C2×#22852
#170662
#170622
#170462
#151742
#150152
#148212
#148202
#147342
#146842
#143052
#141282
#138682